Question: In the figure above, triangles ABC and CDE are equilateral and line segment AE has length 25. What is the sum of the perimeters of the two triangles?

Answer: 75

1. The given conditions are: triangles ABC and CDE are equilateral triangles. According to the properties of equilateral triangles, every side in the triangle is equal. Therefore, side lengths AB=BC=CA, CD=DE=EC.

2. According to the figure, the length of line segment AE is 25, and line segments AC and CE share a common side, so AC + CE = AE.

3. Let the side length of equilateral triangle ABC be x, and let the side length of equilateral triangle CDE be y. Since the triangles are equilateral, AE = AC + CE, so x + y = 25.

4. According to the perimeter formula for equilateral triangles, the perimeter of triangle ABC is 3 times the side length, i.e., 3 * x = 3x.

5. Similarly, the perimeter of triangle CDE is also 3 times the side length, i.e., 3 * y = 3y.

6. The sum of the perimeters of the two triangles is 3x + 3y = 3 * (x + y) = 3 * 25 = 75.

7. Through the above reasoning, the final answer is: 75.

Definition of Equilateral Triangle

An equilateral triangle is a triangle in which all three sides have the same length, and all interior angles are equal (each measuring 60°).

<note>Triangle ABC and triangle CDE are both equilateral triangles</note>. In triangle ABC, <note>sides AB, BC, and CA are of equal length</note>, and <note>angles BAC, ABC, and BCA are equal in measure, each being 60°</note>. In triangle CDE, <note>sides CD, DE, and EC are of equal length</note>, and <note>angles DCE, CDE, and DEC are equal in measure, each being 60°</note>.


Question:
The figure above shows five lines. If l || m, which of the following is NOT equal to 90?

(A) r
(B) s
(C) t
(D) u
(E) v

Answer: r

1. Given that l || m, according to the properties of parallel lines, if two parallel lines are cut by a transversal, then corresponding angles are equal, that is, ∠s and ∠u are equal.

2. According to the figure, ∠u is a right angle, equal to 90°, so ∠s is also equal to 90°.

3. The figure shows that ∠v is a right angle, equal to 90°.

4. Given that ∠v is 90°, because of the principle of corresponding angles of parallel lines, ∠t is also 90°.

5. Since there is an angle of 89°, its corresponding angle ∠r is also 89°.

6. The final conclusion is: ∠r is an angle not equal to 90°.

平行线; 同位角; 内错角

Definition of Parallel Lines

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

<note>Line l and line m</note> are located in the same plane, and they do not intersect. Therefore, according to the definition of parallel lines, <note>line l and line m are parallel lines</note>. These two parallel lines are intersected by a third line <note>transversal</note>, forming several pairs of corresponding angles.

Definition of Perpendicular Lines

Two lines are said to be perpendicular if and only if the angle formed by their intersection is 90 degrees.

<note>The angle U formed by the intersection of lines l and m is 90 degrees</note>, therefore, according to the definition of perpendicular lines, <note>the line is perpendicular to m</note>.

Parallel Postulate 2 of Parallel Lines

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

In the figure of this problem, it is known that <note>l || m</note>, and they are intersected by a third line, forming the following geometric relationships: 1. Corresponding <note>angles: angle s and angle u are equal</note>. #### These relationships illustrate that when two parallel lines are intersected by a third line, <note>corresponding angles are equal, alternate interior angles are equal, and same-side interior angles are supplementary</note>. ##

Definition of Corresponding Angles

Two angles are referred to as corresponding angles if they are formed when two lines, a and b, are intersected by a third line, c, and both angles lie on the same side of the intersecting line c and on the same side of the lines a and b.

In the figure of this problem, <note>two parallel lines l and m</note> are intersected by <note>a transversal</note>, where <note>angle t and angle v are on the same side of the transversal</note>, <note>on the same side of the intersected lines l and m</note>, therefore <note>angle t and angle v are corresponding angles</note>. Corresponding angles are equal, that is, <note>angle t is equal to angle v</note>.


Question: What is the value of x in the figure above?

Answer: 72

1. Given that the degrees of the three angles at the bottom of the triangle are respectively

b^

circ, 126^

circ, and

a^

circ. According to the triangle angle sum theorem: the sum of the three interior angles of a triangle is

180^

circ, it can be concluded that the two angles at the bottom and 126° also form a triangle, so there is an equation

b + 126 + a = 180.

2. Find the sum of the two angles at the bottom of the triangle: from the equation

b + 126 + a = 180,

 solve for

b + a = 54.

 This is the relationship between the sum of the two angles at the bottom.

3. Given that one of the top angles of the large triangle is

x^

circ, the other two angles are respectively

(a+b)^

circ.

4. According to the triangle angle sum theorem, we have

b + a + b + a + x = 180.

5. Using the conclusion obtained in step 2

b + a = 54,

 substitute into the formula in step 4:

x = 180 - 2(54) = 180 - 108.

6. Solve for

x = 72.

 This is the final solution obtained through the above reasoning.

三角形的外角; 普通三角形

Triangle Angle Sum Theorem

The sum of the interior angles of any triangle is 180°.

<note>In the diagram of this problem, the three interior angles of the small triangle are \(a^\text{°}\), \(b^\text{°}\), and \(126^\text{°}\)</note>. According to the Triangle Angle Sum Theorem, <note>\(a^\text{°} + b^\text{°} + 126^\text{°} = 180^\text{°}\)</note>. <note>The three interior angles of the large triangle are \((a+b)^\text{°}\), \((a+b)^\text{°}\), and \(x^\text{°}\)</note>. According to the Triangle Angle Sum Theorem, <note>\(2a^\text{°} + 2b^\text{°} + x^\text{°} = 180^\text{°}\)</note>.


Question:
In the figure above, P, Q, and R lie on the same line. P is the center of the larger circle, and Q is the center of the smaller circle. If the radius of the larger circle is 4, what is the radius of the smaller circle?
(A) 1
(B) 2
(C) 4
(D) 8
(E) 16

Answer: 2

1. The given conditions are that points P, Q, R lie on the same line, P is the center of the larger circle, Q is the center of the smaller circle, and the radius of the larger circle is 4. According to the definition, the radius of a circle is the distance from the center to any point on the circle.

2. According to the problem, the distance from point P to point R is the radius of the larger circle, let PR = 4.

3. Since point R is also on the same line PQ, and the smaller circle passes through point P, point R is a point on the smaller circle.

4. The center of the smaller circle is Q, since there is a point P on the smaller circle, and PR is the radius of the larger circle, so PR is greater than the radius of the smaller circle.

####

5. Also according to the figure, if the diameter of the smaller circle is the radius of the larger circle, then 2×QR = 4, i.e., QR = 2.

6. Through the above reasoning, it is finally concluded that the radius of the smaller circle is 2.

Definition of Radius

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

In the figure of this problem, in the large circle, <note>point P is the center</note>, <note>point R is any point on the circle</note>, <note>line segment PR is the line segment from the center to any point on the circle</note>, therefore <note>line segment PR is the radius of the large circle</note>, and <note>PR=4</note>. In the small circle, <note>point Q is the center</note>, <note>point R is any point on the circle</note>, <note>line segment QR is the line segment from the center to any point on the circle</note>, therefore <note>line segment QR is the radius of the small circle</note>.


Question: If the perimeter of the above rectangle is 72, what is the value of x?

Answer: 18

1. The given condition is that the perimeter of the rectangle is 72, the length of the rectangle is x + 3, and the width is x - 3.

2. According to the formula for the perimeter of a rectangle ##Perimeter = 2 * (Length + Width)##, we can substitute the perimeter into the formula to get 2 * ((x + 3) + (x - 3)) = 72.

3. Expand and simplify the formula to get 2 * (2x) = 72.

4. Further simplify to get 4x = 72.

5. Solve this equation 4x = 72 to get x = 72 / 4 = 18.

6. Check the obtained value of x, substituting x = 18, the length of the rectangle is x + 3 = 18 + 3 = 21, and the width is x - 3 = 18 - 3 = 15.

7. Calculate the perimeter with this length and width to see if it meets the condition: ##Perimeter## = 2 * (21 + 15) = 2 * 36 = 72, which meets the requirement.

8. Through the above reasoning, the final answer is 18.

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>The length of the rectangle is x + 3, and the opposite side is also x + 3</note>, <note>The width of the rectangle is x - 3, and the opposite side is also x - 3</note>. <note>All four vertices of the rectangle are right angles (90 degrees)</note>.

Formula for the Perimeter of a Rectangle

The perimeter of a rectangle is equal to twice the sum of the lengths of its longer side (length) and its shorter side (width), which can be expressed as \( P = 2(l + w) \).

<note>In the figure of this problem, the long side of the rectangle is x + 3, and the short side is x - 3</note>, according to the formula for the perimeter of a rectangle, <note>the perimeter of the rectangle is equal to twice the sum of the lengths of its long side and short side, i.e., P = 2((x + 3) + (x - 3))</note>.


Question: In the triangle above, which of the following must be true?

Answer: p>r

1. Given triangle MNQ, ∠M = 80°, ∠Q = 30°. According to the triangle angle sum theorem (the sum of the interior angles of a triangle is 180°), we can find that ∠N is 180° - 80° - 30° = 70°.

2. Given the length of side MN is 4, the length of side MQ is 4 + r, and the length of side NQ is 4 + p.

3. According to the sine rule (in any triangle, the ratio of a side to the sine of its opposite angle is equal), we can derive: MN/sin(∠Q) = MQ/sin(∠N) = NQ/sin(∠M).

4. Substituting the known values, we get 4/sin(30°) = (4 + r)/sin(70°) = (4 + p)/sin(80°).

5. Calculating sin(30°) = 0.5, sin(70°) and sin(80°) can be approximated or calculated to be approximately 0.9397 and 0.9848 respectively.

6. Through the equation 4/sin(30°) = (4 + p)/sin(80°), we further derive 8 = (4 + p)/0.9848, solving for 4 + p ≈ 7.8784, we can find p ≈ 3.8784.

7. Similarly, using the equation 4/sin(30°) = (4 + r)/sin(70°), we get 8 = (4 + r)/0.9397, solving for 4 + r ≈ 7.5181, we can find r ≈ 3.5181.

8. Comparing, we find that (4 + p) is greater than (4 + r). Thus, we conclude that p > r.

9. Through calculation and derivation, we confirm that the answer p > r is necessarily correct.

普通三角形; 三角形的外角

Triangle Angle Sum Theorem

In the diagram of this problem, the sum of the interior angles <note>∠M</note>, <note>∠N</note>, and <note>∠Q</note> of triangle <note>MNQ</note> is 180 degrees, specifically expressed as <note>∠M + ∠N + ∠Q = 180°</note>, that is, 80° + 70° + 30° = 180°.

Sine Theorem

In any triangle, the ratio of the length of each side to the sine of its opposite angle is constant and equal to the diameter of the circumcircle, i.e., \( \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = 2r = D \) (where \( r \) is the radius of the circumcircle, and \( D \) is its diameter).

In <note>any triangle MNQ</note>, the sides <note>MN, MQ, and NQ correspond to angles ∠Q, ∠N, and ∠M</note> respectively. According to the <note>Sine Theorem</note>, the ratio of the length of each side to the sine of its opposite angle is equal to the <note>diameter of the circumscribed circle</note>, that is: <note>MN/sin(∠Q)=MQ/sin(∠N)=NQ/sin(∠M) = 2r = D</note> (where <note>r is the radius of the circumscribed circle, and D is the diameter</note>).


Question: What is the perimeter of the trapezoid above?

Answer: 80

1. In the trapezoid given in the problem, the upper base and lower base are 20 and unknown respectively, the other two sides are 17 and 15 respectively, and the two bases are parallel with a right angle. Let the upper base be AB, i.e., AB = 20, and the right-angle side BC = 15, ##hypotenuse DA## = 17, find the perimeter of the trapezoid.

2. Since the trapezoid has a right angle, we can use the Pythagorean theorem to calculate the length of ##lower base CD##. ##Draw a perpendicular line from point A to CD, intersecting CD at point E##.

3. In the ##right triangle ADE##, ##AE## is equal to the length of the side parallel to the trapezoid, so ##AE## = BC = 15.

4. According to the Pythagorean theorem, ##DA^2 = AE^2 + DE^2##, where ##DA## = 17, ##AE## = 15, therefore: 17^2 = 15^2 + ##DE^2##, 289 = 225 + ##DE^2##, ##DE^2## = 64, thus ##DE## = 8.

5. Therefore, the length of ##lower base CD## is obtained, ##CD## = AB + ##DE## = 20 + 8 = 28.

6. The perimeter of the trapezoid is equal to the sum of the four sides, i.e., AB + BC + CD + AD = 20 + 15 + ##28 + 17##.

7. After calculation, the perimeter of the trapezoid is 80.

8. The final result is option (D) 80.

梯形; 线段; 直角三角形

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, in quadrilateral ABCD, <note>side AB and side DC are parallel</note>, while <note>side AD and side BC are not parallel</note>. Therefore, according to the definition of a trapezoid, <note>quadrilateral ABCD is a trapezoid</note> because it <note>has only one pair of parallel sides</note>.

Pythagorean Theorem

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

In the right triangle <note>ADE</note>, <note>∠AED is a right angle (90 degrees)</note>, sides DE and <note>AE</note> are the legs, and <note>side AD</note> is the hypotenuse. Therefore, according to the Pythagorean Theorem, <note>DA^2 = AE^2 + DE^2</note>, that is, <note>17^2 = 15^2 + DE^2</note>.


Question: In the figure above, AB, CD, and EF intersect at P. If r = 90, s = 50, t = 60, u = 45, and w = 50, what is the value of x?

Answer: 65

1. Given lines AB, CD, and EF intersect at point P, and the degrees of each angle are provided.

2. According to the triangle angle sum theorem, in triangle BPF, ∠BPF = 180° - ∠s - ∠t = 180° - 50° - 60° = 70°. Similarly, in triangle ADP, according to the triangle angle sum theorem, ∠APD can be found as 180° - 45° - 50° = 85°.

According to the vertical angle theorem, ∠APD = ∠BPC.

3. Since ∠EPC + ∠BPC + ∠BPF = 180°, substituting the values, we get: ∠CPE = 180° - 70° - 85° = 25°. In triangle CPE, according to the triangle angle sum theorem, x = 180° - ∠CPE - 90° = 90° - 25° = 65°.

4. Through the above reasoning, the final answer is 65°.

Definition of Vertical Angles

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

In the figure of this problem, two intersecting lines <note>CD and AB intersect at point P</note>, forming four angles: <note>angle BPC, angle BPD, angle DPA, and angle CPA</note>. According to the definition of vertical angles, <note>angle CPB and angle APD are vertical angles</note>, <note>angle BPD and angle APC are vertical angles</note>. Since vertical angles are equal, <note>angle APD = angle CPB</note>, <note>angle BPD = angle APC</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>Ray PE</note> rotates around the endpoint <note>P</note> until it forms a straight line with the initial side, forming <note>straight angle PEF</note>. According to the definition of a straight angle, the measure of a straight angle is 180 degrees, that is, <note>angle PEF = 180 degrees</note>.

Triangle Angle Sum Theorem

<note>Angle C, Angle E, and Angle PCE are the three interior angles of triangle PCE</note>, according to the Triangle Angle Sum Theorem, <note>Angle PCE + Angle E + Angle C = 180°</note>. <note>Angle B, Angle F, and Angle BPF are the three interior angles of triangle PBF</note>, according to the Triangle Angle Sum Theorem, <note>Angle B + Angle F + Angle BPF = 180°</note>. <note>Angle A, Angle D, and Angle APD are the three interior angles of triangle PAD</note>, according to the Triangle Angle Sum Theorem, <note>Angle A + Angle D + Angle APD = 180°</note>.


Question:
In the figure above, which of the following is greatest?

(A) a
(B) b
(C) c
(D) d
(E) e

Answer: b和d

1. Given angles a, b, c, d, e satisfy b° + 60° = 180°, e° + 70° = 180°, thus we get b = 120, e = 110. Also, since a and the 60° angle are vertical angles, we get a = 60.

2. According to the interior angle sum theorem, the sum of the three interior angles of a triangle is 180°, therefore angle c° + 60° + 70° = 180°, we get angle c = 50.

3. Thus d = 180 - c = 120.

4. From the above results, the angle size relationship is b = d > e > a > c.

5. Therefore, b and d are the greatest.

三角形的外角; 普通三角形; 内错角

<note>In a triangle, the angles 60 degrees, 70 degrees, and angle C are the three interior angles of the triangle</note>, according to the Triangle Angle Sum Theorem, <note>60 degrees + 70 degrees + angle C = 180°</note>.

Definition of Vertical Angles

In the figure of this problem, <note>angle a and the 60-degree angle form vertical angles</note>, and since <note>vertical angles are equal</note>, therefore <note>angle a = 60 degrees</note>.

Definition of Linear Pair of Angles

Two angles are in a linear pair if they share a common side, and their other sides are extensions of one another in opposite directions.

<note>Angle c and angle d share a common side</note>, and their other sides are extensions in opposite directions, so <note>angle c and angle d are supplementary adjacent angles</note>. Similarly, <note>the 70-degree angle and angle e share a common side</note>, and their other sides are extensions in opposite directions, so <note>the 70-degree angle and angle e are supplementary adjacent angles</note>.


Question: In the above figure, the side AC of triangle ABC lies on the line l. What is x in terms of k?

Answer: 60-k

1. According to the definition of adjacent supplementary angles, ∠BCA = 180° - 120° = 60°.

2. Since ∠(120 - k) is an exterior angle of triangle ABC, we have: ∠(120 - k) = ∠B + ∠BCA = x + 60°.

3. Solving the equation, we get: x = 60 - k.

普通三角形; 三角形的外角; 线段

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 <note>formed by the extension of side CA and side AB of triangle ABC's interior angle ∠BAC</note> is called the exterior angle of interior angle ∠BAC.

Exterior Angle Theorem of Triangle

An exterior angle of a triangle is equal to the sum of the measures of the two non-adjacent interior angles.

<note>In triangle ABC, angle BAD is an exterior angle of the triangle</note>, <note>angles B and BCA are the two interior angles that are not adjacent to the exterior angle BAD</note>, according to the Exterior Angle Theorem of Triangle, <note>the exterior angle BAD is equal to the sum of the two non-adjacent interior angles B and BCA</note>, that is, <note>angle BAD = angle B + angle BCA</note>.

Definition of Linear Pair of Angles

<note>Angle BCA and angle 120 degrees share a common side BC</note>, their other sides are extensions in opposite directions, so <note>angle BCA and 120 degrees are adjacent supplementary angles</note>.


Question: In the figure above, point O lies on line AB. What is the value of x?

Answer: 144

1. Given that point O lies on line AB, according to the property of linear angles, we get ∠BOA = 180°.

####

##2.## Since these angles are adjacent to each other on line AB with point O as the vertex, we can sum these angles to equal ∠BOA.

##3.## Therefore, (x°/6) + (x°/4) + (x°/3) + (x°/2) = 180°.

##4.## Convert these fractions to have a common denominator for addition, i.e., x/6 + x/4 + x/3 + x/2.

##5.## Find the common denominator, which is 12, and multiply both the numerator and denominator by the common denominator's multiples: (2x/12) + (3x/12) + (4x/12) + (6x/12) = 180°.

##6.## Simplify the numerator and combine like terms, i.e., (2x + 3x + 4x + 6x)/12 = 180°.

##7.## Simplify the denominator and combine to get (15x/12) = 180°.

##8.## Eliminate the denominator 12, yielding: 15x = 2160°.

##9.## Then divide by 15 to get: x = 144°.

##10.## Through the above reasoning, the final answer is x = 144°.

In the figure of this problem, point <note>O is on line AB</note>, angle <note>(x°/6) + angle (x°/4)</note> and angle <note>(x°/3) + angle (x°/2)</note> these two large angles have a <note>common side</note>, their other sides are respectively <note>OA and OB, and they are opposite extensions of each other</note>, therefore angle <note>(x°/6) + angle (x°/4)</note> and angle <note>(x°/3) + angle (x°/2)</note> these two large angles are adjacent supplementary angles, the sum of these two large angles is 180°, that is <note>(x°/6) + (x°/4) + (x°/3) + (x°/2) = 180°.</note>

Definition of an Angle

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

<note>Angle (x°/6), Angle (x°/4), Angle (x°/3), Angle (x°/2)</note> are geometric figures composed of two rays, which share a common endpoint <note>O</note>. This endpoint <note>O</note> is called the vertex of <note>Angle (x°/6), Angle (x°/4), Angle (x°/3), Angle (x°/2)</note>, and the two rays become the sides of <note>Angle (x°/6), Angle (x°/4), Angle (x°/3), Angle (x°/2)</note>.


Question: In the figure above, the lengths and widths of rectangles A, B, C, and D are all integers. The areas of rectangles A, B, C, and D are all integers. The areas of rectangles A, B, and C are 35, 45, and 36 respectively. What is the area of the entire figure?

Answer: 144

1. Given that the area of rectangle A is 35, let the side lengths of rectangle A be a and b, then we have ab = 35. Since the factors of 35 can be (1,35), (5,7), (7,5), and (35,1), and the length and width of the rectangle are integers, a and b can be 5 and 7, or 7 and 5, or 35 and 1, or 1 and 35.

2. Given that the area of rectangle B is 45, let the side lengths of rectangle B be b and c, then we have bc = 45. Since the factors of 45 can be (1,45), (3,15), (5,9), (9,5), (15,3), and (45,1), and the value of b should be consistent with rectangle A, therefore, b is 1 when c is 45, or b is 5 when c is 9.

3. Given that the area of rectangle C is 36, let the side lengths of rectangle C be c and d, then we have cd = 36. Since the factors of 36 can be (1,36), (2,18), (3,12), (4,9), (6,6), (9,4), (12,3), (18,2), and (36,1), and the value of c should remain consistent, thus choosing c as 9 when d is 4.

4. Due to the choices of b and c, we select b = 5, c = 9 to remain consistent, thus d = 4.

5. Because the analysis of the length and width of D needs to be consistent with the adjacent side lengths of A and C, we also specify it as 7 x 4, with an area of 28.

6. The area of rectangle D is (7 x 4) = 28.

7. The total area of the entire figure is A + B + C + D, combined as 35 + 45 + 36 + 28.

9. Combining the above steps, we obtain the total area of the entire figure as 144.

矩形; 普通四边形; 平行线

Definition of Rectangle

<note>Quadrilateral A, B, C, D is a rectangle</note>, its <note>interior angles are all right angles (90 degrees)</note>, and <note>each pair of opposite sides of the rectangle are parallel and equal in length</note>.

Formula for the Area of a Rectangle

The area of a rectangle is equal to its length multiplied by its width.

<note>In rectangle A, sides 5 and 7 are the length and width of the rectangle</note>, so the area of the rectangle = 5 * 7 = 35;
<note>In rectangle B, sides 5 and 9 are the length and width of the rectangle</note>, so the area of the rectangle = 5 * 9 = 45;
<note>In rectangle C, sides 9 and 4 are the length and width of the rectangle</note>, so the area of the rectangle = 9 * 4 = 36;
<note>In rectangle D, sides 7 and 4 are the length and width of the rectangle</note>, so the area of the rectangle = 7 * 4 = 28.


Question: In the figure above, CDE is an equilateral triangle, ABCE is a square with an area of 1. What is the perimeter of the polygon ABCDE?

Answer: 5

1. Given that ABCE is a square and its area is 1, the side lengths of the square are AB = BC = CE = EA = 1.

2. From the problem, triangle CDE is an equilateral triangle, and we know that all sides of an equilateral triangle are equal.

3. Therefore, CD = DE = CE.

4. According to the properties of the square, CE = 1, so CD = DE = 1.

5. The perimeter of polygon ABCDE is AB + BC + CD + DE + EA.

6. Adding the lengths of each side, we get: Perimeter = 1 + 1 + 1 + 1 + 1.

7. From the above reasoning, we finally get the answer as 5.

正方形; 等边三角形; 线段; 五边形

Definition of Square

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

<note>Side AB, side BC, side CE, and side EA are equal</note>, and <note>angle ABC, angle BCE, angle CEA, and angle EAB are all right angles (90 degrees)</note>, so ABCE is a square.

Definition of Equilateral Triangle

An equilateral triangle is a triangle in which all three sides have the same length, and all interior angles are equal (each measuring 60°).

<note>Triangle CDE is an equilateral triangle</note>. <note>Sides CD, DE, and CE are of equal length</note>, and <note>angles CDE, DCE, and CED are equal, each being 60°</note>. Since <note>CE = 1 (square side length)</note>, therefore <note>CD = DE = 1</note>.


Question:
In the figure above, point B lies on AC. If x and y are integers, which of the following is a possible value of x?

(A) 30
(B) 35
(C) 40
(D) 50
(E) 55

Answer: 30

1. According to the problem statement, we know point B lies on line AC, ##let## angle ABE = x degrees, angle EBC = ##3y## degrees.####

2. According to the ##definition of supplementary angles##, ##∠ABC## is 180 degrees. Therefore, we can derive the equation x + ##3y## = 180.

3. Given that x and y are integers, this means the values of x and y must be positive integers and satisfy x + ##3y## = 180.

4. We need to find the possible integer values of x. Therefore, we can verify the given options to see which values satisfy x = 180 - ##3y## while ensuring x and y are integers.

5. The multiple-choice options are 5 values: 30, 35, 40, 50, 55. Calculate x = 180 - ##3y## for each option and verify the suitable values.

6. By calculating and verifying all options, check each option x, under x = 180 - ##3y##, the result is an integer. Therefore, the answer from the options is x=30, which satisfies the condition, y=##50##, meeting the integer requirement.

直线; 邻补角; 对顶角

Definition of Linear Pair of Angles

<note>Angle ABE and angle EBC have a common side BE</note>, <note>their other sides BA and BC are extensions in opposite directions</note>, so angle ABE and angle EBC are adjacent supplementary angles, therefore x + ##3y## = 180 degrees.


Question: In isosceles triangle ABC above, AM and CM are the angle bisectors of angle BAC and angle BCA. What is the measure of angle AMC?

Answer: 110

1. Given that triangle ABC is an isosceles triangle, where ∠ABC = 40°, AB = BC, according to the properties of isosceles triangles, ∠BAC = ∠BCA in △ABC.

2. Let ∠BAC = ∠BCA = θ°, since the sum of the interior angles of a triangle is 180°, we have ∠BAC + ∠ABC + ∠BCA = 180°, that is, θ° + 40° + θ° = 180°. Combining like terms, we get 2θ + 40 = 180.

3. Solving 2θ + 40 = 180, we find 2θ = 140, that is, θ = 70°, so ∠BAC = ∠BCA = 70°.

4. Since AM and CM are the angle bisectors of ∠BAC and ∠BCA respectively, according to the properties of angle bisectors, ∠BAM = ∠MAC = 35°, and ∠BCM = ∠MCA = 35°.

5. Therefore, we can find ∠AMC = 180 - ##(∠MAC + ∠MCA)##. Substituting the known values, ∠AMC = 180 - (35° + 35°) = 180 - 70 = 110°.

6. Through the above reasoning, the final answer is 110°.

Definition of Isosceles Triangle

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

<note>In triangle ABC, side AB and side BC are equal</note>, therefore triangle ABC is an isosceles triangle.

Definition of Angle Bisector

An angle bisector is a line that originates from the vertex of an angle and divides the angle into two congruent angles.

<note>The vertex of angle BAC is point A</note>, from <note>point A a line AM is drawn</note>, this line divides <note>angle BAC into two equal angles</note>, namely <note>angle BAM and angle MAC are equal</note>. Therefore, <note>line AM is the angle bisector of angle BAC</note>. Similarly, <note>the vertex of angle BCA is point C</note>, from <note>point C a line CM is drawn</note>, this line divides <note>angle BCA into two equal angles</note>, namely <note>angle BCM and angle MCA are equal</note>. Therefore, <note>line CM is the angle bisector of angle BCA</note>.

Triangle Angle Sum Theorem

The sum of the interior angles of any triangle is 180°.

Original: <note>In triangle ABC</note>, angle BAC, angle ABC, and angle BCA are the three interior angles of triangle ABC. According to the Triangle Angle Sum Theorem, <note>angle BAC + angle ABC + angle BCA = 180°</note>.##<note>In triangle MAC</note>, angle MAC, angle AMC, and angle MCA are the three interior angles of triangle MAC. According to the Triangle Angle Sum Theorem, <note>angle MAC + angle AMC + angle MCA = 180°</note>.##

Properties of Isosceles Triangle

In an isosceles triangle, the angles opposite the equal sides are equal, and the sides opposite the equal angles are congruent.

In the <note>isosceles triangle ABC</note>, <note>sides AB and BC are equal</note>. Therefore, according to the properties of an isosceles triangle, the angles opposite the equal sides are equal, that is, <note>angle BAC = angle BCA</note>.


Question: In the figure above, what is the value of x ?

Answer: 50

1. Given that two lines are drawn inside the large triangle, the two lines intersect outside the triangle forming a small triangle. The vertical angles of the two interior angles of the small triangle are 55 degrees and 75 degrees respectively. According to the equality of vertical angles, the two interior angles of the small triangle are 55 degrees and 75 degrees respectively.

2. According to the triangle angle sum theorem, the sum of the interior angles of the small triangle is 180 degrees. That is, x = 180 - 55 - 75. Calculating the value of x gives 50 degrees.

3. Through the above reasoning steps, the final answer is 50.

三角形的外角; 普通三角形

Triangle Angle Sum Theorem

<note>In the small triangle, the 55-degree angle, the 75-degree angle, and angle x are the three interior angles of the triangle</note>. According to the Triangle Angle Sum Theorem, <note>x + 55 degrees + 75 degrees = 180°</note>.

Definition of Vertical Angles

The two interior angles in the <note>small triangle</note> respectively form <note>vertical angles with the 55-degree angle and the 75-degree angle marked in the figure</note>, so <note>the two interior angles are respectively equal to 55 degrees and 75 degrees</note>.


Question: The figure above is composed of 25 small triangles that are congruent and equilateral. If the area of △DFH is 10, what is the area of △AFK?

Answer: 62.5

1. From the figure, it is known that △DFH is composed of 4 small triangles. Given that the area of △DFH is 10, and since the congruent small triangles have equal areas, the area of a single small triangle can be calculated as 10/4=2.5.

2. Because △AFK is composed of 25 congruent small triangles, the area of △AFK is 2.5*25=62.5.

等边三角形; 平移; 普通三角形

Definition of Equilateral Triangle

In the figure of this problem, <note>the small triangle, △DFH, and △AFK are all equilateral triangles</note>. <note>The three sides of the small triangle are of equal length, and all three interior angles are 60°</note>. <note>The sides DF, DH, and FH of △DFH are of equal length, and the angles ∠DFH, ∠FDH, and ∠DHF are all 60°</note>. <note>The sides AF, AK, and FK of △AFK are of equal length, and the angles ∠AFK, ∠FAK, and ∠KAF are all 60°</note>.


Question: In the above triangle ABC, AB = AC, E is the midpoint of segment AB, and D is the midpoint of segment AC. If AE = x and ED = 4, find the length of BC.

Answer: 8

1. Given that AB = AC, and E and D are the midpoints of segments AB and AC respectively, according to the definition of midpoint, we get AE = EB = x, AD = DC = x.

####

##2.## In ##triangle ABC##, according to the ##midline theorem (points E and D are the midpoints of AB and AC)##. AB = AC, so the length of BC is 2*ED=8.

####

等腰三角形; 中点; 线段

Midpoint of a Line Segment

A midpoint of a line segment is the point that divides the line segment into two equal parts.

In the figure of this problem, <note>the midpoint of line segment AB is point E</note>, <note>the midpoint of line segment AC is point D</note>. According to the definition of the midpoint of a line segment, <note>point E divides line segment AB into two equal parts</note>, that is, <note>the lengths of line segments AE and EB are equal</note>; <note>point D divides line segment AC into two equal parts</note>, that is, <note>the lengths of line segments AD and DC are equal</note>. <note>That is, AE = EB = x, AD = DC = x</note>.

Triangle Midline Theorem

In a triangle, if a line segment connects the midpoints of two sides, then this line segment is parallel to the third side and is equal to half the length of the third side.

In the diagram of this problem, in triangle ABC, point E is the midpoint of <note>side AB</note>, point D is the midpoint of <note>side AC</note>, and segment ED connects these two midpoints. According to the Triangle Midline Theorem, segment <note>ED is parallel to the third side BC</note> and is equal to half of the third side <note>BC</note>, i.e., <note>ED || BC</note>, and <note>ED = 1/2 * BC</note>. Therefore, <note>BC = 2 * ED = 2 * 4 = 8</note>.


Question: In the figure above, points A and B are located on a circle with center O. If x = 80, what is the value of z?

Answer: 80

1. Given that points A and B are on the circle with center O, therefore, OA = OB, which means triangle OAB is an isosceles triangle.

2. According to the properties of isosceles triangles, we know angle A is equal to angle B, that is, z = x = 80 degrees.

3. Hence, we conclude that z = 80 degrees.

Definition of Isosceles Triangle

<note>Side OA and side OB are equal</note>, therefore <note>triangle OAB is an isosceles triangle</note>.

Properties of Isosceles Triangle

In the figure of this problem, <note>isosceles triangle OAB</note>, <note>sides OA and OB are equal</note>. Therefore, according to the properties of isosceles triangles, <note>the angles opposite the equal sides are equal</note>, i.e., <note>angle A = angle B</note>.

Definition of Radius

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

In the figure of this problem, <note>circle O</note>, <note>point O is the center of the circle</note>, <note>point A is any point on the circle</note>, <note>segment OA is the segment from the center to any point on the circle</note>, therefore <note>segment OA is the radius of the circle</note>. Similarly, <note>point O is the center of the circle</note>, <note>point B is any point on the circle</note>, <note>segment OB is the segment from the center to any point on the circle</note>, therefore <note>segment OB is the radius of the circle</note>.


Question: In the figure above, the radius of the circle centered at O is 7, AB is the diameter, and AC = BC. What is the area of triangle ABC?

Answer: 49

1. Given conditions: The center of the circle is O, the radius is 7, AB is the diameter, AC = BC, and triangle ABC is a right triangle, ∠ACB = 90°.

2. According to the theorem of the angle subtended by a diameter, this theorem states: If the vertex of an angle is on the circle and the two sides of the angle pass through the endpoints of the diameter, then the angle is a right angle. In this problem, ∠ACB is the angle subtended by the chord AB, and AB is the diameter, so ∠ACB is a right angle.

3. In the right triangle ABC, AB is the diameter and equals twice the radius, so AB = 2 * 7 = 14.

4. Triangle ABC is an isosceles right triangle because AC = BC (isosceles), and ∠ACB = 90° (right angle). ####

5. Since AC = BC and ∠ACB = 90°, ##according to the Pythagorean theorem##, we know: AC and BC are equal in length, denoted as x, ## AB ^2 = x ^2 + x ^2 ##, the length of the hypotenuse AB is x√2 = 14.

6. To find the specific value of the side length x, x√2 = 14, we get x = 14/√2 = 14√2 / 2 = 7√2.

7. According to the ##triangle area formula (using sine function)##, area = 1/2 * AC * BC * sin(∠ACB). Since ∠ACB = 90°, sin90° = 1. Substituting AC = BC = 7√2, we get area = 1/2 * 7√2 * 7√2 * 1 = 1/2 * 49 * 2 = 49.

8. Through the above reasoning, the final answer is 49.

直角三角形; 等腰三角形; 圆; 圆周角; 线段

Definition of Diameter

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

<note>AB is the diameter</note>, connecting <note>the center O</note> and <note>points A and B on the circumference</note>, with a length of <note>2 times the radius</note>, i.e., AB = 14.

Definition of Isosceles Triangle

In the figure of this problem, in triangle ABC, <note>side AC and side BC are equal</note>, therefore <note>triangle ABC is an isosceles triangle</note>.

Definition of Right Triangle

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

<note>In triangle ABC, angle ACB is a right angle (90 degrees)</note>, therefore triangle ABC is a right triangle. <note>Side AC and side BC are the legs, and side AB is the hypotenuse</note>.

Corollary to the Inscribed Angle Theorem 2: The Angle Subtended by the Diameter

The angle subtended by the diameter of a semicircle is a right angle; conversely, if an inscribed angle is 90°, then the chord subtending that angle is a diameter.

<note>In the diagram of this problem, in circle O, the angle subtended by the diameter AB at the circumference, ∠ACB, is a right angle (90 degrees)</note>.

Definition of Isosceles Right Triangle

An isosceles right triangle is a triangle with two sides of equal length and one angle measuring 90 degrees.

<note>Triangle ABC is an isosceles right triangle</note>, where <note>angle ACB is a right angle (90 degrees)</note>, and <note>sides AC and BC are the equal legs of the right triangle</note>.

Triangle Area Formula (Using Sine Function)

The area \( S \) of any triangle can be expressed as \( S = \frac{1}{2} \times a \times b \times \sin(C) \), where \( a \) and \( b \) are the lengths of two sides, and \( C \) is the included angle between these two sides.

In <note>triangle ABC</note>, <note>side AC and side BC are respectively a and b, and angle ACB is the included angle between these two sides C</note>. According to the triangle area formula, <note>the area S of triangle ABC</note> can be expressed as <note>S = (1/2) * a * b * sin(C), that is, S = (1/2) * AC * BC * sin(ACB)</note>.


Question: In the figure above, lines l and m are not parallel. Which of the following cannot be the value of x?

Answer: 90

1. Given conditions: lines l and m are not parallel, and there is a line k perpendicular to line l, with the intersection point on line l forming a right angle. If line k intersects line m below the intersection point at an angle of x degrees.

2. When l and m are parallel, according to the parallel postulate 2, corresponding angles are equal, so x=90 degrees.

3. Since lines l and m are not parallel, x cannot be 90 degrees.

垂线; 邻补角; 对顶角; 内错角; 同旁内角

Definition of Perpendicular Lines

Two lines are said to be perpendicular if and only if the angle formed by their intersection is 90 degrees.

<note>Line k</note> and <note>line l</note> intersect to form a <note>90-degree angle</note>, so according to the <note>definition of perpendicular lines</note>, <note>line k and line l are perpendicular to each other</note>.

Parallel Postulate 2 of Parallel Lines

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

<note>Two parallel lines l and m</note> are intersected by <note>a third line k</note>, forming the following geometric relationship: 1. <note>Corresponding angles: the angle marked with a right angle in the figure is equal to angle x</note>.

Definition of Corresponding Angles

Two angles are referred to as corresponding angles if they are formed when two lines, a and b, are intersected by a third line, c, and both angles lie on the same side of the intersecting line c and on the same side of the lines a and b.

<note>Two parallel lines l and m</note> are intersected by <note>a line k</note>, where <note>the angle marked with a right angle and angle x are on the same side of the intersecting line k, on the same side of the intersected lines l and m</note>, therefore <note>the angle marked with a right angle and angle x are corresponding angles</note>. Corresponding angles are equal, that is <note>x=90 degrees</note>.


Question: Four lines pass through the center of the rectangle as shown in the figure. What fraction of the area of the rectangle is the shaded part?

Answer: 1/4

1. Observe the figure given in the problem. According to the known conditions, there are four lines intersecting at the center of the rectangle, dividing the rectangle into several triangles.

2. From the structure of the figure, it can be decomposed into eight triangles, among which, ##the shaded triangles are staggered##.

##3. In the rectangle, the diagonals divide the rectangle into four triangles with equal areas##.

4. Based on the symmetry in the figure, we assume the length of the rectangle is a and the width is b, then its total area is a*b.

##5. Since the rectangle is a centrally symmetric figure, the lines passing through the center of the rectangle divide the figure into pairs of symmetric shapes. Additionally, the other two lines divide the upper and lower large triangles into six smaller triangles, with pairs of equal areas##.

6. Using the symmetry strategy and uniform tiling method, ##we get that the area of the shaded part equals the area of the lower large triangle##.

##7. Since the diagonals divide the rectangle into four large triangles with equal areas, the area of the shaded part is a*b/4##.

8. Through the above reasoning, the final answer is 1/4.

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 (≅).

The original text: <note>A quadrilateral is a rectangle</note>, its interior angles are all <note>right angles (90 degrees)</note>, and <note>the opposite sides are parallel and equal in length</note>. The center of the rectangle is <note>the intersection of the four lines</note>, and <note>the opposite sides are a and b, respectively</note>.

Symmetry

Symmetry refers to a geometric figure or pattern remaining invariant under certain operations such as rotation, reflection, or translation.

<note>In the figure of this problem, the shape has symmetry. Specifically, the shape is symmetric about the</note> intersection point of the diagonals <note>of the rectangle, which is the center of symmetry for the rectangle. Each part of the rectangle has a symmetric corresponding part on the other side of the center of symmetry.</note>

Property of Diagonals in a Rectangle

In a rectangle, the diagonals are equal in length and bisect each other.

The <note>diagonals of a rectangle</note> are equal and <note>bisect each other</note>, that is, the <note>intersection point of the two diagonals</note> is the <note>midpoint of the two diagonals</note>.


Question: Point O is the center of both circles in the figure above. If the circumference of the large circle is 36 and the radius of the small circle is half of the radius of the large circle, what is the length of the darkened arc?

Answer: 4

1. Given conditions: The circumference of the large circle is 36, and the radius of the small circle is half of the radius of the large circle. Let the radius of the large circle be R, and the radius of the small circle be R/2.

2. According to the circumference formula C = 2πr, the circumference of the large circle is 36, from which we can find the radius R of the large circle. Calculation is as follows: 36 = 2πR, thus R = 36 / 2π = 18 / π.

3. From the radius R of the large circle, we know the radius of the small circle is R/2, that is 18 / π / 2 = 9 / π.

4. The problem requires calculating the length of the darkened arc on the small circle, and it is known that the central angle of the darkened arc is 80°.

5. According to the arc length formula L = rθ (where r is the radius and θ is the central angle in radians), we need to convert 80° to radians. Using the formula to convert degrees to radians: 80° * π / 180° = 4π / 9.

6. Substitute the radius of the small circle and the radians into the arc length formula to get the arc length L: L = (9 / π) * (4π / 9) = 4.

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

Circumference Formula of Circle

The circumference of a circle is the length around the circular boundary. It can be calculated using the formula \( C = 2\pi r \), where \( C \) represents the circumference and \( r \) represents the radius.

<note>In the figure of this problem, in the large circle, point O is the center of the circle, and the segment from O to the circumference of the large circle is the radius R</note>. According to the circumference formula of the circle, the circumference of the circle <note>C is equal to 2π multiplied by the radius R, i.e., C=2πR</note>. Given that the circumference of the large circle is 36, we can find <note>the radius of the large circle R=36/2π=18/π</note>.

Formula for the Length of an Arc of a Sector

The length \( L \) of the arc of a sector is equal to the central angle \( \theta \) (measured in radians) multiplied by the radius \( r \): \( L = \theta r \).

In the figure of this problem, the arc length of the dark arc part of the small circle is required to be calculated. The small circle's <note>radius is 9/π</note>, <note>central angle is 80°</note>. <note>80° converted to radians is 4π/9</note>, so according to the <note>formula for the length of an arc of a sector L=rθ</note>, the arc length is <note>L=(9/π)*(4π/9)=4</note>.

Definition of Radius

<note>Point O is the center of both the large circle and the small circle</note>. Let the radius of the large circle be R, then the radius of the small circle is R/2. Given that the circumference of the large circle is 36, we get <note>the radius of the large circle R=18/π</note>, thus we find <note>the radius of the small circle is R/2=9/π</note>.


Question: In the figure above, three lines intersect at a point. If f = 85 and c = 25, what is the value of a?

Answer: 70

1. According to the problem statement, three lines intersect at one point. It is known that ∠f = 85°, ∠c = 25°.

2. Since ∠f, ∠a, and ##∠g## are a set of angles at the intersection point, they satisfy the sum of angles being 180°, i.e., ∠f + ∠a + ##∠g## = 180°.

3. ##According to the definition of a straight angle,## substituting the known data, we get 85° + a + 25° = 180°.

4. Solving this equation, we get a = 180° - 85° - 25°.

5. Calculating, we get a = 70°.

6. Through the above reasoning, the final answer is 70°.

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>In the figure of this problem, the ray between angle a and angle b rotates around the endpoint (intersection of three right angles) to form a straight line with the initial side, forming a straight angle</note>. According to the definition of a straight angle, the measure of a straight angle is 180 degrees, that is <note>∠f + ∠a + ∠g = 180°</note>.

Definition of Vertical Angles

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

In the diagram of this problem, three lines intersect at a point, forming six angles: ∠a, ∠b, ∠c, <note>∠g</note>, ∠e, and ∠f. According to the definition of vertical angles, ∠a and <note>∠e</note> are vertical angles, ∠b and <note>∠f</note> are vertical angles, and ∠c and <note>∠g</note> are vertical angles. Since vertical angles are equal, ∠a = <note>∠e</note>, ∠b = <note>∠f</note>, and ∠c = <note>∠g</note>.


Question: In the triangle ABC above, the bisector of ∠BAC is perpendicular to BC at point D. If AB = 6 and BD = 3, what is the measure of ∠BAC?

Answer: 60

1. Given that the bisector of ∠BAC, AD, is perpendicular to BC, and BD = 3, it indicates that AD is the height of ##triangle ABC##. Thus, △ABD and △ACD are right triangles.

2. Since AD is the bisector of ∠BAC, we have ∠BAD = ∠CAD.

3. In the right triangle ABD, given AB = 6 and BD = 3, according to the Pythagorean theorem, we have: AD = √(AB^2 - BD^2) = √(6^2 - 3^2) = √(36 - 9) = √27 = 3√3.

4. In the right triangle ABD, according to the sine theorem: sin(∠BAD) = opposite side/hypotenuse = BD/AB = 3/6 = 1/2.

5. By the inverse sine theorem, ∠BAD = 30°.

6. Since ∠BAD = ∠CAD, we have ∠BAC = ∠BAD + ∠CAD = 30° + 30° = 60°.

7. Through the above reasoning, the final answer is 60°.

普通三角形; 垂线; 直角三角形

Definition of Angle Bisector

An angle bisector is a line that originates from the vertex of an angle and divides the angle into two congruent angles.

<note>The vertex of angle BAC is point A</note>, from <note>point A a line AD is drawn</note>, this line divides <note>angle BAC into two equal angles, i.e., ∠BAD and ∠CAD are equal</note>. Therefore, <note>line AD is the angle bisector of angle BAC</note>.

Definition of Right Triangle

<note>In the figure of this problem, in triangle ABD, angle ADB is a right angle (90 degrees)</note>, therefore triangle ABD is a right triangle. <note>Sides AD and BD are the legs, and side AB is the hypotenuse</note>. Similarly, <note>in triangle ACD, angle ADC is a right angle (90 degrees)</note>, therefore triangle ACD is a right triangle. <note>Sides AD and DC are the legs, and side AC is the hypotenuse</note>.

Pythagorean Theorem

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

<note>△ABD is a right triangle</note>, <note>AB is the hypotenuse, BD is one of the legs</note>, according to the Pythagorean Theorem: <note>AB##^2## = AD##^2## + BD##^2##</note>, i.e., <note>AD = √(AB##^2## - BD##^2##)</note> = √(6##^2## - 3##^2##) = √(36 - 9) = √27 = 3√3.

Sine Theorem

In any triangle, the ratio of the length of each side to the sine of its opposite angle is constant and equal to the diameter of the circumcircle, i.e., \( \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = 2r = D \) (where \( r \) is the radius of the circumcircle, and \( D \) is its diameter).

In the figure of this problem, in the right triangle ABD, <note>the sides BD, AD, and AB correspond to the angles ∠BAD, ∠ABD, and ∠BDA respectively</note>. According to the Sine Theorem, <note>the ratio of the length of each side to the sine of its opposite angle is equal and is equal to the diameter of the circumscribed circle</note>, that is: <note>BD/sin(∠BAD) = AD/sin(∠ABD) = AB/sin(∠BDA) = 2r = D (where r is the radius of the circumscribed circle, and D is the diameter)</note>.


Question: In the figure above, point P is on line l. What is the value of x?

Answer: 36

1. In the figure, point P is located on line l. Suppose point E is to the right of point P on line l, and point F is to the left of point P on line l. There are five angles, \( \angle APF, \angle APB, \angle BPC, \angle CPD, \angle DPE \), each equal to \( x \) degrees.

2. Since \( \angle FPE \) is a straight angle, it follows that \( \angle FPE = 180^{\circ} \). Therefore, the sum of the five angles \( \angle APF + \angle APB + \angle BPC + \angle CPD + \angle DPE = 180^{\circ} \).

3. Using the above equation to represent the angle measures, we have:

4. \[ x + x + x + x + x = 180^{\circ} \]

5. Simplifying the above equation, we get \( 5x = 180^{\circ} \).

6. Dividing the entire equation to solve for a single x, we get \( x = \frac{180^{\circ}}{5} \).

7. After calculation, we find that \( x = 36^{\circ} \).

8. Therefore, the answer to this problem is: option (E).

Definition of Straight Angle

<note>The ray PE rotates around the endpoint P to form a straight line with the initial side</note>, forming <note>straight angle FPE</note>. According to <note>the definition of straight angle</note>, the measure of a straight angle is 180 degrees, that is <note>angle FPE = 180 degrees</note>.


Question: The circle above has an area of 25*π and is divided into 8 congruent regions. What is the perimeter of one of these regions?

Answer: 10+5/4*\pi

1. Given that the area of the circle is 25*π, according to the formula for the area of a circle A = ##πr^2##, we get: ##πr^2##= 25π.

2. After canceling out π, we get ##r^2##= 25. Therefore, r = √25 = 5. The radius r of the circle is 5.

3. The circle is divided into 8 congruent sectors. The central angle corresponding to each sector is 360° / 8 = 45°.

4. For each sector, its perimeter consists of two radii and one arc. Since the radius of the circle is 5, the sum of the two radii is 5 + 5 = 10.

5. Calculate the arc length of the sector. The formula for the arc length is: (central angle/360°) * 2πr. Hence, the arc length is (45°/360°) * 2 * π * 5 = 1/8 * 2 * π * 5 = 5/4 * π.

6. Therefore, the perimeter of each region is the sum of the lengths of the two radii and the arc length, which is 10 + 5/4 * π.

7. Through the above reasoning, the final answer is 10 + 5/4 * π.

圆; 弧; 圆心角; 扇形; 线段

Definition of Circle

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

In the figure of this problem, in the circle, <note>point O is the center of the circle</note>, <note>the radius is 5</note>. In the figure, <note>all points that are at a distance of 5 from point O are on 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>Each sector is composed of two radii and the arc between them</note>. The circle is divided into 8 congruent sectors, <note>each sector's central angle is 45°</note>. <note>Each sector's two radii are from the center O to two points on the circumference</note>, <note>the arc is the arc between these two points</note>.

Area Formula of a Circle

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

The original text: <note>圆的面积为25π</note>, according to the area formula of a circle <note>A = πr^2</note>, where <note>A表示面积</note>, <note>π是圆周率</note>, <note>r是圆的半径</note>. Given <note>A = 25π</note>, therefore <note>25π = πr^2</note>, we can deduce <note>半径r = √25 = 5</note>.

<note>The central angle is 45° (expressed in radians as π/4)</note>, <note>the radius is 5</note>. According to the formula for the length of an arc of a sector, the arc length L is equal to the central angle θ multiplied by the radius r, i.e., <note>L = θ * r</note>. Therefore, <note>the arc length of each sector l = (π/4) * 5 = 5/4 * π</note>.


Question: In the figure above, if l || m and r = 91, then t + u =

Answer: 178

1. According to the given conditions and the figure in the problem, the given line l is parallel to line m (l || m), and ∠r = 91°. Therefore, we can find the adjacent supplementary angle ∠v on the same side of line n as ∠r: ∠v = 180° - ∠r = 89°.

2. In the current analysis of the figure, line n is the transversal of lines l and m. Therefore, according to the parallel lines postulate 2, corresponding angles are equal, i.e., ∠v = ∠t = 89°.

3. Since ∠u and ∠t are vertical angles, ∠u = ∠t = 89° = 180°.

4. The sum of the two angles is: ∠t + ∠u = 89° + 89° = 178°.

12. Through the above reasoning, the final answer is 178.

平行线; 内错角; 同旁内角; 对顶角; 邻补角

Definition of Vertical Angles

<note>Line n intersects line m at point O</note>, forming four angles. According to the definition of vertical angles, <note>∠u and ∠t are vertical angles</note>, and since the angles of vertical angles are equal, <note>∠u = ∠t</note>.

Parallel Postulate 2 of Parallel Lines

<note>Line l is parallel to line m (l || m)</note>, <note>transversal n intersects the two parallel lines</note>. Therefore, <note>∠v and ∠t are corresponding angles</note>, <note>∠v = ∠t</note>.

Definition of Corresponding Angles

<note>Line l is parallel to line m (l || m)</note>, and the transversal n intersects the two parallel lines. Among them, <note>angle v and angle t are on the same side of the transversal n, on the same side of the intersected lines l and m</note>, therefore <note>angle v and angle t are corresponding angles</note>. Corresponding angles are equal, that is, <note>angle v is equal to angle t</note>.


Question: The two semicircles in the figure above have centers R and S, respectively. If RS = 12, what is the total length of the darkened curve?

Answer: 12*\pi

1. Given RS = 12, which is the distance between the centers of the two semicircles. According to the properties of semicircles, RS is equal to the sum of the radii of the two semicircles. Suppose the radius of the lower semicircle is r and the radius of the upper semicircle is R, then r + R = 12.

2. According to the properties of the geometric figure, we see that the total arc length of the lower semicircle and the upper semicircle constitutes the total length of the darkened curve. Let the points where the lower semicircle and l intersect be A and B, and the points where the upper semicircle and l intersect be B and C. Therefore, calculate the sum of the arc lengths of these two semicircles.

3. The radius of the lower semicircle is r, so its arc length is θ * r; the radius of the upper semicircle is R, so its arc length is θ * R.

4. Therefore, the total length of the darkened curve is θ * r + θ * R.

5. Since r + R = 12, therefore θ * r + θ * R = θ (r + R) = θ * 12, and since in the figure θ = 180° = π.

6. By calculation, the total length of the darkened curve is 12π.

7. Therefore, through the above reasoning, the final answer is 12π.

Definition of Semicircle

A semicircle is a geometric figure constructed from a diameter and an arc of a circle, meaning it represents one of the two congruent parts into which a circle is divided by its diameter.

<note>In the diagram of this problem, the lower semicircle is centered at R and consists of the diameter ##AB## and the lower arc</note>; <note>the upper semicircle is centered at S and consists of the diameter ##BC## and the upper arc</note>. <note>The diameters ##AB## and ##BC## of the two semicircles are both located on line l</note>.

Definition of Diameter

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

<note>Line segment RS is the distance connecting the centers of two semicircles</note>, and its length is 12. According to the definition, <note>a diameter is a line segment that passes through the center of a circle and has its endpoints on the circle, being the longest chord of the circle, with a length equal to 2 times the radius</note>. Therefore, <note>RS is not a diameter, but the sum of the radii of the two semicircles</note>.

Definition of Radius

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

The center of the lower semicircle is <note>R</note>, and the distance from any point on the semicircle to <note>R</note> is r; the center of the upper semicircle is <note>S</note>, and the distance from any point on the semicircle to <note>S</note> is R. According to the problem statement, <note>RS = r + R = 12</note>.

<note>In sector RAB, the central angle ARB is θ (expressed in radians)</note>, <note>the radius is r</note>. According to the formula for the length of an arc of a sector, the arc length L is equal to the central angle θ multiplied by the radius r, i.e., L = θ * r. <note>In sector SBC, the central angle BSC is θ (expressed in radians)</note>, <note>the radius is R</note>. According to the formula for the length of an arc of a sector, the arc length L is equal to the central angle θ multiplied by the radius R, i.e., L = θ * R.


Question:
If p || q, which the following is equal to 80?
 a. a
 b. b
 c. c
 d. d
 e. e

Answer: e

1. Given that line p is parallel to line q (p || q), according to the property of ##corresponding## angles of parallel lines being equal, we can deduce ## ∠e = 80°##.

####

##2##. Check the options, the angle ∠e in option e matches the result of the previous reasoning, its value is equal to 80°.

Based on the above reasoning, the final answer is e.

平行线; 同位角; 内错角

Definition of Corresponding Angles

Two angles are referred to as corresponding angles if they are formed when two lines, a and b, are intersected by a third line, c, and both angles lie on the same side of the intersecting line c and on the same side of the lines a and b.

<note>In the diagram of this problem, line p and line q are intersected by line z</note>, where <note>the 80° angle and ∠e are on the same side of the intersecting line z and on the same side of the intersected lines p and q</note>, therefore <note>the 80° angle and ∠e are corresponding angles</note>.

Parallel Postulate 2 of Parallel Lines

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

<note>p || q</note>, <note>∠e</note> and <note>80° angle</note> are corresponding angles. According to the corresponding angles postulate of parallel lines, <note>∠e = 80°</note>.


Question: The figure above shows six right triangles. What is the value of x^2 + y^2?

Answer: 21

1. Let the right triangle where √3 is located be ABC, the right triangle where x is located be CDE, the right triangle where √6 is located be GHE, and the right triangle where √10 is located be BFG.

2. In triangles BFG, ABC, and GHE, according to the Pythagorean theorem, we can find the lengths of BG, BC, and GE. BG^2 = (10 + 7) = 17, BC^2 = (3 + 3) = 6, EG^2 = (4 + 6) = 10.

3. In triangle BGE, using the Pythagorean theorem, we can find BE^2 = BG^2 + EG^2 = 27.

4. In triangle BCE, according to the Pythagorean theorem, we can find CE^2 = BE^2 - BC^2 = 21.

5. Finally, in triangle CDE, according to the Pythagorean theorem, we have: CE^2 = x^2 + y^2 = 21.

6. Through the above reasoning, the final answer is 21.

Pythagorean Theorem

In the <note>right triangle ABC</note>, the hypotenuse is <note>BC</note>, and the two legs are <note>√3 and √3</note>. Therefore, according to the Pythagorean Theorem, we have: <note>BC^2 = (√3)^2 + (√3)^2 = 3 + 3 = 6</note>. In the <note>right triangle BFG</note>, the hypotenuse is <note>BG</note>, and the two legs are <note>√10 and √7</note>. Therefore, according to the Pythagorean Theorem, we have: <note>BG^2 = (√10)^2 + (√7)^2 = 10 + 7 = 17</note>. In the <note>right triangle GHE</note>, the hypotenuse is <note>GE</note>, and the two legs are <note>√6 and 2</note>. Therefore, according to the Pythagorean Theorem, we have: <note>GE^2 = (√6)^2 + 2^2 = 6 + 4 = 10</note>. In the <note>right triangle CDE</note>, the hypotenuse is <note>CE</note>, and the two legs are <note>x and y</note>. Therefore, according to


Question:
In the diagram above, what is the sum of the measures of the angles x, y, and z?
a. 180 degrees
b. 360 degrees
c. 540 degrees
d. 720 degrees
e. cannot be determined

Answer: 360

1. Given ∠x, ∠y, and ∠z are the three exterior angles of a triangle.

2. According to the exterior angle sum theorem for polygons, the sum of the exterior angles of any polygon is equal to 360°, i.e., ∠x + ∠y + ∠z = 360 degrees.

3. Based on the above reasoning, the final answer is 360 degrees.

对顶角; 邻补角; 直线

Exterior Angle Sum Theorem of Polygon

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

In the figure of this problem, in the triangle formed by the three lines, the exterior angles at each vertex are <note>∠x, ∠y, and ∠z</note> respectively. According to the <note>Exterior Angle Sum Theorem of Polygon</note>, the sum of these exterior angles is equal to <note>360°</note>, that is, <note>∠x + ∠y + ∠z = 360°</note>.

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

<note>∠x, ∠y, and ∠z form the three exterior angles of a triangle</note>.


Question: In the figure above, ABCDEF is a regular hexagon with its center at point O. What is the value of x?

Answer: 60

1. Connect OC to form the central angle BOC. Given that ABCDEF is a regular hexagon and its center is point O. According to the definition and calculation formula of the central angle, each central angle = 360/6 = 60°, i.e., ∠ = 60°.

2. Since each side of the regular hexagon forms an isosceles triangle with the center line, OB = OC, therefore BOC is an isosceles triangle.

3. According to the properties of isosceles triangles, ∠OBC = ∠OCB.

4. In triangle OBC, according to the triangle angle sum theorem, we can find x = (180 - 60)/2 = 60.

4. Finally, the value of x is 60.

六边形; 正多边形; 圆心角; 对称

Regular Hexagon

A regular hexagon is a hexagon in which all interior angles are equal, and all sides are of the same length.

In the <note>regular hexagon ABCDEF</note>, each interior angle is equal and each side length is equal. Specifically, <note>each interior angle of the regular hexagon is 120 degrees</note>, and each side length is equal, that is, <note>AB=BC=CD=DE=EF=FA</note>.

Definition of Central Angle

A central angle is an angle formed by two radii connecting the center of a regular polygon to two of its adjacent vertices. A central angle can also be described as the angle subtended at the center of a circumscribed circle by any side of the regular polygon.

In the figure of this problem, <note>angle BOC</note> is a <note>central angle</note>, formed by the lines connecting the <note>center O</note> of the regular polygon and the two adjacent <note>vertices B and C</note>. According to the definition of the central angle, <note>angle BOC</note> is the central angle of the regular polygon, and its measure is equal to the <note>circumcircle's central angle</note> corresponding to each side of the regular polygon.

Properties of Isosceles Triangle

In an isosceles triangle, the angles opposite the equal sides are equal, and the sides opposite the equal angles are congruent.

In the <note>isosceles triangle BOC</note>, sides <note>BO</note> and <note>CO</note> are equal. Therefore, according to the properties of isosceles triangles, the angles opposite the equal sides are equal, i.e., <note>∠OBC</note> = <note>∠OCB</note>.

Formulas for the Central Angle and Interior Angle of a Regular Polygon

For a regular polygon with \( n \) sides, the measure of each central angle is given by \( \frac{360^\circ}{n} \). The measure of each interior angle is given by \( \frac{(n - 2) \cdot 180^\circ}{n} \).

<note>The central angle of a regular hexagon ∠BOC = 360° / 6 = 60°</note>; <note>the interior angle ∠ABC = (6-2) * 180° / 6 = 120°</note>.

Definition of Isosceles Triangle

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

In the <note>triangle OBC</note>, <note>sides OB and OC are equal</note>, therefore <note>triangle OBC is an isosceles triangle</note>.

Triangle Angle Sum Theorem

The sum of the interior angles of any triangle is 180°.

<note>In triangle BOC, angles OBC, OCB, and BOC are the three interior angles of triangle BOC</note>, according to the Triangle Angle Sum Theorem, <note>angle BOC + angle OCB + angle OBC = 180°</note>.


Question:
In the figure above, l || m. Which of the following must equal 180?

(A) k + n + r
(B) k + p + s
(C) n + p + s
(D) n + p + t
(E) r + s + t

Answer: k + p + s

1. Given l ∥ m, according to the parallel axiom 2 of parallel lines, alternate interior angles are equal, we get: ∠k = ∠r.

2. According to the definition of a straight angle, we have: ∠r + ∠s + ∠p = 180°.

3. Substituting the conclusion from step one, we get ∠k + ∠s + ∠p = 180°.

4. Therefore, among the given options, (B) k + p + s is the answer.

平行线; 同旁内角; 对顶角

Parallel Postulate 2 of Parallel Lines

In the figure of this problem, <note>line l is parallel to line m</note>, <note>line np is the transversal</note>, forming the following geometric relationship: <note>alternate interior angles ∠k=∠r</note>.

Definition of Alternate Interior Angles

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

<note>Two parallel lines l and m</note> are intersected by a <note>transversal</note>, where <note>angle k and angle r are located between the two parallel lines</note> and on <note>opposite sides of the transversal</note>. Therefore, <note>angle k and angle r are alternate interior angles</note>. Alternate interior angles are equal, i.e., <note>angle k = angle r</note>.


Question: Rectangle ABCD is subdivided into two identical square regions, as in the figure above. If the area of each square is 9, what is the perimeter of ABCD?

Answer: 18

1. Given that the area of each square is 9. Assuming the side length of each square is x, according to the area formula of a square, the side length is the square root of the area of the square, i.e., x = √9 = 3.

2. Since rectangle ABCD is divided into two identical squares, the side length of the two squares is equal to the width of the rectangle, where the width AB = 3.

3. The length of the rectangle is the sum of the side lengths of the two squares, therefore AD = 2 * 3 = 6.

4. Using the perimeter formula of the rectangle, perimeter = 2 * (length + width), substituting the known values, the perimeter of rectangle ABCD is 2 * (6 + 3) = 18.

5. Through the above reasoning, the final answer is 18.

矩形; 正方形; 平行线; 线段

Definition of Square

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

In the figure of this problem, rectangle ABCD is divided into two identical square regions, <note>∠A, ∠B, ∠C, and ∠D are all right angles</note>. Let segment EF intersect side BC at E and side AD at F. According to the problem statement, <note>∠AFE, ∠BEF, ∠CEF, and ∠DFE are also all right angles</note>, <note>sides AF, FE, FB, and BA are equal</note>, <note>sides FD, DC, CE, and EF are equal</note>, so <note>ABFE and FDCE are two squares</note>.

The perimeter of a rectangle is equal to twice the sum of the lengths of its longer side (length) and its shorter side (width), which can be expressed as \( P = 2(l + w) \).

In <note>rectangle ABCD</note>, <note>side AD</note> and <note>side BC</note> are the <note>long sides of the rectangle</note>, and <note>side AB</note> and <note>side CD</note> are the <note>short sides of the rectangle</note>. According to the <note>formula for the perimeter of a rectangle</note>, the perimeter of the rectangle is equal to twice the sum of its long side length and short side length, i.e., P = 2(<note>AD</note> + <note>AB</note>). Therefore, the <note>perimeter of rectangle ABCD</note> is calculated as perimeter = 2*(6+3)=18.

Formula for the Area of a Rectangle

The area of a rectangle is equal to its length multiplied by its width.

<note>Square ABFE and square FDCE are two equal squares, and they are also special rectangles</note>, so <note>the area of a square = side length * side length</note>. Given that <note>the area of the square is 9</note>, according to the formula for the area of a square, the side length is the square root of the area of the square, that is, <note>x = √9 = 3</note>.


Question: In the rectangle ABCD above, the area of the shaded region is given by \pi*l*w/4. If the area of the shaded region is 7*\pi, find the total area of the non-shaded region in rectangle ABCD (rounded to the nearest integer).

Answer: 6

1. Given that the shaded area of rectangle ABCD is (π*l*w)/4, and the shaded area is 7π.

2. According to the condition, assume the shaded area is actually an ellipse, with the area formula Area = (π*l*w)/4.

3. Let the shaded area (π*l*w)/4 = 7π.

4. By the equation (π*l*w)/4 = 7π, eliminate π and solve for l*w/4 = 7, thus obtaining l*w = 28.

5. The total area of rectangle ABCD is l*w = 28.

6. The unshaded area of rectangle ABCD is the total area of the rectangle minus the shaded area, which is 28 - 7π.

7. Calculate the approximate value of π as 3.14, resulting in 7π ≈ 7 * 3.14 = 21.98.

8. Therefore, the unshaded area is 28 - 21.98 ≈ 6.

9. Through the above reasoning, the final unshaded area is approximately 6.

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>, with its interior angles <note>∠ABC, ∠BCD, ∠CDA, ∠DAB all being right angles (90 degrees)</note>, and <note>side AB is parallel to side CD and of equal length</note>, <note>side BC is parallel to side AD and of equal length</note>.

Area of Ellipse

The formula for the area of an ellipse is A = πab, where a and b are the lengths of the semi-major axis and the semi-minor axis, respectively.

<note>The long axis of the ellipse is half the length of rectangle ABCD, i.e., l/2</note>, <note>the short axis is half the width of rectangle ABCD, i.e., w/2</note>. According to the ellipse area formula S=πab, where <note>a=l/2</note>, <note>b=w/2</note>, the area of the ellipse is (π*(l/2)*(w/2))= (π*l*w)/4.

Formula for the Area of a Rectangle

In the figure of this problem, in rectangle ABCD, <note>side BC and side CD are the length and width of the rectangle</note>, so <note>the area of the rectangle = l * w</note>.


Question:
In the figure above, a shaded polygon which has equal sides and equal angles is partially covered with a sheet of blank paper. If x + y = 80, how many sides does the polygon have?
(A) Ten
(B) Nine
(C) Eight
(D) Seven
(E) Six

Answer: Nine

1. The given condition is an equilateral and equiangular polygon, x and y are part of its exterior angles, x + y = 80.

2. According to the theorem of the sum of interior angles of a quadrilateral, the sum of interior angles of any quadrilateral is equal to 360 degrees.

3. Then the sum of two interior angles of this regular polygon is equal to 360 degrees minus (x + y).

4. Since the problem gives x + y = 80, the value of one interior angle is: (360-80) /2= 140.

5. Therefore, one exterior angle of the regular polygon = 180-140=40 degrees, according to the exterior angle theorem of regular polygons, 360/n =40, we get n=9.

6. Therefore, the polygon has 9 sides.

In the figure of this problem, <note>an interior angle of a polygon and the angle formed by the extension of the adjacent side</note> is called <note>the exterior angle of this interior angle</note>.

In any quadrilateral, the sum of the four interior angles is 360°.

In the figure of this problem, in the quadrilateral, <note>angle x</note>, <note>angle y</note>, and <note>two interior angles of the regular polygon</note> are the four interior angles of the quadrilateral. According to the Sum of Interior Angles of a Quadrilateral Theorem, the sum of these four interior angles is 360°, that is, <note>angle x + angle y + two interior angles of the regular polygon = 360°</note>.

Exterior Angle Theorem for Regular Polygons

The measure of each exterior angle of a regular n-sided polygon is equal to 360/n degrees.

<note>Assuming the polygon is a regular n-sided polygon</note>, then each exterior angle is <note>360/n degrees</note>. Since one exterior angle is <note>40 degrees</note>, therefore <note>360/n = 40 degrees</note>.


Question: What is the area of the figure below?

Answer: 120

1. Given that the length of the base is 20 and the height is 12. According to the area formula for geometric figures, for a triangle, its area is (base × height) / 2, the area of the triangle can be expressed as (20 × 12) / 2.

2. Calculation yields: area = (20 × 12) / 2 = 240 / 2 = 120.

3. Through the above reasoning, the final answer is 120.

等腰三角形; 直角三角形; 线段

Definition of Altitude

An altitude is a perpendicular line segment extending from a vertex of a triangle to the opposite side (or to the extension of the opposite side).

<note>The line segment from the vertex perpendicular to the base (or its extension) is the altitude of the vertex</note>. <note>The line segment forms a right angle (90 degrees) with the base (or its extension)</note>, which indicates that the line segment is the perpendicular distance from the vertex to the base (or its extension). The base length is 20, and the altitude is 12.

Area Formula of a Triangle

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

Original text: <note>The base length of the triangle is 20, and the height is 12</note>. According to the area formula of a triangle, <note>the area of the triangle</note> is equal to <note>the base multiplied by the height and then divided by 2</note>, which means the area = (20 × 12) / 2 = 120.


Question: In the figure above, the smaller circles each have radius 3. They are tangent to the larger circle at points A and C, and are tangent to each other at point B, which is the center of the larger circle. What is the perimeter of the shaded region?

Answer: 12*\pi

1. The given conditions are: the radius of each small circle is 3, and they are tangent to the large circle at points A and C, and tangent to each other at point B, which is the center of the large circle. According to the properties of circles, the tangent at the point of tangency is perpendicular to the radius of the circle.

2. Since point B is the point of tangency for the two small circles, AB = 6 and BC = 6. At the same time, point B is the center of the large circle, so the radius of the large circle is AB = BC = 6.

3. We need to calculate the perimeter of the shaded region. The perimeter of the shaded region is half of the large circle plus half of each of the two small circles. Since the radii of the two small circles are equal, their perimeters are equal, so it is half the perimeter of the large circle plus the perimeter of one small circle.

4. The perimeter of the large circle is 2 * π * 6 = 12π.

5. The perimeter of each small circle is 2 * π * 3 = 6π.

6. The perimeter of the shaded region consists of half the perimeter of the large circle and the perimeter of one small circle. Since the total perimeter of the small circle is 6π, the total perimeter of the large circle is 12π.

7. Based on the above analysis, half the perimeter of the large circle is 12π/2=6π, the perimeter of the small circle part is 2 * 3π=6π, so the perimeter of the shaded region is 6π + 6π = 12π.

8. Therefore, the final answer is that the perimeter of the shaded region is 12π.

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>In the diagram of this problem, in the large circle, point B is the center, and the radius is 6.</note> All points that are 6 units away from point B are on the large circle. In the two small circles, <note>points A and C are the points of tangency with the large circle.</note> The radius of each small circle is 3. <note>The distance from point A to the center of the small circle on the left is 3, and the distance from point C to the center of the small circle on the right is 3.</note>

Definition of Tangent Circles

Two circles are tangent if they intersect at exactly one point, and their tangents at the point of contact coincide.

In the diagram of this problem, <note>the large circle is tangent to the left small circle at point A</note>, <note>the large circle is tangent to the right small circle at point C</note>, <note>the two small circles are tangent at point B</note>. <note>Points A, B, and C are the common tangency points of the two circles</note>. The tangent line is <note>tangent to the two circles at points A, B, and C</note>. Therefore, <note>the large circle is tangent to the left small circle at point A</note>, <note>the large circle is tangent to the right small circle at point C</note>, <note>the two small circles are tangent at point B</note>.

Circumference Formula of Circle

The circumference of a circle is the length around the circular boundary. It can be calculated using the formula \( C = 2\pi r \), where \( C \) represents the circumference and \( r \) represents the radius.

<note>In the large circle, point B is the center</note>, <note>line segments AB and BC are radii r=6</note>. According to the circumference formula of a circle, the circumference C of the circle is equal to 2π multiplied by the radius r, i.e., C=2πr, so the circumference of the large circle is 2 * π * 6 = 12π. <note>In the small circles, points A and C are the centers</note>, <note>the radii of both small circles are r=3</note>. According to the circumference formula of a circle, the circumference C of the circle is equal to 2π multiplied by the radius r, i.e., C=2πr, so the circumference of each small circle is 2 * π * 3 = 6π.


Question: In △PQR above, PR = QR. Which of the following must be true?

Answer: y=z

1. Given that in triangle PQR, PR = QR, according to the properties of an isosceles triangle, we can conclude that angle PQR is equal to angle QPR.

2. Since angle PQR is angle v, and angle QRP is angle u, therefore, based on the conclusion from the first step, we have v = u.

3. According to the exterior angle theorem of triangles, the exterior angle y of triangle PQR is equal to u + x, and z is equal to v + x, thus based on the conclusion from the second step, we can deduce y = z.

4. Through the above reasoning, the final answer is y = z.

等腰三角形; 三角形的外角

Definition of Isosceles Triangle

In the figure of this problem, in triangle PQR, <note>sides PR and QR are equal</note>, therefore <note>triangle PQR is an isosceles triangle</note>.

Properties of Isosceles Triangle

In an isosceles triangle, the angles opposite the equal sides are equal, and the sides opposite the equal angles are congruent.

<note>In the isosceles triangle PQR, sides PR and QR are equal</note>. Therefore, according to the properties of an isosceles triangle, the angles opposite the equal sides are equal, i.e., <note>angle PQR = angle QPR</note>.

Exterior Angle Theorem of Triangle

An exterior angle of a triangle is equal to the sum of the measures of the two non-adjacent interior angles.

In the figure of this problem, in triangle PQR, <note>angle y is an exterior angle of the triangle</note>, <note>angle u and angle x are the two interior angles not adjacent to exterior angle y</note>. According to the Exterior Angle Theorem of Triangle, <note>exterior angle y is equal to the sum of the two non-adjacent interior angles u and x</note>, that is, <note>angle y = angle u + angle x</note>. <note>Angle z is an exterior angle of the triangle</note>, <note>angle v and angle x are the two interior angles not adjacent to exterior angle z</note>. According to the Exterior Angle Theorem of Triangle, <note>exterior angle z is equal to the sum of the two non-adjacent interior angles v and x</note>, that is, <note>angle z = angle v + angle x</note>.


Question: What is the value of x?

Answer: 65

1. Given conditions: In a right triangle, one leg is 25, the other leg is 60, and the hypotenuse is x.

2. According to the Pythagorean theorem (the square of the hypotenuse of a right triangle is equal to the sum of the squares of the two legs), let the legs be a and b, and the hypotenuse be c, then we have c^2 = a^2 + b^2.

3. In this problem, a = 25, b = 60, c = x, then applying the Pythagorean theorem: x^2 = 25^2 + 60^2.

4. Calculating: x^2 = 625 + 3600.

5. Further calculation: x^2 = 4225.

6. Taking the square root: x = √4225.

7. Calculating the value of √4225: x = 65.

8. Therefore, through the above reasoning, the final answer is 65.

Definition of Right Triangle

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

In the figure of this problem, the triangle is a <note>right triangle</note>, with the <note>right angle at the bottom left corner</note>. One <note>leg of the right angle is 25</note>, the other <note>leg of the right angle is 60</note>, and the corresponding <note>hypotenuse is x</note>.

Pythagorean Theorem

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

In a <note>right triangle</note>, <note>one leg is 25</note>, <note>the other leg is 60</note>, <note>the hypotenuse is x</note>. According to the Pythagorean Theorem, <note>## x^2 = 25^2 + 60^2 ##</note>. Calculating this gives: <note>## x^2 ## = 625 + 3600 = 4225</note>, therefore <note>x = √4225 = 65</note>.


Question: What is the length of c in the triangle above?

Answer: 80

1. Given the condition of a right triangle, where one leg length is 60, the hypotenuse length is 100, let the other leg be c. According to the Pythagorean theorem, the square of the hypotenuse equals the sum of the squares of the two legs, i.e., ##100^2= 60 ^2+ c^2##.

3. Substitute the given conditions into the Pythagorean theorem to obtain the equation: 10000 = 3600 + ##c^2##.

7. Through the above reasoning, the final answer is 80.

Definition of Right Triangle

A triangle contains a <note>90-degree interior angle</note>, so the triangle is a right triangle. The sides with lengths 60 and c are the legs, and the <note>side with length 100 is the hypotenuse</note>.

Pythagorean Theorem

In a <note>right triangle</note>, the square of the <note>hypotenuse</note> is equal to the sum of the squares of the <note>two legs</note>. Given that <note>one leg is 60</note> and the <note>hypotenuse is 100</note>, according to the Pythagorean Theorem, we can obtain: ##<note>100^2= 60^2 + c^2</note>##.


Question: In the figure above, AB = 6 and BC = 8. What is the area of triangle ABC?

Answer: 12*\sqrt{3}

1. Given that AB = 6, BC = 8, and angle ABC = 60°, according to the problem, we can conclude that this is a triangle situation.

2. Based on the triangle area formula S = (1/2) * a * b * sin(C), where a and b are the lengths of two sides and C is the included angle, we can use it to solve for the area of triangle ABC.

3. Substitute the given side lengths AB = 6 and BC = 8, and angle ABC = 60° into the area formula, we get S = (1/2) * 6 * 8 * sin(60°).

4. Since sin(60°) = √3/2, substituting we get S = (1/2) * 6 * 8 * (√3/2) = 12√3.

5. Through the above reasoning, the final answer is 12√3.

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, <note>triangle ABC</note> is a geometric figure formed by <note>three non-collinear points A, B, and C</note> and their connecting <note>line segments AB, BC, and CA</note>. <note>Points A, B, and C are the three vertices of the triangle</note>, and <note>line segments AB, BC, and CA are the three sides of the triangle</note>, where <note>AB = 6, BC = 8</note>, and <note>angle ABC = 60°</note>.

Definition of Sine Function

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

<note>sin(∠ABC)</note> is the sine value applicable to non-right triangles, where <note>∠ABC = 60°</note>, so <note>sin(60°) = √3/2</note>.

The area \( S \) of any triangle can be expressed as \( S = \frac{1}{2} \cdot a \cdot b \cdot \sin(C) \), where \( a \) and \( b \) are the lengths of two sides, and \( C \) is the angle between these two sides.

In <note>triangle ABC</note>, <note>sides AB and BC are 6 and 8 respectively</note>, <note>angle ABC is the included angle between these two sides, which is 60°</note>. According to the triangle area formula, the area S of triangle ABC can be expressed as <note>S = (1/2) * 6 * 8 * sin(60°)</note>, which is <note>S = (1/2) * 6 * 8 * (√3/2) = 12√3</note>.


Question:
In the figure above, AC = 6 and BC = 3. Point P (not shown) lies on AB between A and B such that CP ⊥ AB. Which of the following could be the length of CP?
(A) 2
(B) 4
(C) 5.25
(D) 7
(E) 8

Answer: 5.25

1. Suppose ΔACP and ΔBCP are two right triangles, line CP is their common perpendicular side and is perpendicular to line AB.

2. According to the problem, take segment AP = x and PB = AB - x, and AC = 6, BC = 3.

3. In ΔACP, apply the Pythagorean theorem: AC^2 = AP^2 + CP^2, that is, 6^2 = x^2 + CP^2, resulting in 36 = x^2 + CP^2.

4. In ΔBCP, apply the Pythagorean theorem: BC^2 = PB^2 + CP^2, that is, 3^2 = (6 - x)^2 + CP^2, simplifying to 9 = (6 - x)^2 + CP^2.

5. Solving the two equations, we calculate x = 5.25.

6. Through the above reasoning, the final answer is 5.25.

垂线; 直角三角形; 普通三角形; 线段; 点

Definition of Right Triangle

In the figure of this problem, in triangle ACP, <note>angle APC is a right angle (90 degrees)</note>, thus triangle ACP is a right triangle. <note>Side AP and side CP are the legs</note>, <note>side AC is the hypotenuse</note>. In triangle BCP, <note>angle BPC is a right angle (90 degrees)</note>, thus triangle BCP is a right triangle. <note>Side BP and side CP are the legs</note>, <note>side BC is the hypotenuse</note>.

Pythagorean Theorem

In the figure of this problem, in the right triangle ACP, <note>∠APC is a right angle (90 degrees)</note>, <note>sides AP and CP are the legs</note>, <note>side AC is the hypotenuse</note>, so according to the Pythagorean Theorem, <note>AC^2 = AP^2 + CP^2</note>, that is, 6^2 = x^2 + CP^2; similarly, in the right triangle BCP, <note>∠BPC is a right angle (90 degrees)</note>, <note>sides BP and CP are the legs</note>, <note>side BC is the hypotenuse</note>, so according to the Pythagorean Theorem, <note>BC^2 = PB^2 + CP^2</note>, that is, 3^2 = (6 - x)^2 + CP^2.


Question: What is the length of a in the triangle above?

Answer: 15*\sqrt{2}

1. Given that the triangle is a right triangle, and the two non-right angles are both 45°, this means the triangle is an isosceles right triangle.

2. ##According to the Pythagorean theorem, in an isosceles right triangle, the hypotenuse a^2 = 15^2 + 15^2##.

3. ##Solving the equation, we get## the length of the hypotenuse is 15√2.

4. Therefore, through reasoning, we finally obtain that the length of the hypotenuse a is 15√2.

等腰三角形; 直角三角形

In a <note>right triangle</note>, both legs are <note>15</note>, and the <note>hypotenuse is a</note>, so according to the Pythagorean Theorem, <note>a^2 = 15^2 + 15^2</note>, thus <note>a = 15√2</note>.

Definition of Isosceles Right Triangle

An isosceles right triangle is a triangle with two sides of equal length and one angle measuring 90 degrees.

<note>The triangle is an isosceles right triangle</note>, where <note>the lower left corner is a right angle (90 degrees)</note>, <note>the two sides of length 15 are equal right-angle sides</note>.


Question: In the figure above, if the area of triangle CAF is equal to the area of rectangle CDEF, what is the length of segment AD?

Answer: 7

1. Given that the area of triangle CAF is equal to the area of rectangle CDEF, where the length of rectangle CDEF is 15 and the height is 7, thus the area is 15×7=105.

2. Let the length of AD be x. According to the properties of a right triangle, CAF is a right triangle, and the area formula is 1/2×CA×CF.

3. Since DE is equal to CF, which is equal to the length of the rectangle, 15, CF is 15.

4. From the figure, we can deduce that CA is equal to the height of the rectangle, 7, plus the length of AD, x. Therefore, CA=7+x.

5. Based on the given conditions and calculations, we find that the area of triangle CAF is 1/2×(x+7)×15, and the area of rectangle CDEF is 105. Since they are equal, 1/2×(x+7)×15=105.

6. Solving the equation, 1/2×15×(x+7)=105, simplifying to 7.5×(x+7)=105.

7. Dividing both sides of the equation by 7.5, we get x+7=14.

8. Thus, we find x=14-7=7, and therefore the length of AD is 7.

9. Through the above reasoning, we finally conclude that the answer is 7.

直角三角形; 矩形; 垂线

Definition of Right Triangle

<note>In the figure of this problem, in triangle CAF, ##∠ACF## is a right angle (90 degrees), so triangle CAF is a right triangle</note>. Side CA and <note>##side CF## are the legs</note>, <note>##side AF## is the hypotenuse</note>.

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 CDEF is a rectangle</note>, <note>the interior angles ∠CDE, ∠DEF, ∠EFC, ∠FCD are all right angles (90 degrees)</note>, and <note>side CF is parallel and equal in length to side DE</note>, <note>side CD is parallel and equal in length to side EF</note>.

Area Formula of a Triangle

<note>In the diagram of this problem, in triangle CAF, side CF is the base, ##segment AC## is the height</note>. According to the area formula of a triangle, the area of triangle CAF is equal to the base CF multiplied by <note>##height AC##</note> and then divided by 2, i.e., Area = (CF * <note>##AC##</note>) / 2.

Formula for the Area of a Rectangle

The area of a rectangle is equal to its length multiplied by its width.

In the figure of this problem, in rectangle CDEF, side <note>CD</note> and side <note>DE</note> are the length and width of the rectangle, so the area of the rectangle = <note>CD × DE</note>.


Question: Semicircular arcs AO and OB divide the circle above with center O into two regions. If the length of diameter AB is 12, what is the area of the shaded region?

Answer: 18*\pi

1. Given that the center of the circle is O and the length of the diameter AB is 12, according to the definition of the diameter of a circle, the radius of the circle is half of the diameter. Therefore, the radius r of the circle is 6.

2. The formula for the area of a circle is ##πr^2##. Substituting r, we get the area of the circle as ##π(6)^2## = 36π.

3. ##Since O is the center of the large circle, the radius of the semicircle AO = the radius of the semicircle BO = 1/2 * the radius of the large circle r = r/2. The two semicircular arcs AO and OB respectively form two equal areas with the segments AO and OB, each area being a semicircle with radius = r/2##.

4. ##Therefore, the area of the shaded region = the area of the semicircle with radius r - the area of semicircle OA + the area of semicircle OB = half of the area of the large circle = 18π##, that is, the area of the shaded region is 18π.

圆; 弧; 旋转; 对称

Definition of Diameter

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

<note>AB is the diameter</note>, connecting <note>the center O</note> and <note>points A and B on the circumference</note>, with a length of <note>2 times the radius</note>, i.e., <note>AB = 2r</note>.

Definition of Radius

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

<note>In the figure of this problem, in circle O, point O is the center of the circle, and points A and B are any points on the circle</note>, <note>line segments OA and OB are segments from the center O to any point on the circle</note>, therefore <note>line segments OA and OB are the radii of the circle</note>.

Area Formula of a Circle

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

In the figure of this problem, <note>with O as the center of the circle, the radius of the circle is 6</note>. According to the area formula of a circle, <note>the area A of the circle is equal to the circumference π multiplied by the square of the radius 6, that is, A = ##π(6)^2## = 36π</note>.

Definition of Semicircular Arc

A semicircular arc is a curve formed with a diameter as its chord, positioned entirely on one side of the chord.

<note>Semicircular arc AO and semicircular arc OB</note> are located on either side of <note>diameter AB</note> and connect the center O through <note>points A and B</note>. According to the definition of a semicircular arc, semicircular arc AO and semicircular arc OB cover <note>half of the circumference, that is, 180 degrees</note>.