Question:
In square JKLM, diagonals $\overline{JL}$ and $\overline{MK}$ intersect at point N, m $\angle $ LNK=(8x+2)°, and m $\angle $ MJN=(5x–2y)°.

What are the values of x and y?

x= $\Box$

y= $\Box$

Answer:
x=11
y=5

1. Given that quadrilateral JKLM is a square, according to the properties of the diagonals of a square, we can determine that its diagonals are perpendicular to each other, i.e., \\(\overline{JL} \\perp \\overline{MK}\\), so \\(\angle LNK = 90^\\circ\\).

2. The problem also states \\(\angle LNK = (8x + 2)^\\circ\\), thus we can set up the equation: (8x + 2) = 90.

3. Solving the equation from step 2:

   (1) Transposing: 8x + 2 - 2 = 90 - 2, we get 8x = 88.

   (2) Dividing both sides by 8: x = 11.

4. One of the interior angles of quadrilateral JKLM is \\(\angle MJK = 90^\\circ\\), and \\(\\overline{JL}\\) is the diagonal, therefore \\(\angle MJN = \\frac{90^\\circ}{2} = 45^\\circ\\).

5. The problem states \\(\angle MJN = (5x - 2y)^\\circ\\), based on the conclusion in step 4, set up the equation: (5x - 2y) = 45.

6. Solving the equation from step 5:

   (1) Given x = 11, substitute into the equation: (5(11) - 2y) = 45.

   (2) Calculation: 55 - 2y = 45.

   (3) Transposing: -2y = 45 - 55.

   (4) Simplifying: -2y = -10.

   (5) Dividing both sides by -2: y = 5.

7. Through the above reasoning, the final answer is x = 11 and y = 5.

Definition of Square

A quadrilateral is a square if and only if all four sides are congruent (≅) and all four interior angles are right angles (90°).

<note>Quadrilateral JKLM is a square</note>, therefore <note>\(JK = KL = LM = MJ\)</note>, and <note>\( \angle JKL = \angle KLM = \angle LMJ = \angle MJK = 90^\circ \)</note>.

Definition of an Angle

An angle is a geometric figure formed by two rays sharing a common endpoint. This common endpoint is called the vertex of the angle, and the two rays are referred to as the sides of the angle.

<note>Angle \(\angle LNK\)</note> is a geometric figure composed of <note>rays \(\overline{NL}\) and \(\overline{NK}\)</note>, which share a common endpoint <note>N</note>. This common endpoint <note>N</note> is called the <note>vertex of angle \(\angle LNK\)</note>, and <note>rays \(\overline{NL}\) and \(\overline{NK}\)</note> are called the <note>sides of angle \(\angle LNK\)</note>. <note>Angle \(\angle MJN\)</note> is a geometric figure composed of <note>rays \(\overline{JM}\) and \(\overline{JN}\)</note>, which share a common endpoint <note>J</note>. This common endpoint <note>J</note> is called the <note>vertex of angle \(\angle MJN\)</note>, and <note>rays \(\overline{JM}\) and \(\overline{JN}\)</note> are called the <note>sides of angle \(\angle MJN\)</note>.

Properties of Diagonals in a Square

The diagonals of a square are the line segments that connect opposite vertices. The diagonals of a square are equal in length, and they bisect each other perpendicularly.

In the figure of this problem, in the square JKLM, <note>the diagonals JL and MK are the segments connecting opposite corners</note>. According to the properties of the diagonals in a square, <note>JL and MK are equal</note>, and <note>JL and MK bisect each other perpendicularly</note>, forming four 90-degree angles at their intersection point. Therefore, <note>JL = MK</note>, and <note>they are perpendicular to each other at the intersection point</note>.


Question:
In rectangle STUV, diagonals $\overline{SU}$ and $\overline{TV}$ intersect at point W, ST=36, TU=15, and SW=2.5x+2.

What is the value of x?

x= $\Box$

Answer: x=7

1. Given STUV is a rectangle, according to the properties of a rectangle, the diagonals are equal and bisect each other, thus SU=TV, and point W is the midpoint, i.e., SW=WU.

2. The diagonals of a rectangle can be found using the lengths of the sides of the rectangle by applying the Pythagorean theorem, because in a rectangle, the diagonal can be considered as the hypotenuse of a right triangle.

3. In the right triangle STU, the legs ST=36 and TU=15, using the Pythagorean theorem: ST^2 + TU^2 = SU^2.

4. Calculating, we get 36^2 + 15^2 = SU^2 (1296 + 225 = SU^2), so SU^2 = 1521.

5. Solving SU^2 = 1521 gives SU = sqrt(1521) = 39.

6. Since the diagonals bisect each other and SW=WU, SU = 2SW, i.e., 39 = 2SW.

7. Given SW = 2.5x + 2, then 2(2.5x + 2) = 39.

8. Expanding the equation, we get 5x + 4 = 39.

9. Solving 5x + 4 = 39, we get 5x = 35.

10. Further solving 5x = 35, we can find x = 7.

11. Through the above reasoning, the final answer is x = 7.

Definition of Rectangle

A quadrilateral is called a rectangle if and only if each of its interior angles is a right angle (90°) and its opposite sides are both parallel (∥) and congruent (≅).

<note>Quadrilateral STUV is a rectangle</note>, with its interior angles <note>∠STU, ∠TUV, ∠UVS, and ∠VST all being right angles (90 degrees)</note>, and <note>side ST is parallel and equal in length to side UV</note>, <note>side SV is parallel and equal in length to side TU</note>.

Definition of Line Segment

A line segment is a portion of a straight line that includes two endpoints and all the points between them.

In the figure of this problem, the line segments are: <note>ST</note>, <note>TU</note>, <note>UV</note>, <note>VS</note>, <note>SU</note>, and <note>TV</note>. A line segment is a straight line with two endpoints. <note>The endpoints of line segment ST are S and T</note>, <note>The endpoints of line segment TU are T and U</note>, <note>The endpoints of line segment UV are U and V</note>, <note>The endpoints of line segment VS are V and S</note>, <note>The endpoints of line segment SU are S and U</note>, <note>The endpoints of line segment TV are T and V</note>. Each line segment has a specific length, for example, <note>ST = 36</note> and <note>TU = 15</note>.

Pythagorean Theorem

In a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the two legs.

In the diagram of this problem, the diagonal of rectangle STUV can be regarded as <note>the hypotenuse of right triangle STU</note>. <note>The legs are ST = 36 and TU = 15</note>, using the Pythagorean Theorem: <note>ST^2 + TU^2 = SU^2</note>, we can calculate <note>the length of the diagonal SU of the rectangle</note>.


Question:
The diagram shows quadrilateral JKLM and diagonal $\overline{JL}$ . Sides $\overline{JK}$ and $\overline{ML}$ are parallel and congruent.

To prove that a quadrilateral with one pair of parallel and congruent sides is a parallelogram, Lindsey first wants to prove that △JKL≅△LMJ. Which additional statements can Lindsey use in her proof? Select all that apply.

-  $\angle $ KJL≅ $\angle $ MJL
-  $\angle $ JKL≅ $\angle $ LMJ
- $\overline{KL}$ ≅ $\overline{ML}$
- $\overline{JK}$ ≅ $\overline{KL}$
- $\overline{JL}$ ≅ $\overline{JL}$
-  $\angle $ KJL≅ $\angle $ MLJ
With the statements you selected above, which congruence theorem will Lindsey use to prove that △JKL≅△LMJ?

- Angle-Angle-Side
- Angle-Side-Angle
- Side-Angle-Side
- Side-Side-Side

Answer:
- \$\overline{JL}\$ ≅ \$\overline{JL}\$
-  \$\angle \$ KJL≅ \$\angle \$ MLJ

- Side-Angle-Side

1. Given that line segments \overline{JK} and \overline{ML} are parallel and congruent, according to the 'Alternate Interior Angles Theorem', we have \angle KJL ≅ \angle MLJ.

2. Based on the given conditions and \overline{JL} being the common side of \triangle JKL and \triangle LMJ, according to the 'Reflexive Property', we have \overline{JL} ≅ \overline{JL}.

3. From the above reasoning, we can deduce that \triangle JKL and \triangle LMJ have two pairs of corresponding sides equal and the included angle equal. Therefore, according to the 'Side-Angle-Side Congruence Theorem (SAS Congruence Theorem)', we can conclude \triangle JKL ≅ \triangle LMJ.

4. Since \triangle JKL ≅ \triangle LMJ, the quadrilateral JKLM has one pair of opposite sides that are parallel and congruent. Thus, the properties of quadrilateral JKLM meet the definition of a parallelogram, so quadrilateral JKLM is a parallelogram.

Definition of Parallel Lines

Two lines in the same plane that do not intersect are called parallel lines.

Segments <note>\overline{JK} and \overline{ML}</note> are located in the same plane, and they do not intersect. Therefore, according to the <note>definition of parallel lines</note>, <note>segments \overline{JK} and \overline{ML}</note> are parallel lines.

Definition of Congruent Triangles

Two triangles are congruent if and only if their corresponding sides are equal and their corresponding angles are equal.

<note>Triangles \triangle JKL and \triangle LMJ are congruent triangles</note>, the corresponding sides and corresponding angles of triangle \triangle JKL are equal to those of triangle \triangle LMJ, namely: <note>side \overline{JK} = side \overline{ML}</note>, side \overline{JL} = side \overline{JL}, <note>side \overline{KL} = side \overline{MJ}</note>, and the corresponding angles are also equal: <note>angle \angle JKL = angle \angle LMJ</note>, angle \angle KJL = angle \angle MJL, angle \angle JLK = angle \angle MLJ.

Definition of Parallelogram

A quadrilateral is a parallelogram if and only if it has two pairs of opposite sides that are both parallel (∥) and congruent (≅).

<note>Quadrilateral JKLM is a parallelogram</note>, sides <note>\overline{JK} and \overline{ML} are parallel and equal</note>, sides <note>\overline{JM} and \overline{KL} are parallel and equal</note>.

Definition of Alternate Interior Angles

When a straight line (referred to as a transversal) intersects two parallel lines, the angles that are located between the two parallel lines but on opposite sides of the transversal are called alternate interior angles.

<note>Two parallel lines \overline{JK} and \overline{ML} are intersected by the line \overline{JL}<note>, where<note>angle \angle KJL and angle \angle MLJ are located between the two parallel lines and on opposite sides of the intersecting line \overline{JL}<note>, thus<note>angle \angle KJL and angle \angle MLJ are alternate interior angles<note>. Alternate interior angles are equal, that is<note>\angle KJL \cong \angle MLJ<note>.

Triangular Congruence Theorem (SAS)

If two sides and the included angle of one triangle are equal to two sides and the included angle of another triangle, then the two triangles are congruent.

<note>Side JK is equal to side ML</note>, <note>side JL is equal to side JL</note>, and <note>the included angle ∠KJL is equal to the included angle ∠MLJ</note>, therefore, according to the Triangular Congruence Theorem (SAS), these two triangles are congruent.


Question:
The diagram shows quadrilateral ABCD with diagonals $\overline{AC}$ and $\overline{BD}$ that bisect each other at point E.

To prove that a quadrilateral is a parallelogram when its diagonals bisect each other, Regan first wants to prove that △AEB≅△CED and △AED≅△CEB. Which congruence theorem will Regan use?

- Side-Angle-Side
- Angle-Side-Angle
- Angle-Angle-Side
- Side-Side-Side
After proving that △AEB≅△CED and △AED≅△CEB, how can Regan prove that ABCD is a parallelogram? Complete the statement.
Because △AEB≅△CED and  $\Box$ ,  $\angle $ ABE≅ $\angle $ CDE. So, by the  $\Box$ ,  $\Box$ . Similarly, since △AED≅△CEB and therefore  $\angle $ ADE≅ $\angle $ CBE,  $\Box$ . So, ABCD is a parallelogram since its opposite sides are parallel.

Answer:
- Side-Angle-Side
Because △AEB≅△CED and corresponding parts of congruent triangles are congruent,  \$\angle \$ ABE≅ \$\angle \$ CDE. So, by the Converse of the Alternate Interior Angles Theorem,  \$\overline{AB}\$ ∥ \$\overline{DC}\$ . Similarly, since △AED≅△CEB and therefore  \$\angle \$ ADE≅ \$\angle \$ CBE,  \$\overline{AD}\$ ∥ \$\overline{BC}\$ . So, ABCD is a parallelogram since its opposite sides are parallel.

1. It is given in the problem that the diagonals AC and BD of quadrilateral ABCD bisect each other at point E. Therefore, according to the definition of segment bisector, we have segment AE ≅ CE and segment BE ≅ DE.

2. Observing that ∠AEB and ∠CED are vertical angles, according to the theorem of vertical angles being equal, we get ∠AEB ≅ ∠CED.

3. From the above two points, △AEB and △CED have two pairs of congruent sides and the included angle between them. According to the Side-Angle-Side (SAS) congruence theorem, △AEB ≅ △CED.

4. Similarly, ∠AED and ∠CEB are vertical angles, and according to the theorem of vertical angles being equal, we get ∠AED ≅ ∠CEB.

5. Again using segment AE ≅ CE and segment BE ≅ DE, △AED and △CEB also have two pairs of congruent sides and the included angle between them. According to the Side-Angle-Side (SAS) congruence theorem, △AED ≅ △CEB.

6. Since corresponding parts of congruent triangles are congruent, we get ∠ABE ≅ ∠CDE.

7. According to the converse of the alternate interior angles theorem, if the alternate interior angles formed by a transversal are equal, then the two lines cut by the transversal are parallel. Therefore, ∠ABE ≅ ∠CDE leads to segment AB ∥ segment DC.

8. Similarly, since ∠ADE ≅ ∠CBE, and according to the converse of the alternate interior angles theorem, it leads to segment AD ∥ segment BC.

9. Since two pairs of opposite sides of quadrilateral ABCD are parallel, according to the definition of a parallelogram, quadrilateral ABCD is a parallelogram.

Through the above reasoning, the final conclusion is: ABCD is a parallelogram.

Definition of Parallelogram

<note>Quadrilateral ABCD is a parallelogram</note>, <note>side AB is parallel and equal to side DC</note>, <note>side AD is parallel and equal to side BC</note>.

Definition of Vertical Angles

Vertical angles are the pair of opposite angles formed when two lines intersect. These angles are equal in measure.

In the figure of this problem, <note>two intersecting lines AC and BD intersect at point E</note>, forming four angles: <note>∠AEB, ∠CED, ∠AED, and ∠CEB</note>. According to the <note>definition of vertical angles, ∠AEB and ∠CED are vertical angles, ∠AED and ∠CEB are vertical angles</note>. Since <note>vertical angles are equal</note>, therefore <note>∠AEB=∠CED, ∠AED=∠CEB</note>.

Triangular Congruence Theorem (SAS)

<note>Side AE is equal to side CE</note>, <note>side BE is equal to side DE</note>, and <note>angle AEB is equal to angle CED</note>, therefore according to the Triangular Congruence Theorem (SAS), these two triangles are congruent. Similarly, in triangles AED and CEB, <note>side AE is equal to side CE</note>, <note>side BE is equal to side DE</note>, and <note>angle AED is equal to angle CEB</note>, therefore according to the Triangular Congruence Theorem (SAS), these two triangles are congruent.

Definition of Congruent Triangles

In this problem, triangles <note>AEB</note> and <note>CED</note> are congruent triangles, with corresponding sides and angles of triangle <note>AEB</note> equal to those of triangle <note>CED</note>, namely: side <note>AE = CE</note>, side <note>BE = DE</note>, side <note>AB = CD</note>, and corresponding angles are also equal: angle <note>AEB = CED</note>, angle <note>BAE = DCE</note>, angle <note>ABE = CDE</note>. Similarly, triangles <note>AED</note> and <note>CEB</note> are congruent triangles, with corresponding sides and angles of triangle <note>AED</note> equal to those of triangle <note>CEB</note>, namely: side <note>AE = CE</note>, side <note>DE = BE</note>, side <note>AD = CB</note>, and corresponding angles are also equal: angle <note>AED = CEB</note>, angle <note>DAE = BCE</note>, angle <note>ADE = CBE</note>.

Parallel Postulate 2 of Parallel Lines

If two parallel lines are cut by a transversal, then the corresponding angles are equal, the alternate interior angles are equal, and the consecutive interior angles on the same side of the transversal are supplementary.

<note>In the diagram of this problem, two parallel lines xx and xx are intersected by a third line xx</note>, forming the following geometric relationships:
1. <note>Corresponding angles: angle xxx and angle xxx are equal</note>.
2. <note>Alternate interior angles: angle xxx and angle xxx are equal</note>.
3. <note>Same-side interior angles: angle xxx and angle xxx are supplementary, that is, angle xxx + angle xxx = 180 degrees</note>.
These relationships illustrate that when two parallel lines are intersected by a third line, corresponding angles are equal, alternate interior angles are equal, and same-side interior angles are supplementary.


Question:
The surface area of a cylinder is 140𝜋 square millimeters. The cylinder is shown below.

What is the height of the cylinder?

 $\Box$  millimeters

Answer: 9 millimeters

1. Given the surface area of the cylinder is 140π square millimeters, the radius of the cylinder is 5 millimeters.

2. Use the formula for the surface area of the cylinder: S = 2πr^2 + 2πrh, where S is the surface area, r is the radius, and h is the height.

3. Substitute the known values into the formula: 140π = 2π(5)^2 + 2π(5)h.

4. Calculate 2π(5)^2 to get 50π. Therefore, the formula becomes 140π = 50π + 10πh.

5. Rearrange the equation: 140π - 50π = 10πh.

6. Calculate 140π - 50π to get 90π. Therefore, we have 90π = 10πh.

7. Divide both sides by 10π to get h = 9.

8. Through the above reasoning, the final answer is 9 millimeters.

Cylinder

A cylinder is a geometric solid consisting of two parallel and congruent circular bases and a curved lateral surface connecting the bases.

<note>The cylinder consists of two parallel and identical circular bases and a lateral surface</note>. The bases are <note>two identical circles</note>, with <note>a radius of 5 millimeters</note>, and their <note>centers are on the same line</note>. The lateral surface is a rectangle, and when unfolded, <note>its height is equal to the height h of the cylinder</note>, and <note>its width is equal to the circumference of the circle</note>.

Surface Area Formula for a Cylinder

The surface area of a cylinder is equal to the sum of the areas of the two bases and the lateral surface area. The total surface area (SA) is given by the formula: SA = 2πr² + 2πrh, where r is the radius of the base circle, and h is the height of the cylinder.

The geometric principle <note>Surface Area Formula for a Cylinder</note> is S = 2πr^2 + 2πrh, where <note>r = 5 millimeters</note>, <note>S = 140π square millimeters</note>.

Area Formula of a Circle

The area of a circle is given by the formula A = πr², where A represents the area, π (pi) is the mathematical constant approximately equal to 3.14159, and r is the radius of the circle.

The original text: <note>The base of the cylinder is a circle</note>, the radius of the circle <note>is 5 millimeters</note>, according to the area formula of a circle, the area of the circle A is equal to <note>pi π</note> multiplied by the square of <note>radius 5</note>, that is, A = π(5)² = 25π square millimeters.


Question:
A triangular prism is shown below.

What is the surface area of the triangular prism?

 $\Box$  square millimeters

Answer: 888 square millimeters

1. Given that the triangular base of the triangular prism has two sides each of length 10 mm, and the third side is 16 mm. According to the problem, the base triangle is an isosceles triangle, thus the two shorter sides are each 10 mm, and the base side is 16 mm.

2. Using the given base side of 16 mm and height of 6 mm, we use the triangle area formula to calculate the base area: A = 1/2 * base * height = 1/2 * 16 mm * 6 mm = 48 square mm.

3. Calculate the perimeter of the base triangle using the perimeter formula: P = a + b + c. Substituting the side lengths: P = 10 mm + 10 mm + 16 mm = 36 mm.

4. Since the geometric figure is a triangular prism, the height of the prism is 22 mm. Using the triangular prism surface area formula: S = Ph + 2B. Where P is the base perimeter, h is the prism height, and B is the base area.

5. Substitute the values: S = 36 mm * 22 mm + 2 * 48 square mm = 792 square mm + 96 square mm = 888 square mm.

6. Through the above reasoning, the final answer is 888 square mm.

Definition of Isosceles Triangle

A triangle is an isosceles triangle if and only if it has at least two congruent (≅) sides.

The original: <note>Both sides of the base triangle are 10 millimeters</note>, <note>the base is 16 millimeters</note>, therefore this triangle is an isosceles triangle.

Surface Area Formula of a Prism

The surface area formula for a prism is given by: \( S = Ph + 2B \), where \( P \) denotes the perimeter of the base, \( h \) is the height of the prism, and \( B \) represents the area of the base.

In the figure of this problem, <note>the perimeter of the triangular base is 36 mm</note>, <note>the height of the prism is 22 mm</note>, <note>the area of the base is 48 square mm</note>, so the surface area is <note>36 mm * 22 mm + 2 * 48 square mm = 888 square mm</note>.

Area Formula of a Triangle

The area of any triangle is equal to its base multiplied by its height, divided by 2.

In the figure of this problem, in the triangle at the base of the triangular prism, <note>the side 16mm is the base</note>, <note>the segment 6mm is the height on this base</note>, so the area of the triangle is equal to <note>base 16mm multiplied by height 6mm divided by 2</note>, that is, area = (16 * 6) / 2.


Question:
In parallelogram GHIJ, m $\angle $ G=(5y)°, m $\angle $ H=(5x)°, and m $\angle $ J=(3y)°.

What are the values of x and y?

x= $\Box$

y= $\Box$

Answer:
x=13.5
y=22.5

1. Given that the figure GHIJ is a parallelogram, according to the theorem of supplementary adjacent angles in a parallelogram. Therefore, ∠G and ∠J are adjacent angles and they are supplementary.

2. The sum of supplementary angles is 180°. So, we have the equation ∠G + ∠J = 180°.

3. Substituting the given angle expressions, we have 5y + 3y = 180°.

4. Combining the equations and calculating, we get 8y = 180, solving for y we get y = 22.5.

5. Calculating the specific degree of ∠J, substituting the known ∠J = 3y, then ∠J = 3(22.5) = 67.5°.

6. According to the properties theorem of parallelograms, in a parallelogram, opposite angles are equal. That is ∠H and ∠J are equal, i.e., ∠H = ∠J.

7. Given the conditions ∠H = 5x and ∠J = 67.5°, we get the equation 5x = 67.5.

8. Solving the equation, we get x = 67.5 ÷ 5 = 13.5.

9. Through the above reasoning, the final answer is x = 13.5, y = 22.5.

Definition of Parallelogram

<note>Quadrilateral GHIJ is a parallelogram</note>, <note>side GH is parallel and equal to side IJ</note>, <note>side GJ is parallel and equal to side HI</note>.

Properties of Parallelogram Theorem

In a parallelogram, opposite angles are equal, opposite sides are equal, and the diagonals bisect each other.

In the figure of this problem, in the parallelogram <note>GHIJ</note>, <note>angles G and I are equal</note>, <note>angles H and J are equal</note>; <note>sides GH and JI are equal</note>, <note>sides GJ and HI are equal</note>; <note>diagonals GI and HJ bisect each other</note>, that is, <note>the intersection point divides diagonal GI into two equal segments</note>, <note>divides diagonal HJ into two equal segments</note>.

In a parallelogram, each pair of adjacent interior angles are supplementary, meaning their sum is 180°.

In <note>parallelogram GHIJ</note>, <note>angle G and angle J are adjacent interior angles</note>, <note>angle H and angle I are also adjacent interior angles</note>. According to the Adjacent Angles Supplementary Theorem of Parallelogram, <note>angle G + angle J = 180°</note>, <note>angle H + angle I = 180°</note>.


Question:
Jen is taking her new tent on a camping trip. The tent is shaped like a triangular prism, shown below. She needs to waterproof the outside surfaces of her tent: the front, the back, the sides, and the bottom.

Each ounce of waterproof spray covers 800 square inches of material. To the nearest ounce, how many ounces of waterproof spray does Jen need?

 $\Box$  ounces

Answer: 34 ounces

1. Given that the triangular base of the triangular prism is an isosceles triangle, where the lengths of the two equal sides are both 70 inches, the length of the base is 84 inches, its height is 56 inches, and the height of the prism is 100 inches.

2. First, find the area of the triangular base. According to the formula for the area of a triangle: Area A = 1/2 × base × height, the base of the triangle is 84 inches, and the height is 56 inches. Therefore, the area A = 1/2 × 84 × 56 = 2,352 square inches.

3. Next, find the perimeter of the triangular base, which has three sides: 70 inches, 70 inches, and 84 inches. According to the formula for the perimeter of a triangle, the perimeter P = 70 + 70 + 84 = 224 inches.

4. Calculate the surface area of the triangular prism. According to the formula for the surface area of a prism: Surface area S = P × height + 2 × base area, where P is the perimeter of the base, 'height' is the height of the prism, and the base area is the area previously found.

5. Substitute the data for calculation: Surface area S = 224 × 100 + 2 × 2,352 = 22,400 + 4,704 = 27,104 square inches.

6. Each ounce of waterproof spray can cover 800 square inches of material. The total area to be sprayed is 27,104 square inches. Therefore, the required number of ounces is 27,104 ÷ 800 = 33.88.

7. According to the rounding principle, round the obtained value 33.88 up to the nearest whole number, resulting in 34.

8. Through the above reasoning, the final answer is 34 ounces.

Definition of Triangle

A geometric figure is called a triangle if it is formed by three non-collinear points and the line segments connecting them.

<note>The base triangle is a geometric figure composed of three non-collinear points and their connecting line segments</note>. The three sides of the base triangle are <note>70 inches, 70 inches, and 84 inches</note>, where <note>the two legs are equal, forming an isosceles triangle</note>. <note>The three vertices of the triangle are the two endpoints of the base and the vertex of the height</note>, <note>the line segments are the two legs and the base</note>.

Definition of Isosceles Triangle

<note>The two sides of the base triangle are 70 inches and 70 inches respectively</note>, they are equal, therefore <note>the base triangle is an isosceles triangle</note>.

Definition of Prism

A prism is a polyhedron with two parallel and congruent polygonal bases, and all other faces are parallelograms.

The entire <note>tent</note> is a <note>triangular prism</note>, with the base being <note>two identical isosceles triangles</note>, and the sides being <note>three rectangles</note>.

Surface Area Formula of a Prism

<note>Base triangle perimeter P = 224 inches</note>, <note>height H = 100 inches</note>, <note>base area A = 2,352 square inches</note>. <note>Surface area S = 224 × 100 + 2 × 2,352 = 22,400 + 4,704 = 27,104 square inches</note>.

Area Formula of a Triangle

The original text: <note>84 inches is the base, 56 inches is the height on that base</note>, so the area of the triangle is equal to the base multiplied by the height divided by 2, i.e., Area = (84 * 56) / 2.

Formula for the Perimeter of a Triangle

The perimeter of a triangle is the sum of the lengths of its three sides. Therefore, if the three sides of the triangle are denoted as \( a \), \( b \), and \( c \), then the perimeter \( P \) is given by \( P = a + b + c \).

In the figure of this problem, in the triangle, <note>the three sides are 70 inches, 70 inches, and 84 inches respectively</note>, according to the formula for the perimeter of a triangle, which is <note>Perimeter L = 70 + 70 + 84</note>.


Question:
In circle S,  $\angle $ OSP measures q° and the length of radius $\overline{OS}$ is h.

If circle S were dilated about its center by a scale factor of 1/2, what would the new area of sector OSP be?

- 𝜋h ^ 2q/360
- 𝜋hq/360
- 𝜋hq/1,440
- 𝜋h ^ 2q/1,440

Answer: - 𝜋h ^ 2q/1,440

1. Observe the problem statement, the radius OS of the original circle S is h, and the degree of ∠OSP is q°.

2. Circle S undergoes a scaling transformation centered at its center, with a scaling factor of 1/2. According to the scaling theorem of circles, the radius of the new circle (denoted as S') is 1/2 the radius of the original circle, i.e., O'S' = 1/2 * h.

3. Since the scaling transformation is about the center S, all angles remain unchanged, thus the degree of ∠O'S'P' is still q°.

4. According to the area formula of a circle, the area A of a circle is πr², where r is the radius of the circle. Therefore, the area of the new circle S' is A' = π(1/2h)² = πh²/4.

5. The area formula for a sector is as follows: Sector area = (angle/360) * total area of the circle. Since the angle of arc OSP is q°, the area of sector O'S'P' is K = (q/360) * (πh²/4).

6. Therefore, after calculation, the new area of sector O'S'P' after scaling is K = πh²q/1440.

7. Through the above reasoning, the final answer is πh²q/1,440.

Definition of Circle

A circle is the locus of all points in a plane that are at a fixed distance from a given point.

In <note>circle S</note>, <note>point S is the center</note>, <note>the radius is h</note>. All points in the figure that are <note>at a distance of h from point S</note> are on <note>circle S</note>.

Definition of Radius

A radius is a line segment joining the center of a circle to any point on the circumference of the circle.

In the figure of this problem, in circle S, <note>point S is the center of the circle</note>, <note>point O is any point on the circle</note>, <note>line segment OS is the line segment from the center to any point on the circle</note>, therefore, <note>line segment OS is the radius of the circle</note>, with a length of h.

Definition of Sector

A sector is a figure formed by two radii of a circle and the arc enclosed between them.

In the <note>sector OSP</note>, the <note>radius OS and radius SP</note> are two radii of the circle, and the <note>arc OP</note> is the arc enclosed by these two radii. Therefore, according to the definition of a sector, the figure formed by these two radii and the enclosed <note>arc OP</note> is a sector.

Area Formula of a Circle

In <note>circle S</note>, the radius of the circle is <note>h</note>. According to the area formula of a circle, the area of the circle <note>A</note> is equal to the circle constant <note>π</note> multiplied by <note>the square of the radius h</note>, that is, <note>A = πh²</note>. The area of the new <note>circle S'</note> is <note>A' = π(1/2h)² = πh²/4</note>.

Formula for the Area of a Sector

The area \( X \) of a sector can be calculated using the formula \( X = \frac{\theta}{360} \times \pi \times r^2 \), where \( \theta \) is the measure of the central angle in degrees, and \( r \) is the radius length.

In this problem, <note>the area of the original sector OSP is K = (q/360) * πh²</note>, <note>the area of the new sector O'S'P' is K' = (q/360) * πh²/4 = πh²q/1440</note>.

Circle Scaling Theorem

If a circle undergoes scaling transformation with its center as the reference point and the scaling factor is k, then the radius of the scaled circle is k times the radius of the original circle, and all angles remain equal before and after the scaling.

The circle <note>with radius h</note> is scaled with <note>the center as the center</note>, using a <note>scaling factor of 1/2</note>. After scaling, the circle's <note>radius is 1/2*h</note>, and the <note>angle OSP remains unchanged</note>.


Question:
A prism has a height of 4 centimeters. Its base is shown below.

What is the surface area of the prism?

 $\Box$  square centimeters

Answer: 180 square centimeters

1. Given that the base of the triangular prism is a right triangle with three sides of 5 cm, 12 cm, and 13 cm, and the height of the prism is 4 cm.

2. Based on the given conditions and the converse of the Pythagorean theorem (i.e., if the lengths of the two sides of a triangle are a and b, and the hypotenuse is c, then it satisfies a² + b² = c²), since 5² + 12² = 13², that is 25 + 144 = 169 holds true, this triangle is a right triangle with the legs of 5 cm and 12 cm, and the hypotenuse of 13 cm.

3. The area B of this right triangle can be calculated using the triangle area formula B = 1/2 * base * height, where the base and height are 5 cm and 12 cm respectively, thus B = 1/2 * 5 * 12 = 30 square cm.

4. The perimeter P of this right triangle is 5 + 12 + 13 = 30 cm, which can be used to determine the lateral surface area of the prism.

5. The surface area S of the triangular prism consists of the areas of two triangular bases and three rectangular lateral faces, with the specific formula being S = perimeter P * height h + 2 * base area B, where the height h is 4 cm.

6. Substituting the given conditions S = 30 * 4 + 2 * 30, thus S = 120 + 60 = 180 square cm.

7. Therefore, the surface area of this triangular prism is 180 square cm.

Definition of Prism

<note>In the diagram of this problem, this geometric body is a triangular prism</note>, <note>the base is a right triangle</note>, <note>the three lateral faces are rectangles</note>. <note>The top and bottom faces have the same shape and area</note>, <note>the lateral faces are perpendicular to the base</note>.

Definition of Right Triangle

A triangle is classified as a right triangle if it has one interior angle measuring exactly 90 degrees.

<note>The two sides of the triangle are 5 cm and 12 cm respectively</note>, <note>the angle between these two sides is a right angle (90 degrees)</note>, therefore the triangle is a right triangle. <note>The sides of 5 cm and 12 cm are the legs</note>, <note>the side of 13 cm is the hypotenuse</note>.

Pythagorean Theorem

In a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the two legs.

<note>The two legs of the base triangle are 5 cm and 12 cm long respectively</note>, <note>the hypotenuse is 13 cm long</note>, satisfying <note>5² + 12² = 13²</note> i.e. <note>25 + 144 = 169</note>, thus confirming <note>the base is a right triangle</note>.

Surface Area Formula of a Prism

The area of the base is respectively <note>30 square centimeters</note>, the perimeter of the base is <note>5 + 12 + 13 = 30 centimeters</note>, and the height is <note>4 centimeters</note>. Therefore, the surface area of the prism S = 2 * 30 square centimeters + 30 centimeters * 4 centimeters = 60 square centimeters + 120 square centimeters = 180 square centimeters.

Converse of the Pythagorean Theorem

If the sum of the squares of two sides of a triangle is equal to the square of the third side, then the triangle is a right triangle, and the angle opposite to the longest side is a right angle.

In the figure of this problem, it is known that the three sides of the triangle are a, b, and c, and satisfy <note>c² = a² + b²</note>. Then according to the converse of the Pythagorean Theorem, <note>the triangle is a right triangle</note>.


Question:
What is m $\angle $ DGE?

m $\angle $ DGE= $\Box$ °

Answer: m \$\angle \$ DGE=90°

1. In the circle with center G, ∠EGF, ∠DGE, and ∠DGF are central angles with G as the center. According to the properties of angles in a circle, the sum of the central angles is 360°.

2. Based on the information given in the problem, it is known that ∠EGF = 150° and ∠DGF = 120°. Therefore, we can represent the sum of all three angles with the equation: m ∠EGF + m ∠DGE + m ∠DGF = 360°.

3. Substitute the known angle values into the equation: 150° + m ∠DGE + 120° = 360°.

4. Combine the sum of the known angles: 270° + m ∠DGE = 360°.

5. By subtracting 270° from both sides of the equation, we solve for m ∠DGE: m ∠DGE = 360° - 270° = 90°.

6. Therefore, through the above reasoning, the final answer is 90°.

Definition of Central Angle

An angle formed by two radii connecting two points on the circumference of a circle to the center of the circle is called a central angle.

In the circle with <note>center G</note>, <note>point E</note>, <note>point D</note>, and <note>point F</note> are three points on the circle, and the center is <note>point G</note>. The angle ∠EGF formed by the lines <note>GE and GF</note>, the angle ∠DGE formed by the lines <note>GD and GE</note>, and the angle ∠DGF formed by the lines <note>GD and GF</note> are called central angles.

Angle Properties of a Circle

The total measure of the angles in a complete circular arc is 360 degrees, and the sum of the central angles subtended by the entire circumference of a circle is 360 degrees.

Original text: <note>Arc FE, Arc ED, Arc DF around the circle</note>, the sum of their radians is 360 degrees, and the sum of their corresponding <note>circumferential angles is also 360 degrees</note>, i.e., <note>angle EGF + angle DGE + angle DGF = 360 degrees</note>, then <note>angle DGE = 90 degrees</note>.


Question:
A cone has a slant height of 21 millimeters. Its base is shown below.

What is the surface area of the cone?
Write an exact answer in terms of 𝜋.

 $\Box$ 𝜋/ square millimeters

Answer: 162𝜋 square millimeters

1. According to the problem description, the radius of the base of the cone is 6 mm, and the slant height is 21 mm.

2. The formula for the surface area of a cone is the base area plus the lateral area.

3. The base is a circle, and the formula for the area of a circle is A=πr^2, where r is the radius of the circle. Therefore, the base area A=π*(6^2)=36π square millimeters.

4. The lateral area is the area of the sector formed by unfolding the cone, and the formula is Al=πrℓ, where r is the radius of the base and ℓ is the slant height. Therefore, the lateral area Al=π*6*21=126π square millimeters.

5. The total surface area S of the cone is the base area plus the lateral area, i.e., S=36π+126π=162π square millimeters.

6. Through the above reasoning, the final answer is 162π square millimeters.

Definition of Radius

The radius of the base circle is 6 millimeters, that is, the distance from the center of the circle (the black dot in the figure) to any point on the circumference is 6 millimeters.

Generatrix

The generatrix of a cone is the line segment that joins a point on the circumference of the base to the apex.

In this problem, <note>the generatrix of the cone is 21 millimeters long</note>, that is, <note>the distance from the vertex of the cone to any point on the circumference of the base is 21 millimeters</note>. The generatrix is <note>the line segment from a point on the circumference of the base to the vertex of the cone</note>.

Formula for the Area of a Sector

The original text: <note>The lateral surface area of the cone unfolds into a sector</note>, this <note>sector's radius is the radius of the base circle, 6 millimeters</note>, <note>the arc length of the sector is the slant height of the cone, 21 millimeters</note>. The calculation formula is <note>Al = πrℓ</note>, therefore the lateral surface area is <note>π*6*21 = 126π square millimeters</note>.

Area Formula of a Circle

The area of a circle is given by the formula A = πr², where A represents the area, π (pi) is the mathematical constant approximately equal to 3.14159, and r is the radius of the circle.

<note>The radius of the circle is 6 millimeters</note>, according to the area formula of a circle, the area A of the circle is equal to pi multiplied by the square of the radius 6, that is <note>A = π6²</note>.

Formula for the Surface Area of a Cone

The total surface area of a cone is equal to the sum of the base area and the lateral surface area.

<note>The base of the cone is a circle</note>, with a radius of <note>6 millimeters</note>, and the base area is <note>π*(6^2)=36π square millimeters</note>. <note>The lateral surface of the cone, when unfolded, is a sector</note>, with a radius of <note>slant height 21 millimeters</note>, and the arc length of the sector is equal to <note>the circumference of the base 2π*6=12π millimeters</note>. The lateral area is equal to <note>the area of the sector, which is π*6*21=126π square millimeters</note>. The total surface area of the cone is equal to the base area plus the lateral area, so the total surface area is <note>36π + 126π = 162π square millimeters</note>.


Question:
Look at this shape:Which image shows a reflection?

| A | B | C |
- A
- B
- C

Answer: C

1. Observe the shape of the original figure, such as the shape of one of the quadrilaterals shown in the figure.

2. Look at option image A, it can be observed that the figure has undergone a 90° counterclockwise rotation. This transformation is a rotation, not a reflection.

3. Look at option image B, it can be observed that the figure has moved 2 units in the vertical direction. This transformation is a translation, not a reflection.

4. Look at option image C, it can be observed that the figure and the original figure are symmetrical in the horizontal direction, like a mirror image mapping. This transformation is a reflection.

5. From the above analysis, it can be concluded that option C is the correct answer because it shows a reflection of the original figure.

Rotation Transformation

A rotation involves rotating a geometric figure around a fixed point, known as the center of rotation, through a specified angle. The new coordinates of the points after rotation can be obtained using specific transformation formulas.

<note>The figure in option A is the original figure rotated 90° counterclockwise around the origin</note>, so it is not a reflection.

Definition of Translation

A translation is a geometric transformation where a figure is moved in the plane along a certain direction, without altering its shape and orientation.

The figure in <note>option B</note> is the original figure <note>translated 2 units upward in the vertical direction</note>, so it is not a reflection.

Reflection Transformation

A reflection transformation is a type of geometric transformation that flips a figure over a specific line known as the line of reflection. After the reflection transformation, the coordinates of each point on the figure are changed to the coordinates of its corresponding point symmetrically across the line of reflection.

The figure in <note>option C</note> is the original figure <note>reflected across a vertical axis</note>, so it is the correct answer.

In the diagram of this problem, option C reflects the shape through <note>horizontal reflection</note>, verifying the reflection theorem, that is, <note>the shape and size of the original figure and the reflected figure remain consistent</note>, while <note>the positions correspond to each other</note>. For example, <note>vertex A of the original figure is at the symmetrical position after reflection</note>, that is, <note>coordinates (1,1) map to (7,1)</note>, and so on.


Question:
A cone is shown below.

What is the surface area of the cone?
Write an exact answer in terms of 𝜋.

 $\Box$ 𝜋/ square inches

Answer: 216𝜋 square inches

1. Given the diameter of the base is 18 inches, therefore the radius of the base r is 18/2=9 inches.

2. Given the slant height 𝓁 is 15 inches.

3. The formula for the surface area S of the cone is S=𝜋r^2 + 𝜋r𝓁, where 𝜋r^2 is the area of the base circle, and 𝜋r𝓁 is the lateral area.

4. Calculate the area of the base circle, 𝜋r^2=𝜋×9^2=81𝜋.

5. Calculate the lateral area, 𝜋r𝓁=𝜋×9×15=135𝜋.

6. Add the area of the base circle and the lateral area to get the surface area S=81𝜋+135𝜋=216𝜋.

7. Through the above reasoning, the final answer is 216𝜋 square inches.

Definition of Radius

<note>The radius of the base circle is r</note>, derived from <note>the base diameter of 18 inches</note> to get <note>r=18/2=9 inches</note>. <note>The radius of the base circle refers to the length of the line segment from the center of the circle to any point on the circumference</note>.

Definition of Diameter

A diameter is a line segment that passes through the center of a circle and has its endpoints on the circle. It is the longest chord of the circle, with a length equal to twice the radius of the circle.

<note>The diameter of the base circle is 18 inches</note>, connecting the <note>center of the circle</note> and <note>two points on the circumference</note>, the length is 2 times the <note>radius</note>, that is, <note>diameter = 2r</note>.

Generatrix

<note>The slant height of the cone is 15 inches</note>, <note>the slant height is the generatrix</note>.

Area Formula of a Circle

<note>The radius of the base circle is 9 inches</note>. According to <note>the area formula of a circle</note>, the area of the circle A is equal to pi π multiplied by <note>the square of the radius 9</note>, i.e., A = π9².

Lateral Surface Area of a Cone

The lateral surface area \(A\) of a cone is calculated using the formula \(A = \pi r l\), where \(r\) is the radius of the base circle and \(l\) is the slant height of the cone.

In the figure of this problem, the lateral surface area is <note>πr𝓁</note>, where <note>r=9 inches</note>, <note>𝓁=15 inches</note>, so the lateral surface area is <note>π×9×15=135π</note>.

Formula for the Surface Area of a Cone

In the diagram of this problem, <note>the base of the cone is a circle</note>, with a <note>radius of 9 inches</note>, and a <note>base area of π×9²=81π square inches</note>. <note>The lateral surface of the cone, when unfolded, is a sector</note>, with a <note>radius equal to the slant height of 15 inches</note>, and the <note>arc length of the sector is equal to the circumference of the base 2π×9=18π inches</note>. The lateral area is equal to <note>the area of the sector</note>, which is <note>π×9×15=135π square inches</note>. The cone's <note>total surface area is equal to the base area plus the lateral area</note>, so the <note>total surface area is 81π+135π=216π square inches</note>.


Question:
What is m $\angle $ FEG?

m $\angle $ FEG= $\Box$ °

Answer: m \$\angle \$ FEG=105°

1. According to the circular diagram provided in the problem, it can be seen that point E is the center of the circle, and points F, G, and H are all located on the same circumference. The line segments EF, EG, and EH form three angles centered at E: ∠FEG, ∠GEH, and ∠FEH.

2. It is known that the measure of ∠GEH is 130°, and the measure of ∠FEH is 125°. These angles form a complete circle around the center E, so their sum should be 360°.

3. According to the properties of angles in a circle: in a circle, the sum of angles centered at the circle's center is 360°, so we can obtain the equation: m∠FEG + m∠GEH + m∠FEH = 360°.

4. Substituting the known values m∠GEH = 130° and m∠FEH = 125° into the equation, we get: m∠FEG + 130° + 125° = 360°.

5. Calculate the sum of the known angles on the right side of the equation: 130° + 125° = 255°.

6. Simplify the equation to: m∠FEG + 255° = 360°.

7. By subtracting 255° from both sides, we get m∠FEG = 360° - 255°.

8. Calculate the result of 360° - 255°, obtaining m∠FEG = 105°.

9. Through the above reasoning, the final answer is 105°.

Angle Properties of a Circle

Original: <note>Arc GF Arc HF Arc GH around the circle</note>, the sum of the radians is 360 degrees, and the corresponding <note>sum of the angles at the circumference is also 360 degrees</note>, i.e. <note>angle FEG + angle FEH + angle GEH = 360 degrees</note>, then <note>angle FEG = 105 degrees</note>.

Definition of Central Angle

<note>Points F, G, and H are all located on the circumference</note>, <note>the center of the circle is point E</note>. The angle ∠FEG formed by the lines <note>EF and EG</note>, the angle ∠GEH formed by the lines <note>EG and EH</note>, and the angle ∠FEH formed by the lines <note>EF and EH</note> are all central angles.


Question:
What is m $\overset{\frown}{GH}$ ?

m $\overset{\frown}{GH}$ = $\Box$ °

Answer: m \$\overset{\frown}{GH}\$ =80°

1. Let the circle be circle O, given that the degree of arc FG = 145°, the degree of arc EH = 70°, and the degree of arc EF = 65°.

2. According to the properties of the circle, the sum of the degrees of arcs FG, GH, EH, and EF around the circle is 360°.

3. From the figure, it is known that the circumference arcs FG + GH + EH + EF = 360°.

4. Substitute the known arc degrees into the addition formula, we get: 145° + GH + 70° + 65° = 360°.

5. Combine the known angle values, we have 145° + 70° + 65° = 280°.

6. Simplifying the original equation, we get: GH + 280° = 360°.

7. By subtracting 280°, we deduce: GH = 360° - 280°.

9. Through the above reasoning, the final answer is 80°.

Definition of Arc

An arc is a curve on a circle that is defined by two distinct points on the circle.

In the diagram of this problem, there are <note>two points G and H</note> on the circle, <note>arc GH is a segment of the curve connecting these two points</note>. According to the definition of an arc, <note>arc GH is a segment of the curve between the two points G and H on the circle</note>.

Definition of Central Angle

In the figure of this problem, in the circle, points G, H, E, and F are four points on the circle, <note>the center of the circle is point O</note>. <note>The angle ∠GOH formed by the lines OG and OH</note> is called the central angle. Similarly, <note>the angle ∠EOH formed by the lines OE and OH</note>, <note>the angle ∠EOF formed by the lines OE and OF</note> are also central angles.

Angle Properties of a Circle

The original text: <note>Arc FG, arc GH, arc HE, and arc EF make a full circle</note>, with a total angle of <note>360 degrees</note>.


Question:
What is m $\overset{\frown}{UV}$ ?

m $\overset{\frown}{UV}$ = $\Box$ °

Answer: m \$\overset{\frown}{UV}\$ =140°

1. Given three arcs on the circumference, namely arc TV, arc UV, and arc TU, divided by points V, U, and T. According to the diagram, the degree of arc TV is 90°, and the degree of arc TU is 130°.

2. According to the properties of a circle, the total angle around the circumference is 360°. Thus, we can derive the equation: m(∠TV) + m(∠UV) + m(∠TU) = 360°.

3. Substitute the given information into the equation, we get: 90° + m(∠UV) + 130° = 360°.

4. Combine the known terms in the equation, 90° + 130° = 220°, thus updating the equation to: m(∠UV) + 220° = 360°.

5. Subtract 220° from both sides of the equation, we get: m(∠UV) = 360° - 220°.

7. Through the above reasoning, the final answer is m(∠UV) = 140°.

Definition of Central Angle

In <note>circle O</note>, <note>point V and point T are two points on the circle</note>, <note>the center of the circle is point O</note>. The angle formed by the lines <note>OV and OT</note>, <note>∠VOT</note>, is called the central angle.

Definition of Arc

In the diagram of this problem, there are three arcs on the circumference, namely <note>arc TV</note>, <note>arc UV</note>, and <note>arc TU</note>. <note>Arc TV</note> is a segment of the curve connecting point <note>T</note> and point <note>V</note>, <note>arc UV</note> is a segment of the curve connecting point <note>U</note> and point <note>V</note>, and <note>arc TU</note> is a segment of the curve connecting point <note>T</note> and point <note>U</note>. According to the definition of an arc, <note>arc TV</note>, <note>arc UV</note>, and <note>arc TU</note> are all segments of the curve between two points on the circle. <note>Arc TV measures 90°</note>, <note>arc TU measures 130°</note>.

Angle Properties of a Circle

In the diagram of this problem, <note>arc TV, arc VU, arc TU around the circle</note>, the sum of the radians is <note>360 degrees</note>, and the corresponding <note>sum of the angles at the circumference is also 360 degrees</note>, that is, <note>angle TV + angle UV + angle TU = 360 degrees</note>, then <note>angle VUT = 360 degrees</note>.


Question:
$\overline{AB}$ is shown on the graph below. $\overline{AB}$ is dilated by a scale factor of 1/2 centered at (9,–1) to create $\overline{A'B'}$ .

What is the length of $\overline{A'B'}$ ?
Write your answer as a whole number or as a decimal rounded to the nearest tenth.

 $\Box$  units

Answer: 6.4 units

1. According to the problem, the endpoints of the segment \( \overline{AB} \) are \( A(-8, -5) \) and \( B(2, 3) \). First, we use the distance formula to calculate the length of \( \overline{AB} \).

2. The distance formula is: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

3. Substituting the coordinates, we get: \[ AB = \sqrt{(2 - (-8))^2 + (3 - (-5))^2} = \sqrt{(2 + 8)^2 + (3 + 5)^2} = \sqrt{10^2 + 8^2} = \sqrt{100 + 64} \]

4. Simplifying, we get: \( AB = \sqrt{164} \)

5. According to the problem, \( \overline{AB} \) is scaled by a factor of \( \frac{1}{2} \) relative to the point \( (9, -1) \), resulting in \( \overline{A'B'} \). Therefore, the length of \( \overline{A'B'} \) is \( \frac{1}{2} \) of the length of \( \overline{AB} \).

6. Calculating the length of \( \overline{A'B'} \): \[ A'B' = \frac{1}{2} \times \sqrt{164} = \frac{1}{2} \times 12.806 \approx 6.403 \]

7. Rounding to the nearest tenth, the length of \( \overline{A'B'} \) is: \( 6.4 \) units.

8. Through the above calculations and reasoning, the final answer is \( 6.4 \) units.

Distance Formula Between Two Points

In a Cartesian coordinate plane, the distance \( d \) between two points \((x_1, y_1)\) and \((x_2, y_2)\) can be calculated using the formula \( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \).

<note>The coordinates of point A are \( A(-8, -5) \)</note>, <note>The coordinates of point B are \( B(2, 3) \)</note>, using the distance formula to calculate <note>the length of segment \( \overline{AB} \)</note>: \[ AB = \sqrt{(2 - (-8))^2 + (3 - (-5))^2} = \sqrt{(2 + 8)^2 + (3 + 5)^2} = \sqrt{10^2 + 8^2} = \sqrt{100 + 64} = \sqrt{164} \]

Definition of Scale Factor in Rectangular Coordinate Systems

In a rectangular coordinate system, if a polygon xxx is transformed to xxx by scaling, then the scale factor is defined as the ratio of the length of any side of the scaled polygon to the length of the corresponding side of the original polygon.

<note>After scaling polygon AB, we obtain polygon A'B'</note>, scale factor = <note>length of A'B' / length of AB</note>, i.e., scale factor = 1/2.


Question:
Lester wants to prove that a quadrilateral is a parallelogram if one pair of opposite sides is congruent and parallel. To start his proof, he draws quadrilateral PQRS with $\overline{PQ}$ ∥ $\overline{SR}$ and $\overline{PQ}$ ≅ $\overline{SR}$ . He also draws diagonal $\overline{QS}$ .

Complete the proof that PQRS is a parallelogram.
In quadrilateral PQRS,  $\overline{PQ}$ ∥ $\overline{SR}$  and  $\overline{PQ}$ ≅ $\overline{SR}$ . By the Alternate Interior Angles Theorem,  $\Box$ . Also,  $\overline{QS}$ ≅ $\overline{QS}$  by the  $\Box$ . So, by the  $\Box$  Congruence Theorem, △PQS≅△RSQ.
Since corresponding parts of congruent triangles are congruent,  $\angle $ PSQ≅ $\Box$ . So, by the Converse of the  $\Box$ ,  $\Box$ . So, PQRS is a parallelogram because it has two pairs of parallel sides.

Answer:
In quadrilateral PQRS,  \$\overline{PQ}\$ ∥ \$\overline{SR}\$  and  \$\overline{PQ}\$ ≅ \$\overline{SR}\$ . By the Alternate Interior Angles Theorem,  \$\angle \$ PQS≅ \$\angle \$ RSQ. Also,  \$\overline{QS}\$ ≅ \$\overline{QS}\$  by the Reflexive Property of Congruence. So, by the Side-Angle-Side Congruence Theorem, △PQS≅△RSQ.
Since corresponding parts of congruent triangles are congruent,  \$\angle \$ PSQ≅ \$\angle \$ RQS. So, by the Converse of the Alternate Interior Angles Theorem,  \$\overline{PS}\$ ∥ \$\overline{QR}\$ . So, PQRS is a parallelogram because it has two pairs of parallel sides.

1. In quadrilateral PQRS, it is given that segment PQ is parallel to segment SR (i.e., \overline{PQ} ∥ \overline{SR}), and segment PQ is congruent to segment SR (i.e., \overline{PQ} ≅ \overline{SR}).

2. By adding the diagonal \overline{QS}, two triangles △PQS and △RSQ are formed.

3. According to the 'Alternate Interior Angles Theorem', when parallel lines are intersected by a transversal, the alternate interior angles are equal. Therefore, in this problem, \angle PQS ≅ \angle RSQ.

4. According to the 'Reflexive Property of Congruence', any geometric figure is congruent to itself. Therefore, \overline{QS} ≅ \overline{QS}.

5. Now we know that △PQS and △RSQ have two pairs of congruent corresponding sides, and the angles between these sides are also congruent. Therefore, according to the 'Side-Angle-Side Congruence Theorem', △PQS ≅ △RSQ.

6. Since corresponding parts of congruent triangles are congruent (CPCTC), \angle PSQ ≅ \angle RQS.

7. According to the 'Converse of the Alternate Interior Angles Theorem', when two lines are intersected by a transversal and the alternate interior angles are equal, the two lines are parallel. Therefore, \overline{PS} ∥ \overline{QR}.

8. Therefore, since quadrilateral PQRS has two pairs of parallel opposite sides, it is a parallelogram.

Definition of Parallelogram

A quadrilateral is a parallelogram if and only if it has two pairs of opposite sides that are both parallel (∥) and congruent (≅).

In the figure of this problem, in quadrilateral PQRS, <note>\overline{PQ} ∥ \overline{SR}</note> and <note>\overline{PS} ∥ \overline{QR}</note>, therefore according to the definition of parallelogram, <note>PQRS is a parallelogram</note>.

Definition of Alternate Interior Angles

When a straight line (referred to as a transversal) intersects two parallel lines, the angles that are located between the two parallel lines but on opposite sides of the transversal are called alternate interior angles.

<note>Two parallel lines PQ and SR are intersected by a line QS</note>, where <note>angle PQS and angle RSQ are between the two parallel lines and on opposite sides of the intersecting line QS</note>, therefore <note>angle PQS and angle RSQ are alternate interior angles</note>. Alternate interior angles are equal, that is <note>angle PQS is equal to angle RSQ</note>.

Triangular Congruence Theorem (SAS)

If two sides and the included angle of one triangle are equal to two sides and the included angle of another triangle, then the two triangles are congruent.

In the original text: <note>Triangle PQS and triangle RSQ</note>, <note>side PQ is equal to side SR, side QS is equal to side QS, and the included angle PQS is equal to the included angle RSQ</note>, therefore, according to the Triangular Congruence Theorem (SAS), these two triangles are congruent.

Definition of Congruent Triangles

Two triangles are congruent if and only if their corresponding sides are equal and their corresponding angles are equal.

<note>Triangles PQS and RSQ are congruent triangles</note>, the corresponding sides and angles of triangle PQS are equal to those of triangle RSQ, namely:
<note>Side PQ = Side SR</note>
<note>Side PS = Side RQ</note>
<note>Side QS = Side QS</note>,
and the corresponding angles are also equal:
<note>Angle PQS = Angle RSQ</note>
<note>Angle PSQ = Angle RQS</note>
<note>Angle QPS = Angle QRS</note>.


Question:
$\overline{AB}$ is shown on the graph below. $\overline{AB}$ is dilated by a scale factor of 1/2 centered at (9,–1) to create $\overline{A'B'}$ .

What is the length of $\overline{A'B'}$ ?
Write your answer as a whole number or as a decimal rounded to the nearest tenth.

 $\Box$  units

Answer: 6.4 units

1. Given the endpoints of the line segment AB are A(-8, -5) and B(2, 3). To calculate the length of the line segment A'B', we first need to find the length of AB.

2. According to the distance formula between two points: AB=√{(x2-x1)^2 + (y2-y1)^2}, substituting A(-8, -5) and B(2, 3), we get: √{(2 - (-8))^2 + (3 - (-5))^2}.

3. Calculate the values inside the parentheses: (2 - (-8)) = 10 and (3 - (-5)) = 8. Therefore, we need to calculate √{10^2 + 8^2}.

4. Continue calculating the sum of squares: 10^2 = 100 and 8^2 = 64, thus √{100 + 64} = √{164}.

5. Calculate the approximate value of √{164}, using a calculator we get √{164} approximately equals 12.806. The length of AB is 12.806.

6. The problem requires calculating A'B', given that AB undergoes a scaling transformation with center at (9, -1) and a scaling factor of 1/2.

7. According to the definition of scaling factor in the Cartesian coordinate system, the length of A'B' is the length of AB multiplied by the scaling factor 1/2, that is A'B' = 1/2 * 12.806.

8. Calculate 1/2 * 12.806 = 6.403.

9. Round 6.403 to the nearest tenth, we get the approximate length of A'B' as 6.4 units.

10. Through the above reasoning, the final answer is 6.4 units.

Distance Formula Between Two Points

In the diagram of this problem, <note>the coordinates of point A are (-8, -5)</note>, <note>the coordinates of point B are (2, 3)</note>, therefore <note>the length of AB is calculated using the distance formula</note>, which is <note>AB = √{(2 - (-8))^2 + (3 - (-5))^2}</note> = <note>√{(10)^2 + (8)^2}</note> = <note>√{100 + 64}</note> = <note>√{164}</note>

<note>AB缩放后得到A'B'</note>, Scale factor = <note>Length of A'B'/Length of AB</note>, Scale factor = 1/2.


Question:
What is m $\overset{\frown}{VW}$ ?

m $\overset{\frown}{VW}$ = $\Box$ °

Answer: m \$\overset{\frown}{VW}\$ =75°

1. Confirm the markings and degrees of the four arcs on the circumference: 𝛼TU = 100°, 𝛼UV = 130°, 𝛼TW = 55°, and 𝛼VW (to be solved).

2. According to the properties of angles in a circle, the sum of the degrees of the entire circle is 360°. Based on this, we can set up the equation: 𝛼TU + 𝛼UV + 𝛼VW + 𝛼TW = 360°.

3. Substitute the known degrees into the equation: 100° + 130° + 𝛼VW + 55° = 360°.

4. Perform addition on the known values on the left side of the equation to get 285° + 𝛼VW = 360°.

5. To solve for 𝛼VW, subtract 285° from both sides of the equation to get 𝛼VW = 75°.

6. Through the above reasoning, the final answer is 𝛼VW = 75°.

Angle Properties of a Circle

In the diagram of this problem, <note>arc TU, arc UV, arc VW, arc WT around the circle, the sum of the radians is 360 degrees</note>, <note>the sum of their corresponding central angles is also 360 degrees</note>, that is, <note>angle TOU + angle UOV + angle VOW + angle WOT = 360 degrees</note>, then <note>angle VOW = 75 degrees</note>.


Question:
What is m $\angle $ TSW?

m $\angle $ TSW= $\Box$ °

Answer: m \$\angle \$ TSW=130°

1. First, it is observed that within the circle, angles ∠USV, ∠VSW, ∠TSW, and ∠TSU are all the angles formed by rays emanating from the same point S. According to the properties of angles in a circle, their sum is 360°.

2. According to the given information, m ∠USV = 60°, m ∠VSW = 115°, m ∠TSU = 55°. These angle measures can be directly substituted into the equation from the previous step.

3. Substituting these known angle values into the equation, we get the equation: m ∠USV + m ∠VSW + m ∠TSW + m ∠TSU = 360°, i.e., 60° + 115° + m ∠TSW + 55° = 360°.

4. Adding the known angle values, we get a total of 230°, and our equation becomes: m ∠TSW + 230° = 360°.

5. To find the value of m ∠TSW, subtract 230° from both sides, resulting in m ∠TSW = 360° - 230°.

6. The calculation result is: m ∠TSW = 130°.

7. Through the above reasoning, the final answer is m ∠TSW = 130°.

Original: <note>Arc WT Arc TU Arc UV Arc WV around the circle</note>, the radian sum is <note>360 degrees</note>, the corresponding <note>sum of the angles at the circumference is also 360 degrees</note>, that is, <note>angle WST + angle TSU + angle USV + angle WSV = 360 degrees</note>, then <note>angle WST = 130 degrees</note>.


Question:
$\overline{DE}$ is shown on the graph below. $\overline{DE}$ is dilated by a scale factor of 4 centered at point D to create $\overline{D'E'}$ .

What is the length of $\overline{D'E'}$ ?
Write your answer as a whole number or as a decimal rounded to the nearest tenth.

 $\Box$  units

Answer: 52 units

1. Given point D(1, -8) and point E(6, 4), we need to calculate the length of segment DE.

2. According to the distance formula, the distance between two points (x1, y1) and (x2, y2) is d = √[(x2 - x1)² + (y2 - y1)²].

3. Substituting point D(1, -8) and point E(6, 4) into the distance formula, we have the length of DE as d = √[(6 - 1)² + (4 - (-8))²].

4. Calculation steps: first calculate the difference in x-coordinates (6 - 1)² = 5² = 25; then calculate the difference in y-coordinates (4 - (-8))² = 12² = 144.

5. Adding the two: the length of DE is d = √(25 + 144) = √169.

6. Since √169 = 13, the length of segment DE is 13 units.

7. Now, according to the problem, DE is scaled up by a factor of 4 with center at point D to obtain D'E'.

8. According to the scaling theorem, the length of the segment is scaled to the original length multiplied by the scaling factor, so the length of D'E' is the scaling factor multiplied by the length of DE.

9. Calculating the length of D'E', we have the length of D'E' as d = 4 * 13 = 52.

10. Through the above reasoning, the final answer is that the length of D'E' is 52 units.

Distance Formula Between Two Points

In the figure of this problem, the distance formula used to calculate the distance between <note>point D(1, -8) and point E(6, 4)</note> is <note>DE length</note> as d, which is <note>d=√[(6 - 1)² + (4 - (-8))²]</note>.

<note>After scaling, line DE becomes line D'E'</note>, scale factor = <note>length of D'E'</note> / <note>length of DE</note>, i.e., scale factor = 4.


Question:
Look at this shape:Which image shows a rotation?

| A | B | C |
- A
- B
- C

Answer: A

1. First, observe the given original figure. This is a quadrilateral located in the first quadrant of the Cartesian coordinate plane.

2. Select the figure in option A and observe its changes relative to the original figure.

3. According to the definition of rotational transformation, rotation of a figure refers to rotating the figure around a fixed point by a certain angle. In option A, it is observed that the figure is rotated 90 degrees clockwise relative to the original figure. This can be judged from the change in the orientation of the figure.

4. Select the figure in option B and observe its changes relative to the original figure.

5. Option B shows a symmetrical figure, which conforms to the definition of reflection transformation, i.e., the mirror image of the figure about a symmetry axis. Therefore, option B is not a rotation.

6. Select the figure in option C and observe its changes relative to the original figure.

7. The figure in option C is obtained from the original figure through translation (downward by 2 units and to the right by 1 unit). According to the definition of translation transformation, the position of the figure changes but it does not rotate. Therefore, option C is also not a rotation.

8. Through the above analysis, the change shown in option A conforms to the definition of rotation, i.e., the figure is rotated 90 degrees clockwise relative to its original position, rather than being translated or reflected. Therefore, option A shows a rotation.

9. After the above reasoning, the final answer is A.

Rotation Transformation

<note>The figure in option A is rotated 90 degrees clockwise relative to the original figure</note>. <note>The fixed point is located at the vertex of the original figure</note>, <note>the rotation angle is 90 degrees</note>, so option A conforms to the definition of rotation transformation.

Reflection Transformation

<note>The image of the figure with respect to a certain axis of symmetry</note>, this conforms to <note>the definition of reflection transformation</note>, therefore option B is not a rotation.

Definition of Translation

<note>Option C shows the figure being translated 2 units down and 1 unit to the right</note>. This conforms to the definition of translation, therefore <note>Option C is not a rotation</note>.

Rotation Invariance Theorem

Certain geometric figures will coincide with their original position after being rotated by a specific angle. This property is known as rotational invariance.

<note>The shape and size of the figure in option A remain consistent with the original figure after undergoing a rotational transformation, only the position changes</note>, which conforms to the Rotation Invariance Theorem.

Reflection Transformation

<note>In the figure of this problem, the shape in option B, after undergoing a reflection transformation about the axis of symmetry</note>, conforms to the mirror symmetry theorem, <note>with the shape remaining symmetrical on both sides of the axis of symmetry</note>, but it is not a rotation.

Translation Invariance Theorem

After a translation transformation, the shape and size of the figure remain unchanged, but its position is altered.

In the figure of this problem, <note>the shape in option C undergoes a translation transformation</note>, <note>the shape and size remain consistent with the original figure</note>, <note>only the position changes</note>, which conforms to the <note>Translation Invariance Theorem</note>.


Question:
What is m $\overset{\frown}{RU}$ ?

m $\overset{\frown}{RU}$ = $\Box$ °

Answer: m \$\overset{\frown}{RU}\$ =85°

1. From the information in the figure, we know there are four arcs on the circumference: arc ST, arc TU, arc RU, and arc RS.

2. According to the angle properties of the circle, the sum of the degrees of all arcs in the entire circumference is 360°.

3. Therefore, we can set up the equation: degree of arc ST + degree of arc TU + degree of arc RU + degree of arc RS = 360°.

4. Substitute the known quantities from the figure: this equation can be written as: 70° + 55° + m ⦯RU + 150° = 360°.

5. Simplify the equation: 70° + 55° + 150° + m ⦯RU = 360° => 275° + m ⦯RU = 360°.

6. Thus, we subtract 275° from both sides of the equation to get m ⦯RU = 360° - 275°.

8. Therefore, through the above reasoning, the final answer is 85°.

Definition of Arc

In the figure of this problem, there are points S, T, U, and R on the circle, <note>arc ST is a segment of the curve connecting point S and point T</note>, <note>arc TU is a segment of the curve connecting point T and point U</note>, <note>arc RU is a segment of the curve connecting point R and point U</note>, <note>arc RS is a segment of the curve connecting point R and point S</note>. According to the definition of an arc, <note>arc ST</note>, <note>arc TU</note>, <note>arc RU</note>, and <note>arc RS</note> are segments of the curve between two points on the circle.

Angle Properties of a Circle

The total measure of the angles in a complete circular arc is 360 degrees, and the sum of the central angles subtended by the entire circumference of a circle is 360 degrees.

<note>Arc ST, arc TU, arc RU, and arc RS surround the circle</note>, the sum of their radian measures is <note>360 degrees</note>, and the sum of their corresponding central angles is also <note>360 degrees</note>.


Question:
$\overline{TU}$ is shown on the graph below. $\overline{TU}$ is dilated by a scale factor of 3 centered at (–7,2) to create $\overline{T'U'}$ .

What is the length of $\overline{T'U'}$ ?
Write your answer as a whole number or as a decimal rounded to the nearest tenth.

 $\Box$  units

Answer: 21.6 units

1. Given that the coordinates of point T are (-7, 8) and the coordinates of point U are (-3, 2). Calculate the length of segment TU using the distance formula between two points. The distance formula is: for two points (x1, y1) and (x2, y2), the distance d = √{(x2 - x1)^2 + (y2 - y1)^2}.

2. Substitute the coordinates of point T (-7, 8) and point U (-3, 2) into the distance formula to calculate the length of TU, obtaining: TU = √{((-3) - (-7))^2 + (2 - 8)^2}.

3. Calculate: ((-3) - (-7))^2 = (4)^2 = 16; (2 - 8)^2 = (-6)^2 = 36.

4. Therefore, the length of TU is √{16 + 36} = √{52}.

5. √{52} can be simplified to √{4 * 13} = 2√{13}, taking the approximate value, TU is approximately equal to 7.2.

6. According to the definition of the scaling factor in the Cartesian coordinate system, scaling factor = 3, obtain T'U'.

7. Therefore, the length of T'U' is 3 times the length of TU.

8. Calculate: T'U' = 3 * TU = 3 * 2√{13}.

9. Therefore, the approximate value of T'U' is: 3 * 7.2 = 21.6.

10. Through the above reasoning, the final answer is 21.6 units.

Distance Formula Between Two Points

In the figure of this problem, <note>the coordinates of point T are (-7, 8)</note>, <note>the coordinates of point U are (-3, 2)</note>. Apply the <note>distance formula to calculate the distance between points T and U</note>, that is, <note>the length of segment TU</note>: d(T, U) = <note>√(((-3) - (-7))^2 + (2 - 8)^2)</note> = <note>√(4^2 + (-6)^2)</note> = <note>√(16 + 36)</note> = <note>√52 ≈ 7.2</note>.

<note>TU</note> scaled to <note>T'U'</note>, scale factor = <note>T'U'</note> length / <note>TU</note> length, scale factor = 3.


Question:
Find XZ and the area of △WXY.

Write your answer as an integer or as a decimal rounded to the nearest tenth.
XZ= $\Box$  cmArea= $\Box$  cm ^ 2

Answer: XZ=14.1 cmArea=126.9 cm ^ 2

1. The problem requires finding the length of segment XZ and calculating the area of △WXY.

2. From the figure, it is known that ∠W is 70°, and △WXZ is a right triangle with XZ as one of the legs.

3. According to the definition of the sine function, in the right triangle WXZ, sin∠W = opposite side/hypotenuse, i.e., sin(∠XWZ) = XZ/WX.

4. Substituting the given conditions, sin(70°) = XZ/15.

5. Refer to the sine table or use a calculator, sin(70°) ≈ 0.9397.

6. Substitute into the calculation, obtaining 0.9397 = XZ/15, solving for XZ = 0.9397 × 15.

7. Calculate XZ ≈ 14.0955, so XZ is approximately 14.1 cm (rounded to one decimal place).

8. Next, calculate the area of △WXY. According to the triangle area formula, Area = 1/2·base·height.

9. In △WXY, choose WY as the base and XZ as the height.

10. Substitute the values, Area = 1/2 × 18 × 14.0955.

11. Calculate the area ≈ 126.8585.

12. Rounded to one decimal place, the area of the triangle is approximately 126.9 square cm.

13. Through the above reasoning, the final answer is that the length of XZ is 14.1 cm, and the area of △WXY is 126.9 square cm.

Definition of Sine Function

In a right-angled triangle, the sine of an acute angle is defined as the ratio of the length of the side opposite the angle to the length of the hypotenuse.

<note>In the figure of this problem, in the right triangle WXZ, angle ∠XWZ is an acute angle, side XZ is the opposite side of angle ∠XWZ, and side WX is the hypotenuse</note>. According to the definition of the sine function, <note>the sine value of angle ∠XWZ is equal to the ratio of the opposite side XZ to the hypotenuse WX</note>, that is, <note>sin(∠XWZ) = XZ / WX</note>.

Area Formula of a Triangle

The area of any triangle is equal to its base multiplied by its height, divided by 2.

<note>In triangle WXY</note>, <note>side WY is the base</note>, <note>segment ZX is the height on this base</note>, so <note>the area of triangle WYX is equal to the base WY multiplied by the height ZX divided by 2</note>, i.e., area = <note>(WY * ZX) / 2</note>.


Question:
Look at this shape:Which image shows a translation?

| A | B | C |
- A
- B
- C

Answer: A

1. In a geometry problem, we are asked to identify the result of a given figure after geometric transformations, which include translation, rotation, and reflection.

2. First, we observe the position of the original figure in the grid as given in the problem.

3. Comparing with option A, the figure in option A has only one unit downward change after a horizontal right translation relative to the original figure, without involving any change in direction.

4. By definition, translation is a distance-preserving transformation that changes the position of the figure but not its shape and direction. Therefore, option A satisfies the characteristics of translation.

5. Then we observe option B, the figure changes its shape direction after a 180° rotation, which meets the characteristics of rotation transformation.

6. Finally, looking at option C, the figure flips along the vertical symmetry axis, which meets the characteristics of reflection transformation.

7. Based on the above reasoning, option A shows the translation of the figure, so option A is the correct answer.

Definition of Translation

In the diagram of this problem, <note>the original figure is translated from its position in the grid to a new position</note>. <note>The change in the figure's position is a translation to the right and down by one unit</note>. This movement <note>maintains the shape and orientation of the figure</note>, thus it is a translation.

Rotation Transformation

A rotation involves rotating a geometric figure around a fixed point, known as the center of rotation, through a specified angle. The new coordinates of the points after rotation can be obtained using specific transformation formulas.

<note>In the figure of this problem</note>, the shape in option B <note>has rotated 180° relative to the original shape</note>, thereby <note>changing direction about a fixed point, i.e., a 180° rotation</note>. This transformation changes the direction of the shape, <note>thus it is a rotation</note>.

Reflection Transformation

<note>The figure in option C is reflected along the vertical axis of symmetry</note>, creating a mirror image relative to the original figure. This transformation preserves the shape of the figure but changes its orientation, thus it is a reflection.

Definition of Translation

A translation is a geometric transformation where a figure is moved in the plane along a certain direction, without altering its shape and orientation.

The figure in <note>option A</note> undergoes <note>horizontal right translation and downward translation by one unit</note>, resulting in a change in position without altering the orientation of the figure. The translation distance is determined linearly by the change in grid units, thus the figure's <note>new position corresponds to the original position</note>.

Rotation Transformation

The figure in <note>Option B</note> rotates 180° around <note>a certain point</note> relative to <note>the original figure</note>, causing the direction of the figure to change <note>counterclockwise</note>. The transformation of the figure in the original position is confirmed based on <note>the rotation angle</note>.

Reflection Transformation

A reflection transformation is a type of geometric transformation that flips a figure over a specific line known as the line of reflection. After the reflection transformation, the coordinates of each point on the figure are changed to the coordinates of its corresponding point symmetrically across the line of reflection.

<note>Reflection along the vertical axis in option C</note>, the reflection result of the original figure causes positional changes on both sides of the mirror symmetry axis, at this time <note>the direction of the figure also undergoes corresponding transformation</note>.


Question:
What is m $\angle $ IGJ?

m $\angle $ IGJ= $\Box$ °

Answer: m \$\angle \$ IGJ=55°

1. In the circle, there are four angles: ∠IGJ, ∠JGK, ∠HGK, ∠HGI. These four angles are all the angles around point G, so their sum is equal to 360°.

2. Based on the sum of the central angles of the circle being 360°, we can write the equation: ∠IGJ + ∠JGK + ∠HGK + ∠HGI = 360°.

3. In the figure, it is given that ∠JGK = 65°, ∠HGK = 90°, ∠HGI = 150°. Substitute these angles into the equation: ∠IGJ + 65° + 90° + 150° = 360°.

4. Calculate the sum: 65° + 90° + 150° = 305°.

5. Substitute 305° into the equation to get: ∠IGJ + 305° = 360°.

6. Subtract 305° from both sides, the equation simplifies to: ∠IGJ = 360° - 305°.

7. Calculate to get: ∠IGJ = 55°. Through the above reasoning, the final answer is 55°.

Definition of Central Angle

An angle formed by two radii connecting two points on the circumference of a circle to the center of the circle is called a central angle.

<note>Point I and point J are two points on the circle</note>, <note>the center of the circle is point G</note>. <note>The angle ∠IGJ formed by the lines GI and GJ is called the central angle</note>. Similarly, <note>∠JGK</note>, <note>∠HGK</note>, <note>∠HGI are also central angles</note>, because they are formed by <note>the radii JG, GK, HG and the center G of the circle</note>.

Angle Properties of a Circle

The original text: <note>Arc JI, Arc JK, Arc KH, Arc HI around the circle</note>, the sum of the radians is <note>360 degrees</note>, <note>the sum of their corresponding angles at the circumference is also 360 degrees</note>, that is, <note>angle JGI + angle JGK + angle KGH + angle HGI = 360 degrees</note>.


Question:
Look at this shape:Which image shows a translation?

| A | B | C |
- A
- B
- C

Answer: B

1. Observe the given figure and the three option figures, the goal is to find an option that matches the source figure after a translation transformation.

2. First, look at option A, analyze its changes relative to the source figure: the shape has undergone a rotation, judged as a rotation transformation.

3. Look at option B, observe that the shape has not undergone rotation or reflection relative to the source figure, only moved down relative to the original figure, consistent with the description of translation, so it is a translation.

4. Look at option C, notice that the shape has undergone a mirror reflection relative to the original figure, judged as a reflection transformation.

5. Based on the above analysis, option B is the only option among the three that meets the definition of translation.

6. After the above reasoning, the final answer is option B.

Definition of Translation

In the figure of this problem, the original figure and the figure in option B undergo <note>translation transformation</note> without changing shape and size, only the position changes. It indicates that a certain <note>point P in the original figure moves to the position of point P'</note> after translation. The <note>positions of point P and P' are equidistant and in the same direction</note>, for example, the <note>vertex A of the original triangle moves to the position of A'</note> after translation, and <note>other vertices also translate the same distance and direction accordingly</note>.

Rotation Transformation

The <note>original figure</note> undergoes a <note>rotation transformation</note>, rotating from its original position by a specific angle, with <note>the shape and size of the figure remaining unchanged during the rotation</note>. For example, <note>the original triangle vertices A, B, C rotate to new positions A', B', C'</note>, maintaining <note>the shape and size of the original triangle</note>.

Reflection Transformation

<note>The original figure undergoes a mirror reflection transformation</note> relative to a certain <note>mirror line</note>. For example, <note>the original triangle vertices A, B, and C reflect to new positions A', B', and C'</note>, maintaining the original triangle's shape and size.


Question:
What is the measure of $\overset{\frown}{UX}$ ?

m $\overset{\frown}{UX}$ = $\Box$ °

Answer: m \$\overset{\frown}{UX}\$ =79°

1. According to the given diagram, there is a quadrilateral UWVX inside the circle, and it can be observed that the lengths of chords UX and WV are equal.

2. Based on the information given in the diagram, the length of arc VW is 79°, and the chord WV has the property of being equal to the arc.

3. According to the central angle theorem (if two chords in a circle are equal, then the arcs they subtend are equal), we can deduce that the length of arc UX is equal to the length of arc VW.

4. Therefore, according to the central angle theorem, the length of arc UX is equal to 79°.

5. After the above reasoning, the final answer is 79°.

Definition of Chord

A chord is defined as a line segment that connects any two points on a circle.

In the figure of this problem, in the circle, <note>point U and point X are any two points on the circle</note>, <note>line segment UX connects these two points</note>, so <note>line segment UX is a chord of the circle</note>. Similarly, <note>point W and point V are any two points on the circle</note>, <note>line segment WV connects these two points</note>, so <note>line segment WV is a chord of the circle</note>.

Definition of Inscribed Angle

An inscribed angle is an angle whose vertex lies on the circumference of a circle and whose sides intersect the circle at two points.

In the figure of this problem, in circle O, <note>the vertex V of angle UVW is on the circumference</note>, <note>the two sides of angle UVW intersect circle O at points U and W respectively</note>; <note>the vertex V of angle WVX is on the circumference</note>, <note>the two sides of angle WVX intersect circle O at points W and X respectively</note>; <note>the vertex X of angle VXU is on the circumference</note>, <note>the two sides of angle VXU intersect circle O at points V and U respectively</note>; <note>the vertex U of angle XUW is on the circumference</note>, <note>the two sides of angle XUW intersect circle O at points X and W respectively</note>. Therefore, angles UVW, WVX, VXU, and XUW are all inscribed angles.

Definition of Arc

An arc is a curve on a circle that is defined by two distinct points on the circle.

<note>There are two points V and W on a circle, and arc VW is a segment of the curve connecting these two points</note>; similarly, <note>there are two points U and X on a circle, and arc UX is a segment of the curve connecting these two points</note>. According to the definition of an arc, <note>arc VW is a segment of the curve between two points V and W on a circle</note>, <note>arc UX is a segment of the curve between two points U and X on a circle</note>.

Central Angle Theorem

In the same circle or in congruent circles, arcs corresponding to equal central angles are equal, chords corresponding to equal central angles are equal, and the perpendicular distances from the center to these chords are also equal.

<note>The chords VW</note> and <note>UX in the same circle are equal</note>, then the corresponding arcs <note>arc VW and UX are equal</note>, that is, <note>arc UX = arc VW = 79°</note>.


Question:
The radius of a circle is 5 centimeters. What is the area of a sector bounded by a 180° arc?

Give the exact answer in simplest form.
 $\Box$ 𝜋/ square centimeters

Answer: 25/2𝜋 square centimeters

1. Given the radius of the circle is 5 cm, first calculate the area of the entire circle using the formula A = πr^2.

2. Substitute the radius r = 5 to get A = π * 5^2 = 25π square cm.

3. Given the sector's angle is 180°, which is a semicircle. According to the sector area formula, the sector area K = A * (m/360), where m is the angle and A is the area of the circle.

4. Substitute the circle's area A = 25π and the angle m = 180 to calculate K = 25π * (180/360) square cm.

5. Simplify 180/360 to 1/2, thus K = 25π * 1/2 = 25π/2 square cm.

6. Through the above reasoning, the final answer is 25π/2 square cm.

Area Formula of a Circle

The area of a circle is given by the formula A = πr², where A represents the area, π (pi) is the mathematical constant approximately equal to 3.14159, and r is the radius of the circle.

<note>The radius of the circle is 5 cm</note>. According to the area formula of a circle, the area A of the circle is equal to pi π multiplied by the radius <note>5 squared</note>, that is, <note>A = π5²</note>.

Definition of Sector

A sector is a figure formed by two radii of a circle and the arc enclosed between them.

<note>The sector is composed of a radius r=5 cm, a radius r=5 cm, and the 180° arc between them</note>. According to the definition of a sector, the figure composed of <note>these two radii and the arc between them</note> is a sector.

Formula for the Area of a Sector

The area \( X \) of a sector can be calculated using the formula \( X = \frac{\theta}{360} \times \pi \times r^2 \), where \( \theta \) is the measure of the central angle in degrees, and \( r \) is the radius length.

In the problem figure, <note>the area of the circle A is 25π square centimeters</note>, <note>the radian m is 180°</note>, therefore <note>the area of the sector K is calculated as K = 25π * (180/360) = 25π/2 square centimeters</note>.


Question:
$\overline{BC}$ is shown on the graph below. $\overline{BC}$ is dilated by a scale factor of 2/3 centered at point C to create $\overline{B'C'}$ .

What is the length of $\overline{B'C'}$ ?
Write your answer as a whole number or as a decimal rounded to the nearest tenth.

 $\Box$  units

Answer: 10.2 units

1. Given the coordinates of point B are (-6, -8), and the coordinates of point C are (9, -5), according to the distance formula between two points d=√((x₂-x₁)²+(y₂-y₁)²), where (x₁, y₁) are the coordinates of point B, and (x₂, y₂) are the coordinates of point C.

2. Substitute the coordinates of points B and C into the distance formula, we get BC=√((9 - (-6))² + ((-5) - (-8))²)=√((9 + 6)² + ((-5) + 8)²)=√(15² + 3²).

3. Further calculation gives BC=√(225 + 9)=√234.

4. The problem states that segment BC is scaled by a factor of 2/3 with point C as the center to obtain segment B'C'. According to the definition of the scaling factor in the Cartesian coordinate system, the length of the scaled segment is the product of the original length and the scaling factor, i.e., B'C'=(2/3)·BC.

5. Use √234 to find the approximate value of BC and then scale to get B'C'=2/3·√234.

6. Calculate √234 to be approximately 15.297, then scale to get B'C'=2/3·15.297≈10.198.

7. Round the result to one decimal place, and finally get the length of B'C' to be 10.2 units.

8. After the above reasoning, the final answer is 10.2.

Distance Formula Between Two Points

In a Cartesian coordinate plane, the distance \( d \) between two points \((x_1, y_1)\) and \((x_2, y_2)\) can be calculated using the formula \( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \).

<note>Point B's coordinates are (-6, -8)</note>, <note>Point C's coordinates are (9, -5)</note>. We apply the <note>Distance Formula Between Two Points</note> to calculate the <note>length of segment BC</note>, which is <note>BC = √((9 - (-6))² + ((-5) - (-8))²)</note>.

In a rectangular coordinate system, if a polygon xxx is transformed to xxx by scaling, then the scale factor is defined as the ratio of the length of any side of the scaled polygon to the length of the corresponding side of the original polygon.

<note>In the figure of this problem, the polygon ABC is scaled to obtain A'B'C'</note>, <note>Scale factor = length of B'C' / length of BC</note>, that is, <note>Scale factor = 2/3</note>.


Question:
$\overline{CD}$ is shown on the graph below. $\overline{CD}$ is dilated by a scale factor of 1/3 centered at (3,–5) to create $\overline{C'D'}$ .

What is the length of $\overline{C'D'}$ ?
Write your answer as a whole number or as a decimal rounded to the nearest tenth.

 $\Box$  units

Answer: 4.1 units

1. First, determine the coordinates of the two endpoints C and D of the line segment, which are C(-4,3) and D(8,1) respectively.

2. Use the distance formula between two points to calculate the length of the segment CD: CD = √{(x2 - x1)^2 + (y2 - y1)^2}.

3. Substitute the coordinates of C and D into the distance formula: CD = √{(8 - (-4))^2 + (1 - 3)^2} = √{(8 + 4)^2 + (-2)^2} = √{12^2 + 2^2}.

4. Calculate to get CD = √{144 + 4} = √{148}.

5. Since CD is scaled by a factor of 1/3 from the point (3,-5) to get C'D', the length of C'D' is 1/3 of CD.

6. Calculate to get C'D' = 1/3 * √{148}.

7. Process √{148} according to common square root values to get √{148} ≈ 12.1655.

9. Round the result to the nearest tenth to get C'D' ≈ 4.1.

10. After the above reasoning, the final answer is 4.1 units.

<note>Point C's coordinates are (-4, 3)</note>, <note>Point D's coordinates are (8, 1)</note>. Using the distance formula to calculate <note>CD= √{(8 - (-4))^2 + (1 - 3)^2} = √{(8 + 4)^2 + (-2)^2} = √{12^2 + 2^2} = √{148}</note>.

<note>CD scaled to obtain C'D'</note>, <note>Scale factor = length of C'D' / length of CD</note>, scale factor = 1/3.


Question:
What is m $\angle $ I?

m $\angle $ I= $\Box$ °

Answer: m \$\angle \$ I=99°

1. In the figure, quadrilateral FGHI is a cyclic quadrilateral, where points F, G, H, and I lie on the same circle.

2. According to the properties of a cyclic quadrilateral, in a cyclic quadrilateral, the opposite angles are supplementary, meaning their angle sum is 180°.

3. Specifically, ∠G and ∠I are supplementary angles. This can be expressed as m∠G + m∠I = 180°.

4. It is given in the problem that m∠G = 81°.

5. Substituting m∠G = 81° into the equation in step (3), we get 81° + m∠I = 180°.

6. By subtracting 81° from both sides, we can find m∠I = 180° - 81°.

7. Calculating the above equation: 180° - 81° = 99°.

8. Therefore, through the above reasoning, the final answer is m∠I = 99°.

Cyclic Quadrilateral

A quadrilateral is called a cyclic quadrilateral if and only if all four vertices lie on the circumference of a single circle. This circle is referred to as the circumcircle of the quadrilateral.

In the diagram of this problem, the quadrilateral <note>FGHI's four vertices F, G, H, and I</note> are all on the same circle. This circle is called the <note>circumcircle of quadrilateral FGHI</note>. Therefore, <note>quadrilateral FGHI is a cyclic quadrilateral</note>. According to the properties of a cyclic quadrilateral, it can be concluded that <note>the sum of opposite angles is equal to 180 degrees</note>, i.e., <note>angle FGH + angle FIH = 180 degrees</note>, <note>angle IFG + angle IHG = 180 degrees</note>.

Corollary 3 of the Inscribed Angle Theorem: Diagonal Supplementary Theorem for Cyclic Quadrilateral

In a cyclic quadrilateral (a quadrilateral inscribed in a circle), the sum of each pair of opposite angles is equal to 180°. Additionally, any exterior angle is equal to its interior opposite angle.

In the <note>cyclic quadrilateral FGHI</note>, the vertices of the quadrilateral <note>F, G, H, and I are on the circle</note>. According to the <note>Diagonal Supplementary Theorem for Cyclic Quadrilateral</note>, the sum of each pair of opposite angles in quadrilateral FGHI equals <note>180°</note>. Specifically, <note>∠FGH + ∠FIH = 180°</note>; <note>∠IFG + ∠IHG = 180°</note>.


Question:
$\overline{AB}$ is shown on the graph below. $\overline{AB}$ is dilated by a scale factor of 1/2 centered at (9,–1) to create $\overline{A'B'}$ .

What is the length of $\overline{A'B'}$ ?
Write your answer as a whole number or as a decimal rounded to the nearest tenth.

 $\Box$  units

Answer: 6.4 units

1. Given the coordinates of point A as (-8,-5) and the coordinates of point B as (2,3), to calculate the length of segment AB, we use the distance formula between two points, which is d = √((x2-x1)^2 + (y2-y1)^2).

2. Substituting the coordinates of A and B, d = √((2 - (-8))^2 + (3 - (-5))^2) = √((2 + 8)^2 + (3 + 5)^2) = √((10)^2 + (8)^2).

3. Calculating the square of 10 as 100 and the square of 8 as 64, therefore d = √(100 + 64) = √164.

4. Calculating the value of √164 as 12.8062 (rounded to four decimal places).

5. Given that after scaling through the center (9, -1), segment AB is scaled by a factor of 1/2. According to the definition of the scaling factor in the Cartesian coordinate system, the length of segment A'B' is 1/2 times AB, i.e., A'B' = 1/2 * 12.8062.

6. Calculating A'B' = 6.4031, rounded to one decimal place to get 6.4.

7. Through the above reasoning, the final answer is 6.4 units.

Distance Formula Between Two Points

<note>The coordinates of point A are (-8, -5)</note> and <note>the coordinates of point B are (2, 3)</note>, we need to calculate <note>the length of segment AB</note>. Therefore, d = √((<note>2 - (-8)</note>)^2 + (<note>3 - (-5)</note>)^2), substitute into the formula and calculate to obtain the length of AB.

After scaling <note>polygon ABC</note>, we obtain <note>polygon A'B'C'</note>. The scale factor = <note>length of A'B'</note> / <note>length of AB</note>, i.e., the scale factor = 1/2.


Question:
Write the coordinates of the vertices after a dilation with a scale factor of 3, centered at the origin.

J'( $\Box$ , $\Box$ )

K'( $\Box$ , $\Box$ )

L'( $\Box$ , $\Box$ )

M'( $\Box$ , $\Box$ )

Answer:
J'(-9,-6)
K'(-3,-6)
L'(-3,6)
M'(-9,6)

1. It is known that the transformation is a scaling transformation around the origin, with the scaling center at the origin and the scaling factor being 3.

2. According to the scaling theorem in the Cartesian coordinate system, any point (x, y) will become (x', y') after the scaling transformation, where x' = kx, y' = ky, and k is the scaling factor.

3. Suppose the original coordinates of point J are J(-3, -2). According to the scaling transformation formula, x' = 3 * (-3) = -9, y' = 3 * (-2) = -6. Therefore, the coordinates of J' are (-9, -6).

4. Suppose the original coordinates of point K are K(-1, -2). According to the scaling transformation formula, x' = 3 * (-1) = -3, y' = 3 * (-2) = -6. Therefore, the coordinates of K' are (-3, -6).

5. Suppose the original coordinates of point L are L(-1, 2). According to the scaling transformation formula, x' = 3 * (-1) = -3, y' = 3 * (2) = 6. Therefore, the coordinates of L' are (-3, 6).

6. Suppose the original coordinates of point M are M(-3, 2). According to the scaling transformation formula, x' = 3 * (-3) = -9, y' = 3 * (2) = 6. Therefore, the coordinates of M' are (-9, 6).

7. Through the above reasoning, the final coordinates of the four vertices after scaling are J'(-9, -6), K'(-3, -6), L'(-3, 6), M'(-9, 6).

Scaling Theorem in a Rectangular Coordinate System

In a rectangular coordinate system, the coordinates of a vertex \( x \) \((x, x)\), after being scaled by a factor \( k \), will have the new coordinates \( x' \) \((kx, kx)\).

In the figure of this problem, <note>point J(-3, -2)</note>, <note>point K(-1, -2)</note>, <note>point L(-1, 2)</note>, <note>point M(-3, 2)</note>, after scaling by <note>a factor of 3</note>, <note>the coordinates of point x' are J'(-9,-6), K'(-3,-6), L'(-3,6), M'(-9,6)</note>.


Question:
Write the coordinates of the vertices after a dilation with a scale factor of 2, centered at the origin.

F'( $\Box$ , $\Box$ )

G'( $\Box$ , $\Box$ )

H'( $\Box$ , $\Box$ )

Answer:
F'(-8,10)
G'(8,10)
H'(-10,-8)

1. Given the coordinates of the vertices in figure FGH: F(-4, 5), G(4, 5), H(-5, -4), we need to scale the vertices with the origin (0, 0) as the center and a scale factor of 2.

2. According to the scaling theorem in the Cartesian coordinate system: For any point (x, y) on a figure, after a dilation transformation with the origin as the center and a scale factor of k, its new coordinates are (kx, ky).

3. For point F(-4, 5), its new coordinates are F': Multiply each coordinate component by the scale factor 2, yielding F'(-4 * 2, 5 * 2) = F'(-8, 10).

4. For point G(4, 5), its new coordinates are G': Multiply each coordinate component by the scale factor 2, yielding G'(4 * 2, 5 * 2) = G'(8, 10).

5. For point H(-5, -4), its new coordinates are H': Multiply each coordinate component by the scale factor 2, yielding H'(-5 * 2, -4 * 2) = H'(-10, -8).

6. Through the above reasoning, the coordinates after scaling for each point are F'(-8, 10), G'(8, 10), H'(-10, -8). The triangle formed by these points is similar to the original triangle FGH, with a ratio of 2:1.

7. Upon confirmation, the final answer is F'(-8, 10), G'(8, 10), H'(-10, -8).

In the diagram of this problem, <note>point F(-4, 5)</note>, <note>point G(4, 5)</note>, <note>point H(-5, -4)</note>, after scaling by <note>factor of 2</note>, the coordinates of points <note>F'</note>, <note>G'</note>, <note>H'</note> are <note>F'(-4*2, 5*2)</note>, <note>G'(4*2, 5*2)</note>, <note>H'(-5*2, -4*2)</note>.