Question:
Write the coordinates of the vertices after a dilation with a scale factor of 1/2, centered at the origin.

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

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

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

N'( $\Box$ , $\Box$ )

Answer:
K'(2,-5)
L'(2,2)
M'(4,2)
N'(4,-5)

1. According to the problem statement, rectangle KLMN undergoes a scaling transformation with the origin as the center and a scaling factor of 1/2 (i.e., a similarity transformation).

2. The scaling theorem in the Cartesian coordinate system states that in a transformation, if we scale around a certain point, the coordinates of any point on a shape can be obtained by multiplying each coordinate value by the scaling factor to get the new coordinates.

3. First, consider point K with coordinates K(4, -10). According to the similarity transformation theorem, the coordinates of K' are: (4 * 1/2, -10 * 1/2) = (2, -5).

4. Next, consider point L with coordinates L(4, 4). According to the similarity transformation theorem, the coordinates of L' are: (4 * 1/2, 4 * 1/2) = (2, 2).

5. Now, consider point M with coordinates M(8, 4). According to the similarity transformation theorem, the coordinates of M' are: (8 * 1/2, 4 * 1/2) = (4, 2).

6. Finally, consider point N with coordinates N(8, -10). According to the similarity transformation theorem, the coordinates of N' are: (8 * 1/2, -10 * 1/2) = (4, -5).

7. After the above calculations, the coordinates of the vertices of rectangle KLMN after scaling are K'(2, -5), L'(2, 2), M'(4, 2), N'(4, -5).

<note>Point K(4, -10), L(4, 4), M(8, 4), N(8, -10)</note>, after scaling by a factor of 1/2, the coordinates of <note>point x</note> are <note>K'(2, -5), L'(2, 2), M'(4, 2), N'(4, -5)</note>.


Question:
$\overline{FG}$ is shown on the graph below. $\overline{FG}$ is dilated by a scale factor of 5 centered at the origin to create $\overline{F'G'}$ .

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

 $\Box$  units

Answer: 50 units

1. Given the coordinates of point F are (-8, 9) and the coordinates of point G are (-2, 1).

2. Use the distance formula between two points to calculate the length of segment \overline{FG}, which is \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}, where (x_1, y_1) are the coordinates of point F and (x_2, y_2) are the coordinates of point G.

3. For this problem, substitute the coordinates to get: \overline{FG} = \sqrt{((-2) - (-8))^2 + (1 - 9)^2}.

4. Further calculation: \overline{FG} = \sqrt{6^2 + (-8)^2} = \sqrt{36 + 64} = \sqrt{100}.

5. Thus, the length of \overline{FG} is 10.

6. According to the problem, segment \overline{FG} is scaled by a factor of 5 with the origin as the center to obtain \overline{F'G'}.

7. By the definition of scaling factor in the Cartesian coordinate system, the length of \overline{F'G'} is the length of the original segment \overline{FG} multiplied by the scaling factor.

8. Therefore, the length of \overline{F'G'} is 10 \times 5 = 50.

9. Through the above reasoning, the final answer is 50 units in length.

Coordinates of a Point

In a Cartesian coordinate system, the position of a point is represented by an ordered pair (x, y), where x denotes the horizontal (x-coordinate) and y denotes the vertical (y-coordinate).

In the figure of this problem, <note>point F in the plane rectangular coordinate system is represented by the ordered pair (-8, 9)</note>, where -8 represents the x-coordinate of point F and 9 represents the y-coordinate of point F. The x-coordinate -8 indicates the horizontal position of point F, and the y-coordinate 9 indicates the vertical position of point F. Through this <note>ordered pair (-8, 9)</note>, we can determine <note>the specific position of point F in the coordinate system</note>. <note>Point G in the plane rectangular coordinate system is represented by the ordered pair (-2, 1)</note>, where -2 represents the x-coordinate of point G and 1 represents the y-coordinate of point G. The x-coordinate -2 indicates the horizontal position of point G, and the y-coordinate 1 indicates the vertical position of point G. Through this <note>ordered pair (-2, 1)</note>, we can determine <note>the specific position of point G in the coordinate system</note>.

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 \( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \).

In the figure of this problem, <note>the coordinates of point F are (-8, 9)</note>, <note>the coordinates of point G are (-2, 1)</note>, we need to calculate <note>the distance between point F and point G</note>. According to the distance formula between two points, <note>the distance d between point F and point G</note> can be calculated using the following formula: d = √(((-2) - (-8))^2 + (1 - 9)^2). In the figure, <note>the distance between point F and point G is the length of segment FG</note>.

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.

Original text: <note>Line segment FG is scaled to obtain line segment F'G'</note>, scale factor = <note>length of F'G' / length of FG</note>, that is, scale factor = 5.


Question:
Write the coordinates of the vertices after a dilation with a scale factor of 4, centered at the origin.

Q'( $\Box$ , $\Box$ )

R'( $\Box$ , $\Box$ )

S'( $\Box$ , $\Box$ )

Answer:
Q'(-8,-8)
R'(-8,8)
S'(8,-8)

1. Given points Q(-2, -2), R(-2, 2), and S(2, -2). These points need to be scaled with the origin (0, 0) as the center, and the scaling factor is 4.

2. According to the scaling theorem in the Cartesian coordinate system, when the center is at the origin and the scaling factor is k, the point (x, y) after scaling becomes (kx, ky). In this problem, k=4.

3. Calculate the new coordinates of point Q': multiply the coordinates of Q(-2, -2) by the scaling factor 4, resulting in Q'(-8, -8).

4. Calculate the new coordinates of point R': multiply the coordinates of R(-2, 2) by the scaling factor 4, resulting in R'(-8, 8).

5. Calculate the new coordinates of point S': multiply the coordinates of S(2, -2) by the scaling factor 4, resulting in S'(8, -8).

6. Since the transformation center is at the origin and the scaling factor is greater than 0, the transformed figure is similar to the original figure, and the positions are scaled proportionally. Therefore, the transformed points Q', R', and S' still form the same type of shape.

7. Based on the above reasoning, the final answer is Q'(-8, -8), R'(-8, 8), S'(8, -8).

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 \( x' \) \((kx, kx)\).

Original: <note>Point Q(-2, -2)</note>, <note>Point R(-2, 2)</note>, <note>Point S(2, -2)</note> scaled by <note>factor of 4</note> results in <note>coordinates of Q'R'S'</note> being <note>Q'(-8, -8)</note>, <note>R'(-8, 8)</note>, <note>S'(8, -8)</note>.


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

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

Answer: B

1. The given figure is a trapezoid located above in the Cartesian coordinate plane.

2. First, observe which option shows rotation, where the rotation operation can keep the shape unchanged while changing its position.

3. Image A shows a horizontal flip, where the shape is mirrored in the horizontal direction, not involving rotation.

4. Image B shows a 180° rotation. The center of rotation is outside the figure, and the figure appears in a position symmetric to the center, which matches the characteristics of rotation.

5. Image C shows a translation, where the shape only changes position without involving rotation or reflection.

6. Based on the comparison, Image B achieves a 180° rotation, meeting the problem's requirements.

7. Through the above reasoning, the final answer is Image B.

Definition of Trapezoid

A quadrilateral is a trapezoid if and only if it has exactly one pair of parallel sides.

<note>The upper and lower sides of the quadrilateral are parallel</note>, while the left and right sides are not parallel. Therefore, according to the definition of a trapezoid, the quadrilateral in the figure is a trapezoid because <note>it has exactly one pair of parallel sides</note>.

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>Image B</note> shows <note>trapezoid ABCD</note> rotating <note>180°</note>, its shape remains unchanged but its position changes. After rotation, the vertices of trapezoid ABCD move to corresponding positions, with the new vertex coordinates being <note>(x1', y1')</note>, <note>(x2', y2')</note>, <note>(x3', y3')</note>, <note>(x4', y4')</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>Image A</note> shows the trapezoid <note>ABCD</note> reflected about a certain <note>horizontal line</note>, with the shape <note>symmetrically flipped horizontally</note>, and the new coordinates of its vertices become <note>(x1'', y1''), (x2'', y2''), (x3'', y3''), (x4'', y4'')</note>.

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.

<note>Image C</note> shows the <note>translation of trapezoid ABCD</note>, where the trapezoid moves in a certain direction while maintaining its shape, and the new coordinates of its vertices become <note>(x1 + deltaX, y1 + deltaY)</note>, <note>(x2 + deltaX, y2 + deltaY)</note>, <note>(x3 + deltaX, y3 + deltaY)</note>, <note>(x4 + deltaX, y4 + deltaY)</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.

In the figure of this problem, image B achieves <note>a 180° rotation of trapezoid ABCD</note>. According to the rotation theorem, <note>its shape and size remain unchanged</note>. After rotation, <note>the positions of the vertices of the trapezoid (A', B', C', D') change relative to the original vertices (A, B, C, D), and they are in mirror-symmetric positions</note>.

Translation Invariance Theorem

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

<note>Image C shows the translation of trapezoid ABCD</note>. According to the Translation Invariance Theorem, <note>the translated trapezoid A'B'C'D' remains the same in shape and size as the original trapezoid ABCD</note>, only <note>the position changes, with its vertices translated to new coordinates</note>.


Question:
Write the coordinates of the vertices after a dilation with a scale factor of 1/2, centered at the origin.

A'( $\Box$ , $\Box$ )

B'( $\Box$ , $\Box$ )

C'( $\Box$ , $\Box$ )

Answer:
A'(-2,-3)
B'(0,-3)
C'(0,-5)

1. Given a scaling transformation about the origin, with a scaling factor of 1/2. For this geometric transformation, it is centered at the origin (0, 0).

2. For point A(-4, -6), apply the scaling factor of 1/2. Multiply the x-coordinate and y-coordinate of A by 1/2 respectively to get the coordinates of A'. The calculation is as follows: x = -4 * 1/2 = -2, y = -6 * 1/2 = -3. Therefore, A'(-2, -3).

3. For point B(0, -6), apply the scaling factor of 1/2. Multiply the x-coordinate and y-coordinate of B by 1/2 respectively to get the coordinates of B'. The calculation is as follows: x = 0 * 1/2 = 0, y = -6 * 1/2 = -3. Therefore, B'(0, -3).

4. For point C(0, -10), apply the scaling factor of 1/2. Multiply the x-coordinate and y-coordinate of C by 1/2 respectively to get the coordinates of C'. The calculation is as follows: x = 0 * 1/2 = 0, y = -10 * 1/2 = -5. Therefore, C'(0, -5).

5. After the above scaling transformation, the vertices of the original triangle ABC form a new triangle A'B'C' after the scaling transformation, which is similar to the original triangle ABC.

6. Finally, the coordinates of the vertices after scaling are: A'(-2, -3), B'(0, -3), C'(0, -5).

<note>In the figure of this problem, point A(-4, -6), point B(0, -6), point C(0, -10)</note> scaled by <note>a factor of 1/2</note> results in <note>the coordinates of points A'B'C' being A'(-2, -3), B'(0, -3), C'(0, -5)</note>.


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

P'( $\Box$ , $\Box$ )

Q'( $\Box$ , $\Box$ )

R'( $\Box$ , $\Box$ )

Answer:
P'(-6,-6)
Q'(4,-6)
R'(-4,-4)

1. Given that a scaling operation centered at the origin is required, and the scaling factor is 2. According to the definition of the scaling factor in the Cartesian coordinate system, each vertex's coordinate point needs to be multiplied by the scaling factor.

2. The initial coordinates of point P are (-3, -3). According to the scaling theorem in the Cartesian coordinate system, with the origin (0,0) as the center and a scaling factor of 2, each coordinate of point P is multiplied by 2, resulting in the coordinates of P' being (-3 * 2, -3 * 2)=(-6, -6).

3. The initial coordinates of point Q are (2, -3). Similarly, according to the scaling theorem in the Cartesian coordinate system, with a scaling factor of 2, each coordinate of point Q is multiplied by 2, resulting in the coordinates of Q' being (2 * 2, -3 * 2)=(4, -6).

4. The initial coordinates of point R are (-2, -2). According to the scaling theorem in the Cartesian coordinate system, with a scaling factor of 2, each coordinate of point R is multiplied by 2, resulting in the coordinates of R' being (-2 * 2, -2 * 2)=(-4, -4).

5. After the above scaling transformation, the coordinates of points P, Q, and R become P'(-6, -6), Q'(4, -6), and R'(-4, -4) respectively. The figure maintains a similar shape, with a size twice the original.

6. Therefore, the transformed figure's vertex coordinates are: P'(-6, -6), Q'(4, -6), R'(-4, -4).

Coordinates of a Point

In the figure of this problem, <note>the original coordinate points are P(-3, -3), Q(2, -3), and R(-2, -2)</note>. After the scaling transformation, <note>the new coordinate points are P'(-6, -6), Q'(4, -6), and R'(-4, -4)</note>.

<note>Point P(-3, -3)</note>, <note>Q(2, -3)</note>, <note>R(-2, -2)</note> scaled by <note>a factor of 2</note> results in <note>point P' coordinates as P'(-6, -6)</note>, <note>Q'(4, -6)</note>, <note>R'(-4, -4)</note>.

Original: <note>Polygon RPQ is scaled to obtain polygon R'P'Q'</note>, scale factor = <note>length of R'P' / length of RP</note>, i.e., <note>scale factor = 2</note>.


Question:
$\overline{ST}$ is shown on the graph below. $\overline{ST}$ is dilated by a scale factor of 1.25 centered at the origin to create $\overline{S'T'}$ .

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

 $\Box$  units

Answer: 5.6 units

1. Given the coordinates of the endpoints of the line segment \\overline{ST} are \(S(2,3)\) and \(T(6,1)\), according to the distance formula between two points: \(d = \\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\), we can calculate the length of \\overline{ST}.

2. Substituting the coordinates of the endpoints of the line segment \\overline{ST} \(S(2, 3)\) and \(T(6, 1)\) into the distance formula, we get: \(d = \\sqrt{(6 - 2)^2 + (1 - 3)^2} = \\sqrt{4^2 + (-2)^2} = \\sqrt{16 + 4} = \\sqrt{20}\).

3. Calculating \(\\sqrt{20} = \\sqrt{4 \\times 5} = 2\\sqrt{5}\), according to approximate calculation, \(\\sqrt{5}\) is approximately \(2.236\), and thus \(2\\sqrt{5}\) is approximately \(4.472\).

4. Since \\overline{ST} is scaled from the origin by a scale factor of \(1.25\) to form the new line segment \\overline{S'T'}\), according to the definition of scale factor in the Cartesian coordinate system, we can multiply the length of \\overline{ST} by the scale factor to obtain the length of \\overline{S'T'}.

5. Calculating \\overline{S'T'} = 1.25 \\times 4.472 = 5.59\), rounding to the nearest tenth gives \(5.6\).

6. Through the above reasoning, the final answer is \(5.6\) units.

Distance Formula Between Two Points

In the figure of this problem, the coordinates of point <note>\(S\) are \(S(2,3)\)</note>, the coordinates of point <note>\(T\) are \(T(6,1)\)</note>, according to the distance formula between two points, the length of segment <note>\(\overline{ST}\) is \(d = \sqrt{(6 - 2)^2 + (1 - 3)^2} = \sqrt{4^2 + (-2)^2} = \sqrt{16 + 4} = \sqrt{20}\)</note>.

In the problem diagram, the polygon ST is scaled to obtain the polygon S'T', scaling factor = <note>length of S'T'/length of ST</note>, i.e., scaling factor = 1.25.


Question:
The radius of a circle is 2 meters. What is the area of a sector bounded by a 180° arc?

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

Answer: 2𝜋 square meters

1. Given that the radius of a circle is 2 meters, find the area of a sector defined by a 180° arc.

2. According to the formula for the area of a circle, the area of the circle A = πr^2, where r is the radius. In this problem, r = 2, so A = π * (2)^2.

3. Calculate the area of the circle to get A = 4π square meters.

4. According to the formula for the area of a sector, K = A * (m/360), where A is the area of the circle, and m is the central angle of the sector. In this problem, m = 180°.

5. Substitute the given conditions, K = 4π * (180/360).

6. Simplify the ratio 180/360 to get 1/2, so K = 4π * 1/2.

7. Further simplify to get K = 2π square meters.

8. Through the above reasoning, the final answer is 2π square meters.

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, <note>the radius of the circle r = 2 meters</note>, <note>point O is the center of the circle</note>, <note>point A is any point on the circumference</note>, <note>segment OA is the segment from the center to any point on the circumference</note>, therefore <note>segment OA is the radius of the circle</note>.

Definition of Sector

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

<note>A sector is a figure formed by two radii of a circle and the 180° arc between them</note>. The radius is <note>r=2 meters</note>, and the arc is <note>180°</note>, as shown in <note>the orange part of the figure</note>.

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.

The original text: <note>The center of the circle is point O</note>, two points on the circle <note>A and B are the endpoints of two radii</note>. The angle formed by the lines <note>OA and OB is called the central angle ∠AOB</note>, and its measure is 180°.

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.

In the diagram of this problem, <note>the radius of the circle is 2 meters</note>. According to the area formula of a circle, the area A of the circle is equal to the circumference π multiplied by <note>the square of the radius 2</note>, which is A = π * 2² = <note>4π square meters</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 the figure of this problem, <note>the central angle of the sector m = 180°</note>, <note>the area of the circle A = 4π square meters</note>, therefore <note>the area of the sector K = 4π * (180/360) = 2π square meters</note>.


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

D'( $\Box$ , $\Box$ )

E'( $\Box$ , $\Box$ )

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

Answer:
D'(10,-4)
E'(10,4)
F'(-8,6)

1. Given the coordinates of the vertices of triangle DEF as D(5,-2), E(5,2), and F(-4,3), determine the coordinates of the vertices of the scaled triangle DEF after a scaling transformation with a scale factor of 2 centered at the origin.

2. According to the scaling theorem in the Cartesian coordinate system: if a point P(x, y) is scaled by a factor of k centered at the origin, the new coordinates P'(x', y') are given by x' = kx, y' = ky.

3. Apply the scaling transformation to point D(5,-2):

- Therefore, the scaled coordinates are D'(10,-4).

4. Apply the scaling transformation to point E(5,2):

- Therefore, the scaled coordinates are E'(10,4).

5. Apply the scaling transformation to point F(-4,3):

- Therefore, the scaled coordinates are F'(-8,6).

6. By applying the same scaling transformation to points D, E, and F, we can ensure that the new triangle D'E'F' is similar to the original triangle DEF.

7. Based on the above reasoning, the final coordinates are D'(10,-4), E'(10,4), F'(-8,6).

Coordinates of a Point

<note>The coordinates of the original triangle vertex D are (5, -2)</note>, <note>the coordinates of E are (5, 2)</note>, <note>the coordinates of F are (-4, 3)</note>.

<note>Point D(5, -2)</note>, <note>Point E(5, 2)</note>, <note>Point F(-4, 3)</note> scaled by a factor of 2 results in <note>D' coordinates being (10, -4)</note>, <note>E' coordinates being (10, 4)</note>, <note>F' coordinates being (-8, 6)</note>.


Question:
Write the coordinates of the vertices after a dilation with a scale factor of 1/5, centered at the origin.

A'( $\Box$ , $\Box$ )

B'( $\Box$ , $\Box$ )

C'( $\Box$ , $\Box$ )

D'( $\Box$ , $\Box$ )

Answer:
A'(-2,1)
B'(2,1)
C'(2,2)
D'(-2,2)

1. First, it is known that the vertex coordinates of rectangle ABCD are A(-10,5), B(10,5), C(10,10), D(-10,10), and a similarity transformation is performed with the origin as the center and the scaling ratio as 1/5.

2. According to the scaling theorem in the Cartesian coordinate system, the position of any point (x, y) on the image after transformation is (x', y'), and (x', y') = (r * x, r * y), where r is the scaling ratio.

3. For point A(-10,5), applying the transformation yields A' = (1/5 * -10, 1/5 * 5) = (-2, 1).

4. For point B(10,5), applying the transformation yields B' = (1/5 * 10, 1/5 * 5) = (2, 1).

5. For point C(10,10), applying the transformation yields C' = (1/5 * 10, 1/5 * 10) = (2, 2).

6. For point D(-10,10), applying the transformation yields D' = (1/5 * -10, 1/5 * 10) = (-2, 2).

7. After the above transformation, the transformed vertex coordinates are A'(-2, 1), B'(2, 1), C'(2, 2), D'(-2, 2).

8. Through the above reasoning, the final answer is A'(-2, 1), B'(2, 1), C'(2, 2), D'(-2, 2).

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 ABCD is a rectangle</note>, its interior angles <note>∠DAB, ∠ABC, ∠BCD, ∠CDA are all right angles (90 degrees)</note>, and <note>sides AB and DC are parallel and equal in length</note>, <note>sides AD and BC are parallel and equal in length</note>. After a similar transformation, vertices <note>A'(-2, 1)</note>, <note>B'(2, 1)</note>, <note>C'(2, 2)</note>, and <note>D'(-2, 2)</note> still maintain the properties of a rectangle, that is, <note>the transformed figure is still a rectangle</note>.

<note>Point A(-10,5)</note>, <note>Point B(10,5)</note>, <note>Point C(10,10)</note>, <note>Point D(-10,10)</note>, after scaling by a factor of 1/5, <note>the coordinates of point A' are A'(-2, 1)</note>, <note>Point B'(2, 1)</note>, <note>Point C'(2, 2)</note>, <note>Point D'(-2, 2)</note>.


Question:
Write the coordinates of the vertices after a dilation with a scale factor of 1/4, centered at the origin.

R'( $\Box$ , $\Box$ )

S'( $\Box$ , $\Box$ )

T'( $\Box$ , $\Box$ )

U'( $\Box$ , $\Box$ )

Answer:
R'(-2,-2)
S'(-2,2)
T'(1,2)
U'(1,-2)

1. Given that a scaling transformation is required for rectangle RSTU with the origin as the center, the scaling factor is 1/4.

2. The scaling transformation keeps the origin unchanged and scales the x-coordinate and y-coordinate of other points to 1/4 of their original values, which belongs to the scaling theorem in the Cartesian coordinate system.

3. First, consider point R(-8,-8). According to the scaling theorem in the Cartesian coordinate system, multiply its x-coordinate and y-coordinate by 1/4 to get R'(-2,-2).

4. Next, consider point S(-8,8). Similarly, multiply the x-coordinate and y-coordinate of S by 1/4 to get S'(-2,2).

5. Then, consider point T(4,8). Its x-coordinate and y-coordinate also need to be multiplied by 1/4 to get T'(1,2).

6. Finally, consider point U(4,-8). After calculation, multiply its x-coordinate and y-coordinate by 1/4 to get U'(1,-2).

7. After the above transformations, the coordinates of each point become R'(-2,-2), S'(-2,2), T'(1,2), U'(1,-2), which are the vertex coordinates of the new figure after the similarity transformation with the origin as the center and the scaling factor of 1/4.

8. Based on the above reasoning, the final answer is R'(-2,-2), S'(-2,2), T'(1,2), U'(1,-2).

<note>Point R(-8,-8)</note>, <note>Point S(-8,8)</note>, <note>Point T(4,8)</note>, <note>Point U(4,-8)</note>, after scaling by <note>a factor of 1/4</note> the coordinates are <note>Point R'(-2,-2)</note>, <note>Point S'(-2,2)</note>, <note>Point T'(1,2)</note>, <note>Point U'(1,-2)</note>.

<note>Quadrilateral RSTU is a rectangle</note>, with interior angles <note>∠RST</note>, <note>∠STU</note>, <note>∠TUR</note>, and <note>∠URS</note> all being right angles (90 degrees), and sides <note>RS and TU are parallel and equal in length</note>, sides <note>RU and ST are parallel and equal in length</note>.


Question:
In the graph below, △M'N'O' is the image of △MNO after a dilation.

What are the scale factor and center of the dilation?
Simplify your answers and write them as fractions or whole numbers.

scale factor:  $\Box$

center of the dilation: ( $\Box$ ,  $\Box$ )

Answer:
scale factor: 4
center of the dilation: (9, -3)

1. Given △M'N'O' is the image of △MNO after a dilation centered at a certain point. We need to find the scale factor and the center of dilation.

2. First, use the distance formula to calculate the length of MN. The coordinates of M are (5, -3), and the coordinates of N are (6, -2).

3. According to the distance formula, MN = √[(6-5)^2 + ((-2)-(-3))^2] = √[1^2 + 1^2] = √2.

4. Then calculate the length of M'N'. The coordinates of M' are (-7, -3), and the coordinates of N' are (-3, 1).

5. According to the distance formula, M'N' = √[(-3-(-7))^2 + (1-(-3))^2] = √[4^2 + 4^2] = √32 = 4√2.

6. Determine the scale factor as M'N'/MN = (4√2)/(√2) = 4.

7. Next, find the center of dilation. We know that M, M', and the center of dilation are collinear; N, N', and the center of dilation are collinear; O, O', and the center of dilation are collinear.

8. Draw the line passing through N and N', then draw the line passing through O and O', and find the intersection of the two lines, which is the center of dilation.

9. Verify the intersection point using the given diagram, which is (9, -3).

10. Through the above reasoning, the final answer is: the scale factor is 4, and the coordinates of the center of dilation are (9, -3).

Distance Formula Between Two Points

<note>MN uses the coordinates of point M(5, -3) and point N(6, -2) for calculation</note>, <note>M'N' uses the coordinates of point M'(-7, -3) and point N'(-3, 1) for calculation</note>.

The polygon <note>M'N'O'</note> is scaled to obtain the polygon <note>MNO</note>, <note>Scale Factor = Length of M'N' / Length of MN</note>, which means <note>Scale Factor = 4</note>.


Question:
The triangle D'E'F' is a dilation of the triangle DEF. What is the scale factor of the dilation?

Simplify your answer and write it as a proper fraction, an improper fraction, or a whole number.

 $\Box$

Answer: 1/3

1. First, we need to determine the length of one side of the original triangle DEF. Choose side EF, with endpoints at E (3, 9) and F (-6, 9). Since the y-coordinates of these two points are the same (y-coordinate is 9), EF is a horizontal line segment.

2. Using the distance formula between two points, calculate the length of EF. The formula is: length = √((x2 - x1)^2 + (y2 - y1)^2).

3. Substitute the coordinates of E (3, 9) and F (-6, 9) into the formula to get EF = √((-6 - 3)^2 + (9 - 9)^2) = √((-9)^2 + 0) = √(81) = 9.

4. Similarly, determine the length of the corresponding side E'F' in triangle D'E'F'. The coordinates of E' are (1, 3) and the coordinates of F' are (-2, 3). Since the y-coordinates of these two points are also the same (y-coordinate is 3), E'F' is also a horizontal line segment.

5. Using the same distance formula, calculate the length of E'F'. Substitute the coordinates of E' (1, 3) and F' (-2, 3) into the formula to get E'F' = √((-2 - 1)^2 + (3 - 3)^2) = √((-3)^2 + 0) = √(9) = 3.

6. Triangle D'E'F' is a scaled transformation of triangle DEF. According to the definition of the scaling factor in the Cartesian coordinate system, the scaling ratio factor is the ratio of the lengths of the corresponding sides. Calculate the scaling ratio factor k = E'F' / EF.

7. Substitute the known values to get k = 3 / 9.

8. Simplify the ratio factor: k = 3 / 9 = 1 / 3.

9. Through the above reasoning, the final answer is 1/3.

<note>The length of EF</note> is calculated based on the coordinates of <note>point E(3, 9)</note> and <note>point F(-6, 9)</note> as √((-6 - 3)^2 + (9 - 9)^2), resulting in EF = 9. Similarly, <note>the length of E'F'</note> is calculated based on the coordinates of <note>point E'(1, 3)</note> and <note>point F'(-2, 3)</note> as √((-2 - 1)^2 + (3 - 3)^2), resulting in E'F' = 3.

In the figure of this problem, <note>polygon DEF</note> is scaled to obtain <note>polygon D'E'F'</note>, scale factor = <note>E'F'</note> length / <note>EF length</note>, i.e., scale factor = 1/3.


Question:
The circumscribed circle of △ABC is partially constructed below.

Put the remaining steps of the construction in the correct order.
Mark the intersection of DE and FG with point H.Draw a circle with radius CH centered at H.Draw a circle with radius AB centered at B and a circle with radius AB centered at A.Draw a line through F and G.Mark the intersections of the smaller circle A and circle B with points F and G.

Answer: Draw a circle with radius AB centered at B and a circle with radius AB centered at A.Mark the intersections of the smaller circle A and circle B with points F and G. Draw a line through F and G.Mark the intersection of DE and FG with point H.Draw a circle with radius CH centered at H.

1. Given the construction steps of the circumcircle of △ABC. First, draw a circle with center C and radius AC, then draw a circle with center A and radius AC.

2. Mark the two intersection points of the circles as D and E, and draw a line DE through these two points.

3. Next, draw two small circles with centers at A and B and radius AB respectively.

4. Mark the intersection points of the small circle A and circle B as F and G.

5. Draw a line FG through points F and G.

6. Mark the intersection point of line DE and line FG as H.

7. Draw a circle with center H and radius CH. This will be the circumcircle of △ABC.

8. According to the above steps, we have obtained the circumcircle of △ABC, and H is the circumcenter of △ABC.

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, <note>line segment AC is a part of a straight line, including endpoint A and endpoint C and all points in between</note>. <note>Line segment AB is a part of a straight line, including endpoint A and endpoint B and all points in between</note>. <note>Line segment AC and line segment AB have two endpoints, which are A and C, and A and B respectively</note>, and <note>every point on line segment AC and line segment AB is located between endpoint A and endpoint C, and between endpoint A and endpoint B</note>.

Definition of Circle

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

<note>A circle with center C and radius AC</note>, <note>a circle with center A and radius AC</note>, and <note>a circle with center H and radius CH</note> all satisfy the definition of a circle. The definition of a circle is the set of all points in a plane that are at a fixed distance from a fixed point. This fixed point is called the center, and the fixed distance is called the radius.

Circumcenter of a Triangle

In a triangle, the circumcenter is the point where the three perpendicular bisectors of the sides intersect. This point is equidistant from the three vertices of the triangle.

In the figure of this problem, in triangle ABC, <note>point H is the circumcenter of the triangle</note>. The perpendicular bisectors of the three sides <note>AB, BC, and CA intersect at point H</note>. According to the definition of the circumcenter of a triangle, <note>the distances from point H to the three vertices A, B, and C of triangle ABC are equal</note>.

Properties of the Perpendicular Bisector

Any point on the perpendicular bisector of a line segment is equidistant from the endpoints of the segment.

<note>Line DE passing through points D and E is the perpendicular bisector of segment AC</note>. According to the properties of the perpendicular bisector, <note>points D and E are equidistant from the endpoints A and C of segment AC, i.e., AD = DC and AE = EC</note>.


Question:
The radius of a circle is 4 kilometers. What is the area of a sector bounded by a 90° arc?

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

Answer: 4𝜋 square kilometers

1. Given the radius of the circle is 4 kilometers, according to the formula for the area of a circle A=𝜋r^2, we can find the area of the entire circle.

2. Substitute the radius r=4 kilometers into the formula to calculate the area of the circle: A=𝜋×(4)^2=16𝜋 square kilometers.

3. The problem gives the central angle of the circle as 90°.

4. According to the formula for the area of a sector K=A×(m/360), where A is the area of the circle and m is the measure of the central angle of the sector.

5. Substitute the values A=16𝜋 square kilometers and m=90° into the formula to calculate the area of the sector: K=16𝜋×(90/360)=16𝜋×(1/4)=4𝜋 square kilometers.

6. After the above calculations, the area of the sector is 4𝜋 square kilometers.

Definition of Circle

<note>The center of the circle is the point marked in the diagram</note>, <note>the radius is 4 kilometers</note>. All points in the diagram that are 4 kilometers away from the center are on the circle.

Definition of Central Angle

The center of the circle is <note>point O</note>. The angle formed by the lines <note>OA and OB</note> is called the central angle <note>∠AOB</note>, where <note>A and B are two points on the circle</note>, and <note>∠AOB=90°</note>.

Definition of Sector

<note>A sector is formed by two radii of a circle (each with a length of 4 kilometers)</note> and <note>the arc between them (corresponding to a central angle of 90°)</note>.

Area Formula of a Circle

<note>The radius of the circle is 4 kilometers</note>, according to the area formula of a circle, the area A of the circle is equal to <note>pi π multiplied by the square of the radius 4, i.e., A = π4²</note>.

Formula for the Area of a Sector

<note>In this problem, the formula for the area of a sector</note> is used to calculate the <note>sector area corresponding to a 90° central angle</note>. Substituting <note>the area of the circle A=16π square kilometers</note> and <note>m=90°</note>, we get <note>the sector area is K=16π×(90/360)=4π square kilometers</note>.


Question:
The rectangle F'G'H'I' is a dilation of the rectangle FGHI. What is the scale factor of the dilation?

Simplify your answer and write it as a proper fraction, an improper fraction, or a whole number.

 $\Box$

Answer: 2

1. Given that rectangle FGHI is a scaled version of rectangle F'G'H'I'. First, find the length of side HI in the original rectangle FGHI.

2. The coordinates of point H are (-1, 4), and the coordinates of point I are (-1, 2). According to the distance formula between two points, the length of HI is the absolute value of the difference between their y-coordinates, which is |4-2|=2.

3. Next, find the length of the corresponding side H'I' in the scaled rectangle F'G'H'I'.

4. The coordinates of point H' are (-2, 8), and the coordinates of point I' are (-2, 4). According to the distance formula between two points, the length of H'I' is the absolute value of the difference between their y-coordinates, which is |8-4|=4.

5. Now calculate the scaling factor. According to the definition of the scaling factor in the Cartesian coordinate system, it is the ratio of the length of the corresponding side in the scaled figure to the length of the side in the original figure.

6. Calculate the ratio: H'I'/HI=4/2.

7. Simplify the fraction: 4/2=2.

8. In conclusion, the scaling factor from rectangle FGHI to rectangle F'G'H'I' is 2.

Definition of Rectangle

In the figure of this problem, quadrilateral FGHI is a rectangle, <note>its internal angles ∠FGH, ∠GHI, ∠HIF, ∠IFG are all right angles (90 degrees)</note>, and <note>side FG is parallel and equal in length to side HI</note>, <note>side HG is parallel and equal in length to side IF</note>. Quadrilateral F'G'H'I' is also a rectangle, <note>its internal angles ∠H'G'F', ∠G'F'I', ∠F'I'H', ∠I'H'G' are all right angles (90 degrees)</note>, and <note>side F'G' is parallel and equal in length to side H'I'</note>, <note>side H'G' is parallel and equal in length to side I'F'</note>.

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 \( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \).

In this problem, we use the simplified distance formula to calculate the length of <note>side HI and side H'I'</note> because they have the <note>same x-coordinate</note>. <note>The length of HI is |4 - 2| = 2</note>, <note>the length of H'I' is |8 - 4| = 4</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.

After scaling <note>polygon F'G'H'I'</note>, we obtain <note>polygon FGHI</note>, <note>scale factor = length of H'I' / length of HI</note>, which means the scale factor = 2.


Question:
The rectangle I'J'K'L' is a dilation of the rectangle IJKL. What is the scale factor of the dilation?

Simplify your answer and write it as a proper fraction, an improper fraction, or a whole number.

 $\Box$

Answer: 1/2

1. Observe the rectangle IJKL and its corresponding rectangle I'J'K'L'. The shapes of these two rectangles remain consistent, indicating that I'J'K'L' is a scaled version (similar transformation) of IJKL.

2. To determine the scaling ratio, calculate the length of a side in IJKL and find the length of its corresponding side in I'J'K'L', then find the ratio of the two lengths.

3. Calculate the length of side JK in the original rectangle IJKL. The coordinates of point J are (0, 2), and the coordinates of point K are (0, -10).

4. Since JK is perpendicular to the x-axis, the length is the distance between these two points on the y-axis, which is |2 - (-10)| = 12 units.

5. Calculate the length of the corresponding side J'K' in rectangle I'J'K'L'. The coordinates of point J' are (0, 1), and the coordinates of point K' are (0, -5).

6. Similarly, J'K' is perpendicular to the x-axis, and its length is |1 - (-5)| = 6 units.

7. Set the scaling ratio as the ratio of the lengths of the corresponding sides J'K' and JK, which is the scaling factor 6/12.

8. By simplifying the fraction 6/12, we get the scaling factor 1/2.

9. Through the above reasoning, the final answer is 1/2.

Definition of Rectangle

<note>Quadrilateral IJKL is a rectangle</note>, with its interior angles <note>∠IJK</note>, <note>∠JKL</note>, <note>∠KLI</note>, <note>∠LIJ all being right angles (90 degrees)</note>, and <note>side IJ is parallel and equal in length to side KL</note>, <note>side JK is parallel and equal in length to side IL</note>. <note>Quadrilateral I'J'K'L' is also a rectangle</note>, with its interior angles <note>∠I'J'K'</note>, <note>∠J'K'L'</note>, <note>∠K'L'I'</note>, <note>∠L'I'J' all being right angles (90 degrees)</note>, and <note>side I'J' is parallel and equal in length to side K'L'</note>, <note>side J'K' is parallel and equal in length to side I'L'</note>.

In the problem diagram, the polygon <note>IJKL</note> is scaled to obtain the polygon <note>I'J'K'L'</note>, <note>scale factor = length of J'K' / length of JK</note>, that is, <note>scale factor = 1/2</note>.


Question:
In the graph below, trapezoid G'H'I'J' is the image of GHIJ after a dilation.

What are the scale factor and center of the dilation?
Simplify your answers and write them as fractions or whole numbers.

scale factor:  $\Box$

center of the dilation: ( $\Box$ ,  $\Box$ )

Answer:
scale factor: 3/4
center of the dilation: (4, -5)

1. Given trapezoid G'H'I'J' is the image of trapezoid GHIJ after a dilation transformation. According to the definition of dilation transformation, the length of the segment is scaled by a fixed ratio.

2. Calculate the length of HI. The coordinates of point H are (0, 7), and the coordinates of point I are (4, 7). Since HI is a horizontal segment, the length of HI is |0-4|=4.

3. Calculate the length of H'I'. The coordinates of point H' are (1, 4), and the coordinates of point I' are (4, 4). Since H'I' is also a horizontal segment, the length of H'I' is |1-4|=3.

4. According to the similarity ratio formula, the dilation factor k = H'I'/HI = 3/4. Therefore, the dilation ratio factor is 3/4.

5. To determine the center of dilation, each pair of corresponding points and the center of dilation must be collinear, and the ratio of the distance from the center of dilation to each point of the figure to the distance to the corresponding point is 1/(3/4)=4/3. The center of dilation can be found using the lines connecting G and G', and J and J'.

6. Draw the segment GG' (through G(-4, 3) and G'(-2, 1)) and the segment JJ' (through J(8, 3) and J'(7, 1)).

7. Calculate the equations of the two segments. According to the slope formula, the slope of GG' is (1-3)/(-2-(-4))=-1, and the equation of the line is y=-x-1. The slope of JJ' is (1-3)/(7-8)=2, and the equation of the line is y=2x-13.

8. Solve the intersection of the two line equations. Since the equations of the two segments are different, they intersect at a line parallel to the y-axis, so the intersection is not unique. Therefore, using graphical analysis, the intersection can be found to be approximately (4, -5).

9. Based on the above reasoning, the final answer is that the dilation ratio is 3/4, and the coordinates of the center of dilation are (4, -5).

Definition of Trapezoid

A quadrilateral is a trapezoid if and only if it has exactly one pair of parallel sides.

In the figure of this problem, <note>trapezoid GHIJ</note> and <note>trapezoid G'H'I'J'</note> are the shapes before and after the scaling transformation, respectively, where <note>side GH is parallel to side IJ</note>, <note>side G'H' is parallel to side I'J'</note>, satisfying the definition of trapezoid.

After scaling <note>polygon HIGJ</note>, we obtain <note>polygon H'I'G'J'</note>. The <note>scale factor = H'I'/HI, length of H'I'/length of HI</note>, which means the scale factor = 3/4.


Question:
The rectangle N'O'P'Q' is a dilation of the rectangle NOPQ. What is the scale factor of the dilation?

Simplify your answer and write it as a proper fraction, an improper fraction, or a whole number.

 $\Box$

Answer: 4

1. Determine the corresponding sides of the original rectangle NOPQ and the enlarged rectangle N'O'P'Q'. We choose side NO corresponding to N'O'.

2. Calculate the length of side NO. The coordinates of point N are (-1, -2), and the coordinates of point O are (2, -2). According to the distance formula between two points: d = √[(x2 - x1)^2 + (y2 - y1)^2], so the length of NO is √[(2 - (-1))^2 + (-2 - (-2))^2] = √[3^2 + 0^2] = 3.

3. Calculate the length of side N'O'. The coordinates of point N' are (-4, -8), and the coordinates of point O' are (8, -8). According to the distance formula between two points: d = √[(x2 - x1)^2 + (y2 - y1)^2], so the length of N'O' is √[(8 - (-4))^2 + (-8 - (-8))^2] = √[12^2 + 0^2] = 12.

4. According to the definition of the scaling factor in the Cartesian coordinate system, set the ratio of the scaling factor using the ratio of the length of the corresponding side of the enlarged figure to the length of the corresponding side of the original figure, i.e., N'O'/NO = 12/3.

5. Simplify this ratio, 12/3 = 4.

6. Through the above reasoning, the final answer is 4.

Definition of Rectangle

The rectangle NOPQ is formed by <note>point N(-1, -2)</note>, <note>point O(2, -2)</note>, <note>point P(2, 2)</note>, and <note>point Q(-1, 2)</note>. The rectangle N'O'P'Q' is formed by <note>point N'(-4, -8)</note>, <note>point O'(8, -8)</note>, <note>point P'(8, 8)</note>, and <note>point Q'(-4, 8)</note>. <note>Both rectangles satisfy the condition that opposite sides are equal and parallel</note>.

Distance Formula Between Two Points

Original text: <note>The coordinates of point N are (-1, -2)</note>, <note>The coordinates of point O are (2, -2)</note>, we need to calculate the distance between point N and point O. According to the distance formula between two points, the distance d between point N and point O can be calculated using the following formula: d = √(((-2 - (-1))^2 + (-2 - (-2)))^2). In the diagram, <note>The distance between point N and point O is the length of segment NO</note>. <note>The coordinates of point N' are (-4, -8)</note>, <note>The coordinates of point O' are (8, -8)</note>, we need to calculate the distance between point N' and point O'. According to the distance formula between two points, the distance d between point N' and point O' can be calculated using the following formula: d = √(((8 - (-4))^2 + (-8 - (-8)))^2). In the diagram, <note>The distance between point N' and point O' is the length of segment N'O'</note>.

<note>Polygon NOPQ</note> is scaled to obtain <note>Polygon N'O'P'Q'</note>, scale factor = <note>length of N'O' / length of NO</note>, i.e., scale factor = 4.


Question:
The parallelogram H'I'J'K' is a dilation of the parallelogram HIJK. What is the scale factor of the dilation?

Simplify your answer and write it as a proper fraction, an improper fraction, or a whole number.

 $\Box$

Answer: 1/4

1. Given quadrilateral H'I'J'K' is a dilation of quadrilateral HIJK, first we need to find the length of one side of the original quadrilateral.

2. Take segment \overline{HI}, with endpoints H(-8,-8) and I(-4,-8). Using the distance formula between two points, calculate the segment length HI = \sqrt{(-4 - (-8))^2 + (-8 - (-8))^2} = \sqrt{4^2 + 0} = 4. Therefore, the length of segment \overline{HI} is 4 units.

3. Observe the corresponding segment \overline{H'I'} of the transformed quadrilateral, with endpoints H'(-2,-2) and I'(-1,-2). Using the distance formula between two points, calculate the segment length H'I' = \sqrt{(-1 - (-2))^2 + (-2 - (-2))^2} = \sqrt{1^2 + 0} = 1. Therefore, the length of segment \overline{H'I'} is 1 unit.

4. Calculate the dilation factor ratio, according to the definition of the scaling factor in the Cartesian coordinate system, which is the ratio of the length of the image segment to the length of the original segment.

5. Dilation factor = \frac{H'I'}{HI} = \frac{1}{4}.

6. Through the above reasoning, the final answer is \frac{1}{4}.

Coordinates of a Point

In a Cartesian coordinate system, the position of a point is represented by an ordered pair (x, y), where x denotes the horizontal (x-coordinate) and y denotes the vertical (y-coordinate).

<note>The coordinates of point H are (-8, -8)</note>, <note>The coordinates of point I are (-4, -8)</note>, <note>The coordinates of point H' are (-2, -2)</note>, <note>The coordinates of point I' are (-1, -2)</note>.

Distance Formula Between Two Points

The length of <note>line segment HI</note> is calculated using the distance formula, \( <note>HI</note> = \sqrt{(-4 - (-8))^2 + (-8 - (-8))^2} = \sqrt{4^2 + 0} = 4 \) units; similarly, the length of <note>line segment H'I'</note> is \( <note>H'I'</note> = \sqrt{(-1 - (-2))^2 + (-2 - (-2))^2} = \sqrt{1^2 + 0} = 1 \) unit.

<note>Polygon HIJK is scaled to obtain polygon H'I'J'K'</note>, <note>Scale factor = length of H'I' / length of HI</note>, that is, <note>Scale factor = 1/4</note>.


Question:
In the graph below, rhombus J'K'L'M' is the image of JKLM after a dilation.

What are the scale factor and center of the dilation?
Simplify your answers and write them as fractions or whole numbers.

scale factor:  $\Box$

center of the dilation: ( $\Box$ ,  $\Box$ )

Answer:
scale factor: 2
center of the dilation: (-3, -6)

1. Given that rhombus JKLM maps to rhombus J'K'L'M', find the scale factor and the center of scaling according to the requirements of the problem.

2. Find the length of segment JK, with point J having coordinates (–6, –2) and point K having coordinates (–1, –2). Since segment JK is a horizontal segment, its length is the absolute difference of the two x-values. Calculating, we get JK = |–6 - (–1)| = |–5| = 5.

3. Find the length of segment J'K', with point J' having coordinates (–9, 2) and point K' having coordinates (1, 2). Similarly, since segment J'K' is also a horizontal segment, its length is the absolute difference of the two x-values. Calculating, we get J'K' = |–9 - 1| = |–10| = 10.

4. The scale factor can be determined by the ratio of J'K' to JK, thus the scale factor is J'K'/JK = 10/5 = 2.

5. The method to determine the center of scaling is to find the point that does not move. In this figure, point M and point M' have the same coordinates, both being (–3, –6). Therefore, it can be determined that M is the center of scaling.

Based on the above reasoning, the final answer is: the scale factor is 2, and the center of scaling is (–3, –6).

Definition of Rhombus

A quadrilateral is a rhombus if and only if all four sides are of equal length, and its diagonals are perpendicular bisectors of each other.

<note>Quadrilateral JKLM and quadrilateral J'K'L'M' are both rhombuses</note>. This means that all four of their sides are equal, that is, <note>JK = KL = LM = MJ</note> and <note>J'K' = K'L' = L'M' = M'J'</note>. Furthermore, the <note>diagonals JK and LM of quadrilateral JKLM are perpendicular bisectors of each other</note>, and the <note>diagonals J'K' and L'M' of quadrilateral J'K'L'M' are perpendicular bisectors of each other</note>.

Distance Formula Between Two Points

<note>Line segments JK and J'K' are horizontal line segments</note>, so their distance calculation simplifies to the <note>absolute value of the difference in x-coordinates</note>. <note>JK = |-6 - (-1)| = | -5 | = 5</note>, <note>J'K' = |-9 - 1| = |-10| = 10</note>.

The rhombus <note>JKLM</note> is scaled to obtain the rhombus <note>J'K'L'M'</note>, <note>Scale Factor = Length of J'K' / Length of JK</note>, Scale Factor = 2.


Question:
The diagram shows a convex polygon.

What is the sum of the exterior angle measures, one at each vertex, of this polygon?
 $\Box$ °

Answer: 360°

1. Observe the figure in the problem, this is a convex polygon, and the given condition is to find the sum of the exterior angles at each vertex of this polygon.

2. According to the known geometric principle: the sum of the exterior angles of any convex polygon is 360°. There is no need to consider the number of sides of the polygon.

3. This property applies to all convex polygons, regardless of their shape, as long as it is a convex polygon, the sum of its exterior angles is always 360°.

4. Because the polygon is convex, each exterior angle at each vertex is the supplementary angle of the interior angle at that vertex in the plane of 180°.

5. Based on the above reasoning, there is no need to mark the figure or add auxiliary lines, directly conclude: the sum of the exterior angles of this convex polygon is 360°.

6. After the above reasoning, the final answer is 360°.

Definition of Exterior Angle of a Polygon

An exterior angle of a polygon is the angle formed between one side of the polygon and the extension of an adjacent side.

The angle formed by <note>extending one adjacent side of an interior angle of the polygon</note> is called <note>the exterior angle of this interior angle</note>.

Exterior Angle Sum Theorem of Polygon

For any polygon, the sum of its exterior angles is equal to 360°.

Original text: <note>In this problem diagram, we are considering a convex polygon (triangle)</note>, therefore <note>its exterior angle sum is 360 degrees</note>, applicable to all convex polygons, regardless of the number of sides. <note>As shown, the exterior angle sum of this convex polygon (triangle) is 360 degrees</note>.


Question:
The diagram shows a convex polygon.

What is the sum of the exterior angle measures, one at each vertex, of this polygon?
 $\Box$ °

Answer: 360°

1. Consider a convex polygon, at each vertex we can make an exterior angle.

2. According to the 'Polygon Exterior Angle Sum Theorem', the sum of one exterior angle at each vertex of a polygon is equal to 360°.

3. This conclusion applies to all polygons, regardless of the number of sides.

4. Therefore, the sum of one exterior angle at each vertex of the given convex polygon is also 360°.

5. In summary, the sum of one exterior angle at each vertex of the polygon required by the problem is 360°.

At <note>vertex A</note>, the exterior angle is the angle formed by the interior angle and the extended side. Similarly, at <note>vertex B and vertex C</note>, there are respective exterior angles.

<note>The given polygon is a convex triangle</note>. According to the Exterior Angle Sum Theorem of Polygon, <note>the sum of one exterior angle at each vertex of a convex triangle is 360°</note>. Regardless of the number of sides of the polygon, this theorem applies, therefore, <note>the sum of the exterior angles at each vertex of the given convex polygon is 360°</note>.


Question:
The rectangle K'L'M'N' is a dilation of the rectangle KLMN. What is the scale factor of the dilation?

Simplify your answer and write it as a proper fraction, an improper fraction, or a whole number.

 $\Box$

Answer: 1/3

1. According to the problem statement, the given rectangle K'L'M'N' is a scaled transformation of rectangle KLMN. We need to find the scale factor of this transformation.

2. First, find the length of one side of the original rectangle KLMN. We choose side LM, which connects points L(-6,0) and M(3,0).

3. Use the distance formula between two points to calculate the length of side LM: the known distance formula is: d = √((x2 - x1)^2 + (y2 - y1)^2). Since LM is perpendicular to the x-axis, the difference is only in the x-axis direction, so it can be simplified to: |3 - (-6)| = 9.

4. Next, find the length of the corresponding side L'M' in the image. Side L'M' connects points L'(-2,0) and M'(1,0).

5. Use the distance formula to calculate the length of side L'M': similarly, since L'M' is also perpendicular to the x-axis, the simplified calculation is: |1 - (-2)| = 3.

6. According to the definition of the scale factor in the Cartesian coordinate system, calculate the scale factor through the ratio between the length of one side of the original rectangle and the corresponding side of the transformed rectangle.

7. Compare the length of side L'M' with the length of side LM to obtain the scale factor as 3/9.

8. Simplify the ratio 3/9 to its simplest form, which is 1/3.

9. Through the above reasoning, the final answer is 1/3.

Definition of Rectangle

<note>Quadrilateral KLMN and K'L'M'N' are both rectangles</note>, with interior angles <note>∠KLM, ∠LMN, ∠MNK, ∠NKL and ∠K'L'M', ∠L'M'N', ∠M'N'K', ∠N'K'L' all being right angles (90 degrees)</note>, and <note>side KL is parallel and equal in length to side MN</note>, <note>side LM is parallel and equal in length to side NK</note>, <note>side K'L' is parallel and equal in length to side M'N'</note>, <note>side L'M' is parallel and equal in length to side N'K'</note>.

Distance Formula Between Two Points

<note>Point L(-6,0)</note> and <note>M(3,0)</note> calculate <note>the length of side LM</note> using the distance formula to get |3 - (-6)| = 9. Similarly, using <note>Point L'(-2,0)</note> and <note>M'(1,0)</note> calculate <note>the length of side L'M'</note> using the distance formula to get |1 - (-2)| = 3.

After scaling <note>quadrilateral KLMN</note>, we obtain <note>quadrilateral K'L'M'N'</note>. The scale factor = length of <note>L'M'</note> / length of <note>LM</note>, scale factor = 1/3.


Question:
In the graph below, △R'S'T' is the image of △RST after a dilation.

What are the scale factor and center of the dilation?
Simplify your answers and write them as fractions or whole numbers.

scale factor:  $\Box$

center of the dilation: ( $\Box$ ,  $\Box$ )

Answer:
scale factor: 1/2
center of the dilation: (0, 0)

1. First, use the distance formula between two points to calculate the length of segment RT. The coordinates of point R are (2, -10), and the coordinates of point T are (8, -8). The distance formula is d = √((x2 - x1)^2 + (y2 - y1)^2), substituting in we get RT = √((8 - 2)^2 + (-8 - (-10))^2) = √(6^2 + 2^2) = √(36 + 4) = √40.

2. Next, calculate the length of segment R'T'. The coordinates of point R' are (1, -5), and the coordinates of point T' are (4, -4). Substituting into the distance formula we get R'T' = √((4 - 1)^2 + (-4 - (-5))^2) = √(3^2 + 1^2) = √(9 + 1) = √10.

3. According to the definition of the scaling factor in the Cartesian coordinate system, calculate the scaling factor k, which is the ratio of the corresponding sides of the original triangle, k = R'T'/RT = √10/√40.

4. Calculate the simplified form of the ratio k, by factoring √10/√40 = √(10/40) = √(1/4) = 1/2, obtaining the scaling factor k = 1/2.

5. To determine the scaling center, note that the scaling center, the original figure, and the corresponding points on the image are collinear. Therefore, draw the extension line from R to R' and the extension line from T to T'.

6. According to the line equation, the slope from R to R' is (-5 - (-10))/(1 - 2) = 5/-1 = -5. So R, R' and (0, 0) are aligned. Similarly, calculate the slope from T to T' (-4 - (-8))/(4 - 8) = 4/-4 = -1. It is shown that (0,0) is on the extension line.

7. Through the above reasoning, determine the scaling center to be (0, 0).

8. Combining the above steps, the scaling factor is 1/2, and the center is (0, 0).

When calculating the length of <note>line segment RT</note>, use the coordinates of <note>point R (2, -10)</note> and <note>point T (8, -8)</note> in the formula d = √((8 - 2)^2 + (-8 - (-10))^2) = √(36 + 4) = √40. Similarly, when calculating the length of <note>line segment R'T'</note>, use the coordinates of <note>point R' (1, -5)</note> and <note>point T' (4, -4)</note> in the formula d = √((4 - 1)^2 + (-4 - (-5))^2) = √(9 + 1) = √10.

<note>After scaling polygon RST, we obtain polygon R'S'T'</note>, scale factor = <note>length of R'T' / length of RT</note>, i.e., scale factor = 1/2.


Question:
In the graph below, △L'M'N' is the image of △LMN after a dilation.

What are the scale factor and center of the dilation?
Simplify your answers and write them as fractions or whole numbers.

scale factor:  $\Box$

center of the dilation: ( $\Box$ ,  $\Box$ )

Answer:
scale factor: 3/4
center of the dilation: (5, 0)

1. Given △L'M'N' is the image of △LMN after a similarity transformation. First, we need to find the scale factor of the transformation. We start by analyzing the segment LM.

2. Since segment LM is horizontal, we find the length of LM by calculating the difference in the x-coordinates of point L(-7,-4) and point M(1,-4). The calculation yields LM=|–7–1|=|–8|=8.

3. Next, we calculate the length of the corresponding segment L'M'. Using the difference in the x-coordinates of point L'(–4,–3) and point M'(2,–3), we find L'M'=|–4–2|=|–6|=6.

4. Therefore, the scale factor is L'M'/LM=6/8=3/4.

5. To determine the center of the transformation, geometrically, every point on △L'M'N', its corresponding point on △LMN, and the center of transformation should be collinear.

6. To achieve this, we draw auxiliary lines, connecting point L and L' to form one line, and connecting point N and N' to form another line.

7. The intersection of these two lines is the center of the transformation.

8. After calculation, the coordinates of the intersection point are (5, 0). Therefore, the center of the transformation is (5, 0).

9. Based on the above reasoning, the final answer is that the scale factor is 3/4 and the center of transformation is (5, 0).

After scaling <note>△LMN</note>, we obtain <note>△L'M'N'</note>. The scale factor is <note>L'M' length</note>/LM length, which means the scale factor is 6/8 = 3/4.


Question:
The diagram shows a convex polygon.

What is the sum of the exterior angle measures, one at each vertex, of this polygon?
 $\Box$ °

Answer: 360°

1. Given that the polygon is a convex polygon, according to the properties of convex polygons, each exterior angle and interior angle are adjacent supplementary angles.

2. For any convex polygon, the sum of its exterior angles (the exterior angle at each vertex) is 360°. This is because the exterior angle sum theorem states: the sum of the exterior angles of any polygon (taking one exterior angle at each vertex) is equal to 360°.

3. This theorem can be understood as follows: starting from any vertex of the polygon, walking around the polygon along its edges, the total angle turned is 360°.

4. Since the problem states that the polygon is a convex polygon, we can directly apply the exterior angle sum theorem.

5. Through the above reasoning, we finally arrive at the answer of 360°.

<note>In the diagram of this problem</note>, the angle formed by extending <note>one adjacent side of an interior angle of the polygon</note> is called <note>the exterior angle of this interior angle</note>.

Exterior Angle Sum Theorem of Polygon

<note>Angle A</note>, <note>Angle B</note>, <note>Angle C</note>. According to the Exterior Angle Sum Theorem of Polygon, <note>the sum of these exterior angles is equal to 360°</note>, that is <note>Angle A + Angle B + Angle C = 360°</note>.


Question:
The parallelogram M'N'O'P' is a dilation of the parallelogram MNOP. What is the scale factor of the dilation?

Simplify your answer and write it as a proper fraction, an improper fraction, or a whole number.

 $\Box$

Answer: 5

1. First, we need to calculate the length of segment \\(\overline{NO}\\) in the original parallelogram MNOP. According to the distance formula between two points \\(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\\), the coordinates of point N are (-2, -2) and the coordinates of point O are (1, -2). Substituting into the formula, we get \\(\overline{NO} = \sqrt{(1 - (-2))^2 + ((-2) - (-2))^2} = \sqrt{(3)^2 + (0)^2} = 3\\).

2. Next, calculate the length of the corresponding segment \\(\overline{N'O'}\\) in the parallelogram M'N'O'P' obtained by expanding the original parallelogram MNOP. According to the distance formula, the coordinates of point N' are (-10, -10) and the coordinates of point O' are (5, -10). Substituting into the formula, we get \\(\overline{N'O'} = \sqrt{(5 - (-10))^2 + ((-10) - (-10))^2} = \sqrt{(15)^2 + (0)^2} = 15\\).

3. Now, we can determine the expansion ratio factor of the parallelogram. According to the definition of similar figures, the expansion ratio factor is equal to the ratio of the length of the corresponding sides in the image to the length of the corresponding sides in the original figure. Therefore, the ratio factor is \\(\frac{\overline{N'O'}}{\overline{NO}} = \frac{15}{3}\\).

4. Simplify the ratio \\(\frac{15}{3}\\) to get 5.

5. Through the above reasoning, the final answer is 5.

Distance Formula Between Two Points

In the diagram of this problem, when calculating the length of segment <note>\(\overline{NO}\)</note> in parallelogram MNOP, the coordinates of point <note>N are (-2, -2)</note> and the coordinates of point <note>O are (1, -2)</note>. After substituting into the formula, we get <note>\(\overline{NO} = \sqrt{(1 - (-2))^2 + ((-2) - (-2))^2} = \sqrt{(3)^2 + (0)^2} = 3\)</note>. Similarly, when calculating the length of segment <note>\(\overline{N'O'}\)</note> in parallelogram M'N'O'P', the coordinates of point <note>N' are (-10, -10)</note> and the coordinates of point <note>O' are (5, -10)</note>. After substituting into the formula, we get <note>\(\overline{N'O'} = \sqrt{(5 - (-10))^2 + ((-10) - (-10))^2} = \sqrt{(15)^2 + (0)^2} = 15\)</note>.

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, <note>quadrilateral MNOP is a parallelogram</note>, <note>side MN is parallel and equal to side OP</note>, <note>side NO is parallel and equal to side MP</note>. <note>Quadrilateral M'N'O'P' is also a parallelogram</note>, <note>side M'N' is parallel and equal to side O'P'</note>, <note>side N'O' is parallel and equal to side M'P'</note>.

<note>After scaling polygon MNOP, we get polygon M'N'O'P'</note>, <note>Scale factor = length of N'O' / length of NO</note>, i.e., Scale factor = 5.


Question:
The parallelogram D'E'F'G' is a dilation of the parallelogram DEFG. What is the scale factor of the dilation?

Simplify your answer and write it as a proper fraction, an improper fraction, or a whole number.

 $\Box$

Answer: 3

1. According to the definition of a parallelogram, parallelograms DEFG and D'E'F'G' have sides that are parallel and equal, with unchanged side lengths and parallelism as conditions. For convenience in calculation, we first choose side \overline{FG} and the corresponding side \overline{F'G'} for comparison.

2. Calculate the length of \overline{FG}. Point F=(-3,-3) and point G=(-3,-1), according to the distance formula between two points \sqrt{(x_2-x_1)^2+(y_2-y_1)^2}, the length of side \overline{FG} is \sqrt{((-3)-(-3))^2+((-1)-(-3))^2}=\sqrt{0+4}=2.

3. Calculate the length of \overline{F'G'}. Point F'=(-9,-9) and point G'=(-9,-3), using the same distance formula between two points, the length of side \overline{F'G'} is \sqrt{((-9)-(-9))^2+((-3)-(-9))^2}=\sqrt{0+36}=6.

4. According to the definition of the scaling factor in the Cartesian coordinate system, derive the ratio of the corresponding side lengths of the original figure and the scaled figure, which is the scaling factor. The scaling factor is \frac{\text{length of the scaled figure}}{\text{length of the original figure}}=\frac{6}{2}=3.

5. Finally, the scaling factor can be simplified to a fraction: 6/2=3, which is the final scaling factor.

6. Through the above reasoning, the final answer is 3.

<note>Quadrilateral DEFG is a parallelogram</note>, <note>side DE is parallel and equal to side FG</note>, <note>side EF is parallel and equal to side DG</note>. Correspondingly, <note>quadrilateral D'E'F'G' is also a parallelogram</note>, <note>side D'E' is parallel and equal to side F'G'</note>, <note>side E'F' is parallel and equal to side D'G'</note>.

Distance Formula Between Two Points

In this problem, the lengths of sides <note>\overline{FG}</note> and <note>\overline{F'G'}</note> are calculated using the distance formula between two points in the figure, where the length of side <note>\overline{FG}</note> is <note>\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} = \sqrt{0 + 4} = 2</note>, and the length of side <note>\overline{F'G'}</note> is <note>\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} = \sqrt{0 + 36} = 6</note>.

In the figure of this problem, <note>polygon DEFG is scaled to obtain polygon D'E'F'G'</note>, <note>scale factor = length of F'G' / length of FG</note>, that is, the scale factor = 3.


Question:
Find the area of △WXY.

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

Answer: 226.3 cm ^ 2

1. Given the side lengths of △WXY are WX = 17 cm, WY = 39 cm, XY = 29 cm.

2. According to Heron's formula for the area of a triangle, we first need to calculate the semi-perimeter s. Heron's formula defines the semi-perimeter as s = (WX + WY + XY) / 2.

3. Substitute the given side lengths and calculate the semi-perimeter, yielding s = (17 + 39 + 29) / 2 = 85 / 2 = 42.5 cm.

4. Heron's formula is expressed as: Area = √[s(s - a)(s - b)(s - c)], where a, b, c are the side lengths of the triangle. In this problem, a = WX = 17 cm, b = WY = 39 cm, c = XY = 29 cm.

7. Multiply these values and s together: Area = √[42.5 × 25.5 × 3.5 × 13.5].

8. Calculate and simplify these products: 42.5 × 25.5 × 3.5 × 13.5 = 51207.1875.

9. Calculate the square root to find the area: √51207.1875 ≈ 226.2900....

10. As the problem requires rounding to one decimal place, the final answer is 226.3.

Through the above reasoning, the final answer is 226.3 square centimeters.

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.

In the figure of this problem, the triangle <note>△WXY</note> is a geometric figure composed of three <note>non-collinear points W, X, Y</note> and their connecting line segments <note>WX, WY, XY</note>. Points <note>W, X, Y</note> are the three vertices of the triangle, and line segments <note>WX, WY, XY</note> are the three sides of the triangle.

Heron's Formula

Heron's formula is used to calculate the area of any triangle. The formula is given by: \( A = \sqrt{s(s - a)(s - b)(s - c)} \), where \( s \) is the semi-perimeter, and \( a, b, \) and \( c \) are the lengths of the sides of the triangle.

<note>The semi-perimeter s of triangle △WXY is 42.5 cm</note>, with side lengths <note>WX = 17 cm, WY = 39 cm, XY = 29 cm</note>. Using Heron's formula, substitute into the calculation: Area = √[<note>42.5(42.5 - 17)(42.5 - 39)(42.5 - 29)</note>] = √[<note>42.5 × 25.5 × 3.5 × 13.5</note>] ≈ 226.3 cm².


Question:
Find the area of △UVW.

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

Answer: 728.1 mi ^ 2

1. Given the side lengths of △UVW as UV = 37 miles, VW = 44 miles, UW = 43 miles. Based on the given side lengths, Heron's formula can be used to solve for the area of the triangle.

2. Calculate the semi-perimeter s, the sum of the three sides is 37 + 44 + 43 = 124, divided by 2 to get the semi-perimeter s = 62.

3. Heron's formula is: A = sqrt(s * (s - a) * (s - b) * (s - c)), where s is the semi-perimeter, and a, b, c are the side lengths of the triangle.

4. For △UVW, the parameters in Heron's formula include s = 62, a = 37, b = 44, c = 43.

5. Substitute into Heron's formula to calculate: A = sqrt(62 * (62 - 37) * (62 - 44) * (62 - 43)).

6. Perform the calculation: A = sqrt(62 * 25 * 18 * 19).

7. Continue calculating the product: 62 * 25 * 18 * 19 = 530100.

8. A = sqrt(530100), calculate its square root A ≈ 728.0797.

9. Round to one decimal place to get the area A ≈ 728.1.

10. Through the above reasoning, the final area of △UVW is 728.1 square miles.

In the figure of this problem, use Heron's Formula to calculate the area of <note>△UVW</note>. Given <note>s=62</note>, <note>a=37</note>, <note>b=44</note>, <note>c=43</note>. The calculation process is as follows: A=sqrt(<note>62(62-37)(62-44)(62-43)</note>)=sqrt(62*25*18*19)=sqrt(530100)≈728.0796. Finally, the area of <note>△UVW is 728.1 square miles</note>.


Question:
The diagram shows a convex polygon.

What is the sum of the exterior angle measures, one at each vertex, of this polygon?
 $\Box$ °

Answer: 360°

1. First, we need to understand that the problem requires the sum of the exterior angles at each vertex of a concave polygon.

2. According to the exterior angle theorem, the exterior angle at each vertex of a convex polygon is formed by the interior angle and its supplementary angle at that vertex, which means the sum of the interior angle and its corresponding exterior angle is 180°.

3. To obtain the sum of all exterior angles of a convex polygon, we can use the polygon exterior angle sum theorem, which states: the sum of the exterior angles of any convex polygon is 360°.

4. According to the polygon exterior angle sum theorem, regardless of the number of sides of the polygon, the total sum of the exterior angles does not depend on the number of sides, so for any convex polygon, the sum of the exterior angles at each vertex is 360°.

5. Through the above reasoning, the final answer is 360°.

<note>For each vertex of a convex polygon</note>, the exterior angle can be expressed as <note>the supplementary angle corresponding to the interior angle</note>. If the interior angle is denoted as <note>∠ABC</note>, then its exterior angle can be expressed as <note>∠ABD</note>, where <note>D is the point formed by extending BC</note>.

Exterior Angle Sum Theorem of Polygon

<note>Angle A</note>, <note>Angle B</note>, <note>Angle C</note>. According to the Exterior Angle Sum Theorem of Polygon, the sum of these exterior angles is equal to 360°, that is <note>Angle A + Angle B + Angle C = 360°</note>.


Question:
The diagram shows a convex polygon.

What is the sum of the exterior angle measures, one at each vertex, of this polygon?
 $\Box$ °

Answer: 360°

1. According to the requirements of the problem and the shown figure, this is a convex polygon.

2. According to the exterior angle sum theorem of polygons: In any convex polygon, the sum of the exterior angles at each vertex is equal to 360°.

3. This theorem applies to all convex polygons, regardless of the number of sides of the polygon. Therefore, we do not need to consider the number of sides of the polygon to apply this theorem.

4. Combining the above information, directly applying the exterior angle sum theorem of polygons, it is known that the sum of the exterior angles at each vertex of this convex polygon is 360°.

5. Therefore, through the above reasoning, the final answer is 360°.

In the figure of this problem, an interior angle of the polygon is <note>∠ABC</note> (marked at the vertex), and the angle <note>∠ABD</note> formed by extending the adjacent sides <note>AB</note> and <note>BC</note> of this interior angle is called the exterior angle of the interior angle <note>∠ABC</note>. For example, there is an exterior angle at each vertex of the polygon.

<note>Angle A</note>, <note>Angle B</note>, <note>Angle C</note>. According to the Exterior Angle Sum Theorem of Polygon, the sum of these exterior angles is equal to <note>360°</note>, that is, <note>Angle A + Angle B + Angle C = 360°</note>.


Question:
In the graph below, rhombus K'L'M'N' is the image of KLMN after a dilation.

What are the scale factor and center of the dilation?
Simplify your answers and write them as fractions or whole numbers.

scale factor:  $\Box$

center of the dilation: ( $\Box$ ,  $\Box$ )

Answer:
scale factor: 1/2
center of the dilation: (-4, 2)

1. First, calculate the lengths of the corresponding sides NM and N'M' of the original quadrilateral KLMN and the mapped quadrilateral K'L'M'N'.

2. Points N(-2,-6) and M(8,-6) are on the same horizontal line, and the horizontal distance is |x_2 - x_1| = |8 - (-2)| = 10.

3. The mapped points N'(-3,-2) and M'(2,-2) are also on the same horizontal line, and the horizontal distance is |x_2' - x_1'| = |2 - (-3)| = 5.

4. According to the definition of similar polygons, the scale factor is the ratio of the lengths of the corresponding sides, so N'M' / NM = 5 / 10 = 1/2.

5. Determine the center of scaling. From the properties of similar shapes, we know that each pair of corresponding points and the center point are collinear.

6. Connect N and N' to form a line, and similarly connect M and M'.

7. Calculate the equations of the lines NN' and MM' and find the intersection of these two lines, which is the center of scaling.

8. The equation of the line NN' is (y + 6) / (x + 2) = (-2 + 6) / (-3 + 2) = -4, where b can be arbitrarily chosen, and the intersection point is found by solving for x and y.

9. Similarly, derive the equation of the line MM', and find the intersection point of the two lines.

10. Using the coordinates of the intersection point found above, determine the coordinates of the center of scaling to be (-4, 2).

11. Through the above reasoning, the final answer is that the scale factor of the center of scaling is 1/2, and the center is (-4, 2).

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 ∠KLM</note> is a geometric figure formed by <note>ray LK</note> and <note>ray LM</note>, which share a common endpoint <note>L</note>. This common endpoint <note>L</note> is called the <note>vertex of angle ∠KLM</note>, and <note>ray LK</note> and <note>ray LM</note> are called the <note>sides of angle ∠KLM</note>.

Definition of Rhombus

<note>In quadrilaterals KLMN and K'L'M'N', all sides KL, LM, MN, NK and K'L', L'M', M'N', N'K' are equal</note>, therefore quadrilaterals KLMN and K'L'M'N' are rhombuses. Additionally, in quadrilateral KLMN, <note>diagonals KM and LN are perpendicular bisectors of each other</note>, that is, <note>diagonals KM and LN intersect at point O</note>, and <note>angle KOM is a right angle (90 degrees)</note>, and <note>KO=OM and LO=ON</note>.

<note>Rhombus KLMN scaled to obtain K'L'M'N'</note>, scale factor = <note>length of L'M'</note> / <note>length of LM</note>, i.e., scale factor = 1/2.


Question:
The diagram shows a convex polygon.

What is the sum of the exterior angle measures, one at each vertex, of this polygon?
 $\Box$ °

Answer: 360°

1. Consider the convex polygon given in the figure, we need to find the sum of its exterior angles. The exterior angle at each vertex is obtained by extending the adjacent side and supplementing the interior angle.

2. According to the 'Polygon Exterior Angle Sum Theorem', we know that the sum of the exterior angles of any polygon is 360°, this theorem applies to any polygon regardless of the number of sides.

3. Adding up the exterior angles at all vertices, since the sum of the exterior angles of a convex polygon is always 360°, we can confirm that we do not need to calculate the specific value of each exterior angle.

4. Based on the above reasoning, considering this is a convex polygon, we do not need to further know the number of sides of the polygon and can directly confirm that the sum of the exterior angles is 360°.

5. Through the above reasoning, the final answer is 360°.

In the figure of this problem, the angle formed by one interior angle of a polygon and the extension of the adjacent side is called the exterior angle of this interior angle. For example, <note>the exterior angle at the vertex is formed by extending the adjacent sides</note>.

Exterior Angle Sum Theorem of Polygon

<note>Angle A</note>, <note>Angle B</note>, <note>Angle C</note>, <note>Angle D</note>, <note>Angle E</note>, <note>Angle F</note>. According to the Exterior Angle Sum Theorem of Polygon, the sum of these exterior angles is equal to 360°, that is <note>Angle A + Angle B + Angle C + Angle D + Angle E + Angle F = 360°</note>.


Question:
The trapezoid M'N'O'P' is a dilation of the trapezoid MNOP. What is the scale factor of the dilation?

Simplify your answer and write it as a proper fraction, an improper fraction, or a whole number.

 $\Box$

Answer: 3

1. Given the coordinates of the vertices of trapezoid MNOP as M(0, -3), N(-3, 0), O(3, 0), P(3, -3), calculate the length of side MP based on the coordinates. According to the distance formula between two points, d = √[(x2 - x1)^2 + (y2 - y1)^2], the length of side MP is √[(3-0)^2 + (-3-(-3))^2] = 3 units.

2. Construct the corresponding trapezoid M'N'O'P', where M'(0, -9), N'(-9, 0), O'(9, 0), P'(9, -9). Similarly, calculate the length of the transformed side M'P'. The length of side M'P' is √[(9-0)^2 + (-9-(-9))^2] = 9 units.

3. In the transformation, the shape and angles remain unchanged, and each part of the segment scales by the same factor. Therefore, the scaling factor can be defined using the ratio of the lengths of the points on the line in the Cartesian coordinate system. In this case, the scaling factor is the length ratio M'P'/MP.

4. Divide the lengths of M'P' and MP to obtain their ratio. Scaling factor = length ratio = 9/3.

5. Thus, the transformation scaling factor is 3.

6. Based on the above reasoning, the final answer is 3.

Definition of Trapezoid

<note>In quadrilateral MNOP, side MP and side ON are parallel</note>, while <note>side MN and side OP are not parallel</note>. Therefore, according to the definition of a trapezoid, quadrilateral MNOP is a trapezoid because it has exactly one pair of parallel sides. Similarly, <note>in quadrilateral M'N'O'P', side M'P' and side O'N' are parallel</note>, while <note>side M'N' and side O'P' are not parallel</note>. Therefore, according to the definition of a trapezoid, quadrilateral M'N'O'P' is a trapezoid because it has exactly one pair of parallel sides.

Distance Formula Between Two Points

<note>The coordinates of point M are (0, -3)</note>, <note>the coordinates of point P are (3, -3)</note>, we need to calculate <note>the distance between point M and point P</note>. According to the distance formula between two points, <note>the distance d between point M and point P</note> can be calculated using the following formula: d = √[(3-0)^2 + (-3-(-3))^2]. <note>In the diagram, the distance between point M and point P is the length of segment MP</note>. <note>The coordinates of point M' are (0, -9)</note>, <note>the coordinates of point P' are (9, -9)</note>, we need to calculate <note>the distance between point M' and point P'</note>. According to the distance formula between two points, <note>the distance d between point M' and point P'</note> can be calculated using the following formula: d = √[(9-0)^2 + (-9-(-9))^2]. <note>In the diagram, the distance between point M' and point P' is the length of segment M'P'</note>.

The polygon <note>MNOP</note> is scaled to obtain the polygon <note>M'N'O'P'</note>, with the scale factor = <note>length of M'P' / length of MP</note>, i.e., scale factor = 3.


Question:
Find the area of △VWX.

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

Answer: 178.3 in ^ 2

1. Given the three sides of triangle △VWX: VW = 19 inches, WX = 32 inches, VX = 20 inches.

2. According to Heron's formula, calculating the area of an irregular triangle requires knowing the semi-perimeter and the lengths of the three sides of the triangle.

3. First, calculate the semi-perimeter s, the formula is: s = (VW + WX + VX) / 2, substituting the known side lengths: s = (19 + 32 + 20) / 2 = 71 / 2 = 35.5 inches.

4. Calculate the area A of the triangle, Heron's formula is: A = √[s * (s - VW) * (s - WX) * (s - VX)].

7. Multiply the above values: 35.5 * 16.5 * 3.5 * 15.5, the product is: 31776.9375.

8. Calculate the square root of the product: √31776.9375 ≈ 178.2608.

9. Round to the nearest tenth, the area of the triangle A ≈ 178.3 square inches.

10. Through the above reasoning, the final answer is 178.3.

Definition of Triangle

In the figure of this problem, the triangle <note>△VWX</note> is a geometric figure composed of three non-collinear points <note>V, W, X</note> and their connecting line segments <note>VW, WX, VX</note>. Points <note>V, W, X</note> are the three vertices of the triangle, and line segments <note>VW, WX, VX</note> are the three sides of the triangle.

Heron's Formula

In the diagram of this problem, the triangle <note>WXV, with sides WX, WV, and VX represented as a, b, and c</note>. First, calculate the semi-perimeter of the triangle <note>s, which is s = (a + b + c) / 2</note>. Then, according to Heron's Formula, the area of the triangle <note>A can be expressed as: A = √[s(s - a)(s - b)(s - c)]</note>. Using this formula, the area of the triangle <note>WXV</note> can be determined.


Question:
Find the area of △UVW.

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

Answer: 277.0 mm ^ 2

1. Given that the three sides of triangle △UVW are UV = 22 mm, VW = 28 mm, UW = 27 mm. According to the problem statement, Heron's formula can be used to calculate the area of the triangle. Heron's formula is: Area A = √[s(s - a)(s - b)(s - c)], where s is the semi-perimeter, and a, b, c are the sides of the triangle.

2. First, calculate the semi-perimeter of the triangle, s = (a + b + c) / 2 = (22 + 28 + 27) / 2.

3. Calculate the value of s: s = 77 / 2 = 38.5.

4. Then, substitute into Heron's formula to continue calculating the area A. Using the side lengths a = 22, b = 28, c = 27: A = √[38.5 * (38.5 - 22) * (38.5 - 28) * (38.5 - 27)].

5. Perform the calculations: 38.5 - 22 = 16.5, 38.5 - 28 = 10.5, 38.5 - 27 = 11.5.

6. Continue the calculation: A = √[38.5 * 16.5 * 10.5 * 11.5].

7. Calculate the product: 38.5 * 16.5 * 10.5 * 11.5 = 76706.4375.

8. Calculate the square root to obtain the area: A = √76706.4375 ≈ 277.0.

9. Through the above reasoning, the final answer is 277.0.

<note>a, b, c are the side lengths of the triangle UV = 22 mm, VW = 28 mm, UW = 27 mm</note>, with the semi-perimeter <note>s = 38.5 mm</note>. Substituting into Heron's formula: A = <note>√[38.5 * (38.5 - 22) * (38.5 - 28) * (38.5 - 27)]</note>. Continuing the calculation, the area of the triangle is A = <note>√76706.4375</note> ≈ 277.0 mm².


Question:
The diagram shows a convex polygon.

What is the sum of the exterior angle measures, one at each vertex, of this polygon?
 $\Box$ °

Answer: 360°

1. First, clarify the definition of the exterior angle of a polygon: at each vertex, the exterior angle is the angle formed by extending the interior angle outward. In a convex polygon, the exterior angle at each vertex and its corresponding interior angle form a linear pair, meaning their sum is 180°.

 2. According to the definition of the exterior angle of a polygon and the definition of a straight angle, the sum of the interior angle and the exterior angle at any vertex of a polygon is 180°.

 3. For any convex polygon, according to the polygon interior angle sum theorem, the sum of the interior angles at all vertices is (n-2)×180°, where n is the number of sides of the polygon.

 4. Since the sum of each interior angle and its corresponding exterior angle is 180°, the sum of the interior angles of the polygon can be indirectly determined through the relationship between the 'sum of all angles of the polygon' and the 'sum of the exterior angles': n×180° is the sum of the interior and exterior angles at the vertices of the polygon.

 5. Subtract the sum of the interior angles (n-2)×180° from the sum of the interior and exterior angles at each vertex, which is n×180°, to get the sum of the exterior angles: total sum of exterior angles = n×180° - (n-2)×180°.

 6. By simplifying, we get the sum of the exterior angles = 2×180° = 360°. This shows that regardless of the number of sides of the convex polygon, the sum of the exterior angles is 360°.

 7. From the above reasoning process, we can conclude that the sum of the exterior angles of a convex polygon is 360°.

In the problem diagram, for example, one of the exterior angles can be defined as the angle formed by extending the adjacent side from the vertex. <note>This angle forms a linear pair with the interior angle</note>.

Polygon Interior Angle Sum Theorem

The sum of the interior angles of a polygon is equal to (n - 2) * 180°, where n represents the number of sides of the polygon.

<note>In the figure of this problem, this is a polygon with 5 sides, where 5 represents the number of sides of the polygon</note>. According to the Polygon Interior Angle Sum Theorem, <note>the sum of the interior angles of this polygon is equal to (5-2) × 180°=540°</note>.

Exterior Angle Sum Theorem of Polygon

In the figure of this problem, each vertex of any convex polygon has an exterior angle. According to the <note>Exterior Angle Sum Theorem of Polygon</note>, <note>the sum of these exterior angles is equal to 360°</note>.

Definition of Straight Angle

A straight angle is formed when a ray rotates around its endpoint and the initial side and terminal side lie on the same line but in opposite directions. A straight angle measures 180 degrees.

<note>The internal angle ray and the corresponding external angle ray are on the same straight line</note>, forming an angle of <note>straight angle</note>, <note>measuring 180 degrees</note>.


Question:
Find the area of △WXY.

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

Answer: 137.9 mm ^ 2

1. Given the triangle △WXY with side lengths WX = 29 mm, XY = 21 mm, WY = 14 mm.

2. According to Heron's formula for the area of a triangle, it is given by: Area = √[s(s-a)(s-b)(s-c)], where s is the semi-perimeter, and a, b, c are the side lengths of the triangle.

3. First, calculate the semi-perimeter s: s = (WX + XY + WY) / 2 = (29 + 21 + 14) / 2 = 32 mm.

4. Substitute the calculated semi-perimeter s and the side lengths into Heron's formula: Area = √[32(32 - 29)(32 - 21)(32 - 14)].

5. Calculate each term: 32 - 29 = 3, 32 - 21 = 11, 32 - 14 = 18.

6. Substitute the values into the calculation: Area = √[32 * 3 * 11 * 18].

7. Calculate the product inside the brackets: 32 * 3 * 11 * 18 = 19008.

8. Calculate the square root: Area = √19008 ≈ 137.8695.

9. The final answer rounded to one decimal place is: 137.9.

10. Through the above reasoning, the final answer is 137.9 square millimeters.

Definition of Triangle

<note>Triangle △WXY</note> is a geometric figure composed of <note>three non-collinear points W, X, Y</note> and their <note>connecting line segments WX, XY, WY</note>. <note>Points W, X, Y are the three vertices of the triangle</note>, and <note>line segments WX, XY, WY are the three sides of the triangle</note>.

Heron's Formula

<note>In the figure of this problem</note>, use <note>Heron's formula to calculate the area of △WXY</note>. Substitute the corresponding values as follows: <note>s = 32 mm</note>, <note>a = WX = 29 mm</note>, <note>b = XY = 21 mm</note>, <note>c = WY = 14 mm</note>. The calculation process is as follows: <note>Area = √[32(32-29)(32-21)(32-14)] = √[32 * 3 * 11 * 18] = √19008 ≈ 137.8695 mm²</note>, rounded to <note>137.9 mm²</note>.


Question:
Find the area of △VWX.

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

Answer: 110.8 m ^ 2

1. Given that the side lengths of triangle △VWX are VW=13 meters, WX=20 meters, VX=29 meters.

2. According to Heron's formula for the area of a triangle, the area A=√[s(s-a)(s-b)(s-c)], where a, b, c are the side lengths of the triangle, and s is the semi-perimeter.

3. First, calculate the semi-perimeter s, s=(13+20+29)/2=31.

4. Substitute into Heron's formula to calculate the area of triangle △VWX: A=√[31*(31-13)*(31-20)*(31-29)].

5. Calculate each part of Heron's formula step by step: (31-13)=18, (31-20)=11, (31-29)=2.

7. Calculate the square root: √12276≈110.7971.

8. Round the result to one decimal place, obtaining 110.8.

9. Through the above reasoning, the final answer is 110.8 square meters.

Definition of Triangle

<note>Triangle △VWX is a geometric figure formed by three non-collinear points V, W, X</note> and their connecting line segments <note>VW, WX, VX</note>. <note>Points V, W, X are the three vertices of the triangle</note>, and <note>line segments VW, WX, VX are the three sides of the triangle</note>, with lengths of 13 meters, 20 meters, and 29 meters, respectively.

Heron's Formula

In the diagram of this problem, the triangle <note>VXW</note> has sides <note>WX, VW, and VX</note>, which are denoted as <note>a, b, and c</note> respectively. First, calculate the semi-perimeter of the triangle <note>s</note>, which is <note>s = (a + b + c) / 2</note>. Then, according to Heron's formula, the area of the triangle <note>A</note> can be expressed as: <note>A = √[s(s - a)(s - b)(s - c)]</note>. Using this formula, the area of the triangle <note>WXV</note> can be determined.


Question:
Find the area of △VWX.

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

Answer: 293.8 km ^ 2

1. Given that in △VWX, the lengths of the three sides are: VX = 47 km, WX = 35 km, WV = 19 km.

2. According to Heron's formula, the area A of the triangle can be given by the following formula: A = √[s(s - a)(s - b)(s - c)], where a, b, c are the lengths of the sides of the triangle, and s is the semi-perimeter, i.e., s = (a + b + c) / 2.

3. For △VWX, let: a = 19 km, b = 35 km, c = 47 km.

5. Substitute the side lengths and semi-perimeter into Heron's formula to calculate the area:

9. Substitute the results into the calculation: A = √[50.5 * 31.5 * 15.5 * 3.5].

11. Calculate the square root of the area A: A = √86298.1875 ≈ 293.7655.

12. Round to one decimal place to get A ≈ 293.8.

13. Through the above reasoning, the final answer is 293.8 square kilometers.

Definition of Triangle

<note>Triangle △VWX</note> is a geometric figure composed of three non-collinear points <note>V, W, X</note> and their connecting line segments <note>VX, WX, WV</note>. Points <note>V, W, X</note> are the three vertices of the triangle, and line segments <note>VX, WX, WV</note> are the three sides of the triangle.

Heron's Formula

In the figure of this problem, in triangle WVX, <note>sides WV, WX, and VX are the three sides of the triangle, denoted as a, b, and c</note>. First, calculate the semi-perimeter s of the triangle, which is <note>s = (a + b + c) / 2</note>. Then, according to Heron's formula, the area A of the triangle can be expressed as: <note>A = √[s(s - a)(s - b)(s - c)]</note>. Using this formula, the <note>area of triangle WXV</note> can be determined.


Question:
Find the area of △UVW.

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

Answer: 865.7 ft ^ 2

1. Given that the lengths of the three sides of triangle UVW are UV = 48 ft, UW = 39 ft, WV = 49 ft, the area of the triangle can be found using Heron's formula.

2. First, calculate the semi-perimeter s, s = (UV + UW + WV) / 2 = (48 + 39 + 49) / 2 = 136 / 2 = 68.

3. According to Heron's formula, the area of the triangle A = sqrt(s * (s - UV) * (s - UW) * (s - WV)).

4. Substitute the given values, A = sqrt(68 * (68 - 48) * (68 - 39) * (68 - 49)).

5. Calculate each difference, 68 - 48 = 20, 68 - 39 = 29, 68 - 49 = 19.

6. Substitute the differences, A = sqrt(68 * 20 * 29 * 19).

7. Calculate the product, 68 * 19 = 1292, then 1292 * 29 = 37468, finally 37468 * 20 = 749360.

8. Take the square root, sqrt(749360) ≈ 865.6558.

9. Round to the nearest tenth, 865.6558 ≈ 865.7.

10. Through the above reasoning, the final answer is 865.7 square feet.

Original: <note>在本题图中，三角形UVW中，边WV, WU和UV分别是三角形的三边，记为a, b和c</note>。首先计算三角形的<note>半周长s，即s = (a + b + c) / 2</note>。然后根据<note>海伦公式</note>，三角形的面积<note>A可以表示为：A = √[s(s - a)(s - b)(s - c)]</note>。通过这个公式，可以求出<note>三角形UVW的面积</note>。

Translation: <note>In the diagram of this problem, in triangle UVW, sides WV, WU, and UV are the three sides of the triangle, denoted as a, b, and c</note>. First, calculate the triangle's <note>semi-perimeter s, i.e., s = (a + b + c) / 2</note>. Then, according to <note>Heron's Formula</note>, the area of the triangle <note>A can be expressed as: A = √[s(s - a)(s - b)(s - c)]</note>. Using this formula, the <note>area of triangle UVW</note> can be determined.