Question: Find y.

Answer: 48

1. Let the triangle in the figure be triangle ABC. From the right angle mark at the intersection of the two lines in the figure, it indicates that these two lines are perpendicular to each other. Therefore, angle BAC is a right angle, i.e., ∠BAC=90°. According to the definition of a right triangle, △ABC is a right triangle.

2. According to the triangle angle sum theorem, the sum of the interior angles of a triangle is 180°. Therefore, ∠ABC + ∠ACB + ∠BAC = 180°, i.e., ∠ABC + 30° + 90° = 180°, thus ∠ABC = 60°.

3. Since △ABC is a right triangle and the value of angle ∠ACB is 30°, it conforms to the special property of a 30°-60°-90° right triangle.

4. According to the property of a 30°-60°-90° triangle, in a 30°-60°-90° right triangle, given the length of the right-angle side is 24, the side opposite the 30° angle is the short side, and the side opposite the 90° angle is the hypotenuse, and the hypotenuse = short side × 2.

5. Given the short side is 24, then the hypotenuse is equal to 24 × 2 = 48.

6. Since y is the hypotenuse, therefore y=48.

7. Through the above reasoning, the final value of y is 48.

直角三角形; 正弦; 余弦

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, let the triangle in the figure be <note>triangle ABC</note>. <note>Triangle ABC</note> is a geometric figure composed of <note>three non-collinear points A, B, C</note> and their connecting line segments <note>AC, AB, BC</note>. <note>Points A, B, C</note> are respectively the <note>three vertices</note> of the triangle, and the line segments <note>AC, AB, BC</note> are respectively the triangle's <note>three sides</note>.

Definition of Right Triangle

<note>In a triangle, one of the interior angles is a right angle (90 degrees)</note>, therefore the triangle is a right triangle. <note>Sides 24 and x are the legs</note>, <note>side y is the hypotenuse</note>.

Triangle Angle Sum Theorem

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

<note>The three interior angles of the triangle are ∠BAC, ∠ACB, and ∠ABC.</note> According to the Triangle Angle Sum Theorem, <note>∠BAC + ∠ACB + ∠ABC = 180°.</note> Given <note>∠BAC = 90°</note>, <note>∠ACB = 30°</note>, therefore <note>∠ABC = 180° - 90° - 30° = 60°.</note>

Properties of 30°-60°-90° Triangle

In a 30°-60°-90° triangle, the side opposite the 30-degree angle (the shorter leg) is half the length of the hypotenuse. The side opposite the 60-degree angle (the longer leg) is √3 times the length of the shorter leg.

<note>△ABC is a 30°-60°-90° right triangle</note>。<note>∠ACB = 30°</note>，<note>∠ABC = 60°</note>，<note>∠BAC = 90°</note>。<note>Side BC is the hypotenuse</note>，<note>Side AB is opposite the 30-degree angle</note>，<note>Side AC is opposite the 60-degree angle</note>。According to the properties of a 30°-60°-90° triangle, side AB is equal to <note>half of side BC</note>，<note>side AC is equal to AB times √3</note>。That is: <note>BC = 2 * AB</note>，<note>AC = AB * √3</note>。


Question: Find x.

Answer: 2 \sqrt { 11 }

1. Let the vertex of the angle between the leg X and the hypotenuse 12 be A, let the vertex of the angle between the hypotenuse 12 and the leg 10 be B, and let the vertex of the 90° angle be C.

2. Given that in the right triangle ABC, the hypotenuse AB = 12 and the leg BC = 10, we can use the Pythagorean theorem to find the leg AC.

3. According to the Pythagorean theorem, we get the formula AB² = AC² + BC².

4. Substitute the known conditions: 12² = X² + 10².

5. Calculate the formula: 144 = X² + 100, thus 144 - 100 = 44.

7. Finally, the answer to this geometry problem is x = 2√11.

直角三角形; 垂线; 等腰三角形

Definition of Right Triangle

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

The vertex angle is a right angle (90 degrees), therefore this triangle is a right triangle. Side 10 and <note>side x</note> are the legs, side 12 is the <note>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.

In the context of a <note>right triangle</note>, the <note>square of the hypotenuse</note> is equal to the <note>sum of the squares of the two legs</note>. According to the <note>Pythagorean Theorem</note>, the <note>square of the hypotenuse is equal to the sum of the squares of the two legs</note>, that is <note>12^2 = 10^2 + x^2</note>, therefore <note>x =√(12^2 - 10^2) = √(144 - 100) = √(44) = 2√11</note>.


Question: As shown in the figure, given that lines a and b are intersected by line c, a∥b, ∠1=50°, then ∠2=()

Answer: 50°

1. Given that line a is parallel to line b, and ∠1=50°, ##let ∠2's vertical angle be ∠3##.

2. ##According to the parallel axiom 2 of parallel lines, corresponding angles are equal##, so ∠3=∠1=50°.

3. According to the definition of vertical angles, ##∠2=∠3##.

5. Through the above reasoning, the final answer is ∠2=50°.

平行线; 同位角; 直线

Definition of Vertical Angles

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

In the figure of this problem, <note>two intersecting lines c and b intersect at point ##</note>, forming two angles: <note>angle 2 and angle 3</note>. According to the definition of vertical angles, <note>angle 2 and angle 3 are vertical angles</note>. Since vertical angles are equal, <note>angle 2 = angle 3##</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, two <note>parallel lines a and b</note> are intersected by <note>a third line c</note>, forming the following geometric relationship: <note>corresponding angles: angle 1 and angle 3 are equal</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 a and b</note> are <note>intersected by a line c</note>, <note>angle 1 and angle 3 are on the same side of the intersecting line c and on the same side of the two intersected lines a and b</note>, therefore, <note>angle 1 and angle 3 are corresponding angles</note>. <note>Corresponding angles are equal, that is, angle 1 is equal to angle 3</note>.


Question: Find the area of the regular polygon. Round the result to the nearest tenth.

Answer: 101.8

1. The given figure is a regular octagon. ##According to the definition of a regular polygon, we know it has eight equal side lengths and eight equal interior angles##.

##2. It is known that the length of the line from the center of the regular octagon to one vertex is 6 cm, i.e., the circumradius R of the regular octagon is 6 cm.##

##3. Taking the center of the regular octagon as the center of the circle, draw the circumcircle of the regular octagon, then the radius R of the circumcircle of the regular octagon is 6 cm.##

4. Using the area formula for a regular polygon A =## (1/2) * n * r^2 * sin(360°/n)##, where n is the number of sides and r is the circumradius. In this problem, n = 8, r = 6.

5. Substitute the known conditions into the area formula A = (1/2) * 8 * 6^2 * sin(360°/8) to get A = (1/2) * 8 * 36 * sin(45°).

6. According to sin(45°) = √2/2, we can substitute to get A = (1/2) * 8 * 36 * (√2/2).

7. Simplifying the calculation gives A = 144 * √2/2.

8. Further calculation gives A = 72 * √2.

9. By approximating A ≈ 72 * 1.414, we get A ≈ 101.8236.

10. Rounding off, keeping to one decimal place, we finally get A ≈ 101.8.

11. Through the above reasoning and calculation, the area of the regular octagon is approximately 101.8 square centimeters, rounded to the nearest tenth.

正多边形; 正弦; 余弦

Definition of Regular Polygon

A polygon is a regular polygon if and only if all its sides are of equal length and all its interior angles are equal.

Original: <note>在本题图中，正八边形</note>中，<note>所有边的长度相等，且所有内角相等</note>。因此，该图形是一个正八边形。

Translation: <note>In the figure of this problem, the regular octagon</note>, <note>all sides are of equal length, and all interior angles are equal</note>. Therefore, this figure is a regular octagon.

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>Point O is the center of the circle</note>, the marked vertices of the regular octagon in the figure are any points on the circle, the <note>pink line segment is the line segment from the center of the circle to any point on the circle</note>, therefore this line segment is the radius of the circle.

Area Formula of a Regular Polygon

The area formula for a regular polygon is A = (1/2) * n * r² * sin(360°/n), where n represents the number of sides, and r is the radius of the circumcircle.

Original: <note>正八边形的边数 n = 8</note>，<note>半径 r = 6</note>。根据正多边形的面积公式，正八边形的面积A = (1/2) * n * r² * sin(360°/n)，<note>其中n表示正八边形的边数，r表示正八边形的外接圆半径</note>。代入已知条件可得A = (1/2) * 8 * 6² * sin(360°/8) = (1/2) * 8 * 36 * <note>sin(45°)</note>。

Translation: <note>The number of sides of the regular octagon n = 8</note>, <note>radius r = 6</note>. According to the area formula of a regular polygon, the area of the regular octagon A = (1/2) * n * r² * sin(360°/n), <note>where n represents the number of sides of the regular octagon, and r represents the radius of the circumscribed circle of the regular octagon</note>. Substituting the known conditions, we get A = (1/2) * 8 * 6² * sin(360°/8) = (1/2) * 8 * 36 * <note


Question: Find y.

Answer: 21 \sqrt { 3 }

1. Given that one acute angle of a right triangle is 30°, and the other acute angle is 60°, it can be concluded that this triangle is a ##30°-60°-90°## right triangle. Therefore, the properties of the ##30°-60°-90° right triangle## can be applied.

2. In a ##30°-60°-90°## right triangle, the ratio of the side opposite the 30° angle to the side opposite the 60° angle is 1:√3, and the ratio of the side opposite the 30° angle to the hypotenuse is 1:2.

3. The given condition in the problem is that the length of the side opposite the 30° angle is 21. Therefore, using the properties of the 30-60-90 right triangle, the length of the hypotenuse is ##twice## that, i.e., x = 42.

4. Continuing to use the properties of the ##30°-60°-90°## triangle, given that the side opposite the 30° angle is 21, ##the length of the side opposite the 60° angle (the side where y is located) is √3 times the side opposite the 30° angle##, therefore y = 21√3.

5. Through the above reasoning, the final answer is 21√3.

直角三角形; 正弦; 余弦; 正切

Definition of Right Triangle

In the figure of this problem, <note>one angle in the triangle is 90 degrees</note>, therefore the triangle is a right triangle. <note>The two sides of the right angle are 21 and y</note>, <note>the hypotenuse is x</note>.

Properties of 30°-60°-90° Triangle

<note>Angle 30° is 30 degrees</note>, <note>Angle 60° is 60 degrees</note>, <note>Angle 90° is 90 degrees</note>. <note>Side x is the hypotenuse</note>, <note>Side 21 is the side opposite the 30-degree angle</note>, <note>Side y is the side opposite the 60-degree angle</note>. According to the properties of the 30°-60°-90° triangle, <note>Side x is equal to twice the side 21</note>, <note>Side y is equal to √3 times the side 21</note>. That is: x = 2 * 21, y = 21 * √3.


Question: As shown in the figure, AB is the diameter of ⊙O, point C is on ⊙O, and the tangent to ⊙O at point C intersects the extension of AB at point D. Connect AC. If ∠D = 50°, then the degree of ∠A is ()

Answer: 20°

1. Given that CD is tangent to ⊙O at C, ##connect OC, according to the property of the tangent line of the circle##, we get OC⊥CD, therefore ∠OCD=90°.

2. Given ∠D=50°, from ∠OCD=90°, and the sum of the interior angles of a triangle theorem, we get ∠COD=180°-90°-50°=40°.

3. ##∠COD is an exterior angle of triangle AOC, according to the exterior angle theorem of triangles, we know that ∠COD＝∠OAC＋∠OCA##.

4. ##Since OA and OC are radii of circle O, OA＝OC, triangle AOC meets the definition of an isosceles triangle, according to the property of isosceles triangles, we know that ∠OAC＝∠OCA##.

5.## Therefore ∠COD＝2∠OAC, substituting ∠COD=40° into the above equation, we get ∠A＝∠OAC＝20°##.

圆; 切线; 直角三角形; 圆周角

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>twice the radius</note>, that is, <note>AB = 2 * OA</note>.

Definition of Tangent to a Circle

A line is a tangent to a circle if and only if it has exactly one point of intersection with the circle. This point of intersection is called the point of tangency.

<note>Circle ⊙O and line CD have exactly one common point C</note>, this common point is called <note>tangent point C</note>. Therefore, <note>line CD is the tangent to circle ⊙O</note>.

Definition of Radius

In the circle <note>⊙O</note>, point <note>O is the center</note>, point A and point C are any points on the circle, the line segment <note>OA and the line segment OC are line segments from the center to any point on the circle</note>, therefore the <note>line segment OA and the line segment OC are the radius of the circle</note>.

Definition of Isosceles Triangle

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

In the original text: <note>triangle OAC</note>, <note>sides OA and OC are equal</note>, therefore <note>triangle OAC is an isosceles triangle</note>.

Property of the Tangent Line to a Circle

A tangent line to a circle is perpendicular to the radius that passes through the point of tangency.

In the figure of this problem, in circle O, <note>point C is the point of tangency of line CD with the circle</note>, <note>segment OC is the radius of the circle</note>. According to the property of the tangent line to a circle, <note>the tangent line CD is perpendicular to the radius OC at the point of tangency C</note>, that is, <note>∠OCD=90 degrees</note>.

Triangle Angle Sum Theorem

In the figure of this problem, in triangle OCD, <note>angle OCD</note>, <note>angle D</note>, and <note>angle COD</note> are the three interior angles of triangle OCD. According to the Triangle Angle Sum Theorem, <note>angle OCD + angle D + angle COD = 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.

<note>In the isosceles triangle OAC, sides OA and OC are equal</note>. Therefore, according to the properties of isosceles triangles, the angles opposite the equal sides are equal, that is, <note>angle OAC = angle OCA</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.

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

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.

<note>An interior angle of polygon OAC is ∠AOC</note>, <note>the angle ∠COD formed by extending the adjacent sides AO and OC of this interior angle</note> is called the exterior angle of the interior angle ∠AOC.


Question: If Z P = 4 x - 9 and P Y = 2 x + 5, find Z X.

Answer: 38

1. Given rectangle WXYZ, ##diagonals WY and ZX## intersect at point P, ##according to the properties of the diagonals of a rectangle, the diagonals are equal and bisect each other, so ZP = PY = WP = PX##.

2. The problem also gives ZP = 4x - 9 and PY = 2x + 5. Based on the conclusion from the previous step, we can obtain 4x - 9 = 2x + 5.

3. Moving 2x in the equation 4x - 9 = 2x + 5 to the left side of the equation and the constant to the right side, we get 4x - 2x = 5 + 9.

4. Combining, we get 2x = 14, solving for x, we get x = 7.

5. Substituting the value of x into ZP = 4x - 9, we get ZP = 4(7) - 9 = 28 - 9 = 19.

6. Since #### ZP = PX, therefore ZX = 2 * ZP = 2 * 19 = 38.

7. Through the above reasoning, we finally obtain the value of ZX as 38.

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 (≅).

In the figure of this problem, the quadrilateral WXYZ is a rectangle, with its interior angles <note>∠ZWX, ∠WXY, ∠XYZ, and ∠YZW all being right angles (90 degrees)</note>, and the sides <note>WX and ZY are parallel and equal in length</note>, <note>WZ and XY are parallel and equal in length</note>.

Property of Diagonals in a Rectangle

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

<note>In rectangle WXYZ, side WX is parallel and equal to side ZY</note>, <note>side WZ is parallel and equal to side XY</note>. <note>Diagonals WY and ZX are equal and bisect each other, meaning the intersection point P of diagonals WY and ZX is the midpoint of both diagonals</note>. Therefore, <note>segment WP is equal to segment PX</note>, <note>segment ZP is equal to segment PY</note>.


Question: Use parallelogram J K L M to find m \angle K L M.

Answer: 71

1. Given that quadrilateral JKLM is a parallelogram and ∠K = 109°. According to the theorem of supplementary adjacent angles in a parallelogram, adjacent interior angles in a parallelogram are supplementary, and the sum of the two angles is 180°.

2. In the parallelogram JKLM shown in the figure, ∠K and ∠L are adjacent, so ∠K + ∠L = 180°.

3. Given ∠K = 109°, substitute this value: 109° + ∠L = 180°.

4. Finally, solve for ∠L: ∠L = 180° - 109° = 71°. Therefore, ∠L = 71°.

5. Through the above reasoning, the final answer is ∠KLM = 71°.

平行四边形; 内错角; 同旁内角; 邻补角

Definition of Parallelogram

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

<note>Quadrilateral JKLM is a parallelogram</note>, <note>side JK is parallel and equal to side ML</note>, <note>side JM is parallel and equal to side KL</note>.

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

In <note>parallelogram JKLM</note>, <note>∠JKL</note> and <note>∠KLM</note> are adjacent interior angles, and <note>∠JML</note> and <note>∠MLK</note> are also adjacent interior angles. According to the <note>Adjacent Angles Supplementary Theorem of Parallelogram</note>, ∠JKL + ∠KLM = 180°, ∠JML + ∠MLK = 180°.


Question: Find x.

Answer: 6

1. Given that the three sides of triangle JKL are given in the form of expressions: JK = 3x + 6, KL = 4x, and LJ = 5x - 6.

2. From the figure, it is known that the three sides of triangle JKL have the same markings, which means JK = KL = LJ. According to the definition of an equilateral triangle, triangle JKL is an equilateral triangle.

3. Since JK = KL, we can set up the equation: 3x + 6 = 4x.

4. Solving the equation 3x + 6 = 4x, we get x = 6.

5. To verify the answer, we substitute x = 6 into the expressions for the other two sides: KL = 4x = 24, and LJ = 5x - 6 = 24, which is consistent with JK = 24. Thus, x = 6 is confirmed to be correct.

6. Through the above reasoning, the final answer is x = 6.

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°).

In the figure of this problem, <note>triangle JKL is an equilateral triangle</note>. <note>Sides JK, KL, and LJ are of equal length</note>, and <note>angles JKL, KLJ, and LJK are equal in measure, each being 60°</note>. Specifically, <note>JK = 3x + 6, KL = 4x, LJ = 5x - 6</note>, where each side of the triangle is equal.


Question: As shown in the figure, in Rt△ABC, ∠C=90°, AC=1, BC=2, then the value of \cosB is ()

Answer: \frac{2√{5}}{5}

1. Given in the right triangle ABC, ∠C=90°, AC=1, BC=2.

2. According to the Pythagorean theorem: in the right triangle ABC, the square of the hypotenuse AB is equal to the sum of the squares of AC and BC.

3. From the Pythagorean theorem: AB=√(AC^2 + BC^2) = √(1^2 + 2^2) = √(1 + 4) = √5.

4. According to the definition of the cosine of an acute angle, cos∠B = adjacent side/hypotenuse = BC/AB.

6. Finally, the answer is cos∠B = 2√5/5.

直角三角形; 余弦; 线段

Cosine Function

The cosine function is a trigonometric function that can be defined via the sides of a right triangle. For an angle α, its cosine value is equal to the ratio of the length of the adjacent side to the hypotenuse. In the unit circle, the cosine value is the x-coordinate of the point on the circle that makes an angle α with the positive x-axis.

In <note>right triangle ABC</note>, <note>side BC is the adjacent side of angle ∠ABC</note>, and <note>side AB is the hypotenuse</note>. Therefore, according to the <note>definition of the cosine function</note>, <note>the cosine value of angle ∠ABC is equal to the ratio of the adjacent side BC to the hypotenuse AB</note>, that is, <note>cos(∠ABC) = BC / AB</note>. In the figure, <note>BC=2</note>, <note>AB=√5</note>, therefore <note>cos(∠ABC) = 2/√5 = 2√5/5</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 figure of this problem, in the right triangle ABC, <note>angle ∠ACB is a right angle (90 degrees)</note>, <note>sides AC and BC are the legs</note>, <note>side AB is the hypotenuse</note>, so according to the Pythagorean Theorem, <note>AB##^2##= AC##^2##+ BC##^2##</note>. In the problem, <note>AC=1</note>, <note>BC=2</note>, substituting in gives <note>AB=√(1##^2##+ 2##^2##)=√(1 + 4)=√5</note>, therefore <note>AB=√5</note>.


Question: In \odot P, m \angle M P L = 65 and N P \perp P L. Find m \widehat N M.

Answer: 25

1. Given that in circle P, segment NP is perpendicular to PL, ##according to the definition of perpendicular lines##, ∠NPL is a right angle, and ##∠NPL = 90°##.

2. ##According to the problem statement, ∠MPL=65°, from the figure we can deduce that ∠NPM + ∠MPL=∠NPL, so ∠NPM+65° = 90°, thus ∠NPM = 90°-65° = 25°##.

3. ##According to the property of central angles, the measure of a central angle is equal to the measure of the arc it intercepts, ∠NPM is a central angle, and the arc it intercepts is arc NM, thus ∠NPM = arc NM = 25°##.

##4.## Therefore, we conclude that ##arc NM## = 25°.

##5.## The above is the analysis. Through reasoning, the final answer is 25°.

圆; 垂线; 圆周角; 圆心角; 弧

Definition of Perpendicular Lines

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

In the figure of this problem, <note>the angle ∠NPL formed by the intersection of line NP and line PL is 90 degrees</note>, so according to the definition of perpendicular lines, <note>line NP and line PL are perpendicular to each other</note>.

Definition of Central Angle

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

In the figure of this problem, in circle P, point N and point M are two points on the circle, and the center of the circle is point P. The angle ##∠NPM## formed by the lines <note>PN and PM</note> is called the central angle.


Question: Find x.

Answer: 115

1. Given that the degrees of the three interior angles of the triangle are 40°, 25°, and x, according to the triangle angle sum theorem, the sum of the three interior angles of a triangle is equal to 180°, we can obtain the equation: 40° + 25° + x = 180°.

2. Simplifying, we get: 65° + x = 180°.

3. Solving the equation: move 65° to the right side of the equation, we get: x = 180° - 65°.

4. Calculate the result on the right side, we get: x = 115°.

5. Through the above reasoning, the final answer is 115°.

Triangle Angle Sum Theorem

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

<note>The three interior angles of the triangle are 40°, 25°, and x</note>, according to the Triangle Angle Sum Theorem, <note>40° + 25° + x = 180°</note>.

Definition of Triangle

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

<note>In the figure of this problem, the triangle is a geometric shape formed by three non-collinear points and their connecting line segments.</note> The three points are the three vertices of the triangle, and the three line segments are the three sides of the triangle.


Question: Quadrilateral ABCD is a rectangle. m \angle 2 = 40. Find m \angle 3.

Answer: 40

1. Given quadrilateral ABCD is a rectangle. According to the definition of a rectangle, the opposite sides of a rectangle are parallel and equal in length.

2. According to the parallel axiom 2 of parallel lines, it is known that side AB is parallel to side CD, and they are intersected by the third line AD, corresponding angles are equal, alternate interior angles are equal, and same-side interior angles are supplementary. Among them, ∠2 and ∠3 are on the interior side of line AD and sides AB and CD, so ∠2 and ∠3 are alternate interior angles.

3. It is known that ∠2 and ∠3 are alternate interior angles, so ∠2 and ∠3 are equal, that is, ∠2 = ∠3.

5. Through the above reasoning, the final answer is ∠3 = 40°.

Definition of Rectangle

<note>Quadrilateral ABCD is a rectangle</note>, with interior angles <note>∠DBA, ∠BAC, ∠ACD, ∠CDB all being right angles (90 degrees)</note>, and <note>sides AB and CD are parallel and equal in length</note>, <note>sides AC and BD are parallel and equal in length</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, <note>two parallel lines AB and CD</note> are intersected by <note>a third line AD</note>, forming the following geometric relationships: 1. Corresponding angles: none. 2. Alternate interior angles: <note>angle 2 and angle 3 are equal</note>. 3. Same-side interior angles: none. These relationships illustrate that when two parallel lines are intersected by a third line, the alternate interior angles are equal.

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>Angle 2 and Angle 3 are located between two parallel lines</note>, and <note>on opposite sides of the transversal AD</note>, therefore, <note>Angle 2 and Angle 3 are alternate interior angles</note>. Alternate interior angles are equal, that is, <note>Angle 2 is equal to Angle 3</note>.


Question: As shown in the figure, in ⊙O, ∠AOB=50°, then the degree of the inscribed angle ∠ACB is ()

Answer: 25°

1. Given that in circle O, ∠AOB=50°.

2. According to the inscribed angle theorem, the inscribed angle is half of the corresponding central angle, which refers to ∠ACB and its corresponding central angle ∠AOB.

3. By the inscribed angle theorem, we get ∠ACB=rac{1}{2}∠AOB.

4. Substituting the given condition, ∠ACB=rac{1}{2}×50°=25°.

5. Through the above reasoning, the final answer is 25°.

圆; 圆心角; 圆周角; 弧; 弦

Definition of Central Angle

In the figure of this problem, in the circle <note>O</note>, <note>point A and point B are two points on the circle</note>, <note>the center of the circle is point O</note>. <note>The angle ∠AOB formed by the lines OA and OB</note> is called the central angle. <note>The degree of the central angle ∠AOB is 50°</note>.

Definition of Inscribed Angle

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

<note>In the diagram of this problem, in circle O, the vertex C of angle ∠ACB is on the circumference</note>, <note>the two sides of angle ∠ACB intersect circle O at points A and B respectively</note>. Therefore, angle ∠ACB is an inscribed angle.

Inscribed Angle Theorem

In a circle, an inscribed angle is equal to half of the central angle that subtends the same arc.

In the figure of this problem, in circle O, points A, B, and C are on the circle, <note>arc AB</note> corresponds to the <note>central angle ∠AOB</note>, and the <note>inscribed angle ∠ACB</note>. According to the Inscribed Angle Theorem, ∠ACB is equal to half of the central angle ∠AOB corresponding to the arc AB, that is, <note>∠ACB = 1/2 ∠AOB</note>.


Question: As shown in the figure, quadrilateral ABCD is inscribed in ⊙O, ∠BOD=70°, then the degree of ∠BCD is ()

Answer: 145°

1. Given that quadrilateral ABCD is inscribed in circle O, ∠BOD=70°.

2. According to the ##Inscribed Angle Theorem##, the inscribed angle is equal to half of the central angle that subtends the same arc. Therefore, ∠BAD=1/2×∠BOD=1/2×70°=35°.

3. Since quadrilateral ABCD is a cyclic quadrilateral, according to the ##(Corollary 3 of the Inscribed Angle Theorem) Theorem of Supplementary Opposite Angles of Cyclic Quadrilateral##, the sum of opposite angles is equal to 180°.

4. Hence, ∠BCD=180°-∠BAD=180°-35°=145°.

5. Through the above reasoning, the final answer is 145°.

圆; 圆内接四边形; 圆心角; 圆周角; 弧

Cyclic Quadrilateral

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

In the figure of this problem, <note>the four vertices A, B, C, and D of quadrilateral ABCD are on the same circle O</note>. <note>This circle is called the circumcircle of quadrilateral ABCD</note>. Therefore, quadrilateral ABCD is a cyclic quadrilateral. According to the properties of cyclic quadrilaterals, it can be concluded that <note>the sum of opposite angles is equal to 180 degrees</note>, i.e., <note>angle BAD + angle BCD = 180 degrees</note>, <note>angle ABC + angle ADC = 180 degrees</note>.

Definition of Inscribed Angle

In the figure of this problem, in circle O, <note>the vertex A of angle BAD is on the circumference</note>, <note>the two sides of angle BAD intersect circle O at points B and D respectively</note>. Therefore, <note>angle BAD is an inscribed angle</note>.

Definition of Central Angle

<note>In circle O, points B and D are two points on the circle</note>, <note>the center of the circle is point O</note>. The angle ∠BOD formed by the lines <note>OB and OD</note> is called the central angle.

Inscribed Angle Theorem

In circle O, points A, B, C, and D are on the circle, the arc BD corresponds to the central angle ∠BOD, and the inscribed angle is ∠BAD. According to the Inscribed Angle Theorem, ∠BAD is equal to half of the central angle ∠BOD that corresponds to the arc BD, that is, ∠BAD = 1/2 ∠BOD.

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

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

<note>In the cyclic quadrilateral ABCD, the vertices A, B, C, and D are all on the circle</note>. According to the Diagonal Supplementary Theorem for Cyclic Quadrilateral, the sum of each pair of opposite angles of the quadrilateral ABCD is equal to 180°. Specifically, <note>∠BAD + ∠BCD = 180°</note>; <note>∠ABC + ∠CDA = 180°</note>. Therefore, <note>∠BCD = 180° - ∠BAD = 180° - 35° = 145°</note>.


Question: Place the protractor on a broken glass fragment as shown in the figure, with point A on the semicircle, and the readings at points B and C being 105° and 155° respectively. Then the measure of ∠BAC is ()

Answer: 25°

1. Given that the protractor is placed on a broken glass fragment as shown in the figure, point A is on the semicircle, and the readings of points B and C are 105° and 155° respectively. Let the center of the semicircle be O, and connect OB and OC.

2. According to the problem, point B is at the 105° position, and point C is at the 155° position, then ∠BOC ####= 155° - 105° = 50°.

3. Since ∠BAC and ∠BOC both subtend arc BC, according to the inscribed angle theorem, the inscribed angle subtending the same arc is half of the central angle subtending that arc, thus ∠BAC = 1/2 ∠BOC.

4. Substituting the given condition ∠BOC = 50°, therefore ∠BAC = 1/2 × 50° = 25°.

5. Through the above reasoning, the final answer is 25°.

圆周角; 圆心角; 射线

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, <note>the center of the semicircle is O</note>, and the points <note>A, B, and C are on the semicircle</note>. <note>OB and OC are the radii from the center O to points B and C on the circumference</note>. In the figure, <note>all points that are at a fixed distance (i.e., the radius) from point O are on circle O</note>.

Definition of Central Angle

In the figure of this problem, in circle O, <note>points B and C are two points on the circle</note>, <note>the center of the circle is point O</note>. <note>The angle ∠BOC formed by the lines OB and OC</note> is called the central angle.

Definition of Inscribed Angle

In the diagram of this problem, <note>in circle O</note>, <note>the vertex A of angle BAC is on the circumference</note>, <note>the two sides of angle BAC intersect circle O at points B and C respectively</note>. Therefore, <note>angle BAC is an inscribed angle</note>.

Inscribed Angle Theorem

In the figure of this problem, <note>∠BAC</note> and central angle <note>∠BOC</note> subtend the same arc BC. According to the Inscribed Angle Theorem, <note>∠BAC</note> is equal to half of the central angle <note>∠BOC</note> that subtends arc <note>BC</note>, i.e., <note>∠BAC = 1/2 ∠BOC</note>. Given that <note>∠BOC = 50°</note>, therefore <note>∠BAC = 25°</note>.


Question: m is the perpendicular bisector of X Z, W Z = 14.9. Find W X.

Answer: 14.9

1. Given that m is the perpendicular bisector of XZ, WZ = 14.9.

2. According to the definition of a perpendicular bisector, a perpendicular bisector is a line that passes through the midpoint of a segment and is perpendicular to that segment. Therefore, m passes through the midpoint Y of XZ, and ∠XYW = ∠ZYW = 90°.

3. According to the properties of a perpendicular bisector, any point on the perpendicular bisector is equidistant from the endpoints of the segment.

4. Therefore, the distance from W to X is equal to the distance from W to Z, i.e., WX = WZ.

5. Hence, from the above reasoning, we can conclude that WX = 14.9.

垂直平分线; 中点; 等腰三角形

Definition of Perpendicular Bisector

A line is a perpendicular bisector if and only if it passes through the midpoint of a line segment and is perpendicular to that line segment.

In the figure of this problem, <note>line m passes through the midpoint Y of segment XZ</note>, and <note>line m is perpendicular to segment XZ</note>. Therefore, <note>line m is the perpendicular bisector of segment XZ</note>.

Properties of the Perpendicular Bisector

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

<note>The perpendicular bisector of segment XZ is line m</note>, <note>point W is on line m</note>. According to the properties of the perpendicular bisector, <note>point W is equidistant from the endpoints X and Z of segment XZ</note>, i.e., <note>WX = WZ</note>. Therefore, WX = 14.9.


Question: As shown in the figure, points A, B, C, and D are on ⊙O. DE⊥OA, DF⊥OB, with E and F being the feet of the perpendiculars respectively. If ∠EDF=50°, then the degree of ∠C is ()

Answer: 65°

1. Given that DE is perpendicular to OA, and DF is perpendicular to OB, according to the definition of perpendicularity, we can get ∠OED = 90°, ∠OFD = 90°.

 2. The degree of ∠EDF is 50°, ##according to the theorem of the sum of interior angles of a quadrilateral, in any quadrilateral, the sum of the four interior angles is 360°##. Therefore, the sum of the interior angles of quadrilateral OEDF is ∠OED + ##∠DFO + ∠EOF + ∠EDF = 360°, given that ∠EDF = 50°, ∠OED = 90°, ∠OFD = 90°##.

 3. According to the theorem of the sum of interior angles of a quadrilateral: ##∠EOF## = 360° - (90° + 90° + 50°) = 130°, ##∠EOF is ∠AOB##.

 4. According to the inscribed angle theorem, in the same circle, the degree of the central angle is twice that of the inscribed angle. Therefore, we can get ∠ACB = 1/2 * ∠AOB.

 5. Substituting the values, we get ∠ACB = 1/2 * 130° = 65°.

 ##6##. Through the above reasoning, the final answer is ∠C = 65°.

圆; 圆周角; 垂线; 圆内接四边形; 直角三角形

Definition of Perpendicular Lines

<note>Line DE and line OA intersect to form an angle ∠OED of 90 degrees</note>, <note>Line DF and line OB intersect to form an angle ∠OFD of 90 degrees</note>, therefore according to the definition of perpendicular lines, <note>Line DE and line OA are perpendicular to each other</note>, <note>Line DF and line OB are perpendicular to each other</note>.

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

In the figure of this problem, quadrilateral <note>OEDF</note>, <note>angle OED</note>, <note>angle OFD</note>, <note>angle EDF</note>, and <note>angle EOF</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>∠OED + ∠OFD + ∠EDF + ∠EOF = 360°</note>.

Inscribed Angle Theorem

<note>In the diagram of this problem, points A, B, C, and D are on circle O</note>, the central angle corresponding to <note>arc AB</note> is <note>∠AOB</note>, and the inscribed angle is <note>∠ACB</note>. According to the Inscribed Angle Theorem, <note>∠ACB is equal to half of the central angle ∠AOB corresponding to arc AC</note>, that is, <note>∠ACB = 1/2 ∠AOB</note>.

Definition of Central Angle

<note>In the diagram of this problem, in circle O</note>, point A and point B are two points on the circle, <note>the center of the circle is point O</note>. <note>The angle ∠AOB formed by the lines OA and OB is called the central angle</note>.

Definition of Inscribed Angle

In <note>circle O</note>, the vertex of <note>angle ACB</note> (<note>point C</note>) is on the <note>circumference</note>, and the two sides of <note>angle ACB</note> intersect <note>circle O</note> at <note>point A</note> and <note>point B</note> respectively. Therefore, <note>angle ACB</note> is an <note>inscribed angle</note>.


Question: Find KL.

Answer: 12

##1. Triangle KJL is given as an equilateral triangle, where KJ = KL = JL.

2. Since this is an equilateral triangle, we know that the sides KJ and KL are equal.

3. The side lengths of the triangle are given as KJ = x + 7, JL = 4x - 8.

4. Based on the equality of sides in an equilateral triangle, we can establish the equation KJ = JL. Therefore, x + 7 = 4x - 8.

5. Solving the equation for x, we subtract x from both sides, resulting in: 7 = 3x - 8.

6. To isolate x, we add 8 to both sides: 15 = 3x.

7. Finally, we divide both sides by 3: x = 5.

8. Substituting x = 5 into the expression for KL, which is 4x - 8.

9. Calculating KL: 4(5) - 8 = 20 - 8 = 12.

10. Therefore, the length of side KL is 12.##

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>In the diagram of this problem, triangle KJL is an equilateral triangle</note>. <note>The lengths of side KJ, side KL, and side JL are equal</note>, and <note>the degrees of angle J, angle K, and angle L are equal, each being 60°</note>.


Question: As shown in the figure, quadrilateral ABCD is inscribed in ⊙O. If ∠ABC = 40°, then the degree of ∠ADC is ()

Answer: 140°

1. Given that quadrilateral ABCD is inscribed in ⊙O, according to the property of opposite angles of a cyclic quadrilateral, we get ∠ABC + ∠ADC = 180°.

2. Also given that ∠ABC=40°, substituting into the equation obtained in the previous step, we get ∠ADC = 180° - 40°.

3. Calculating, we get ∠ADC = 140°.

4. Through the above reasoning, the final answer is 140°.

圆内接四边形; 圆周角; 圆

Cyclic Quadrilateral

<note>The four vertices A, B, C, and D of quadrilateral ABCD are all on the same circle O</note>. This circle is called <note>the circumcircle of quadrilateral ABCD</note>. Therefore, <note>quadrilateral ABCD is a cyclic quadrilateral</note>. According to the properties of a cyclic quadrilateral, it can be concluded that <note>the sum of opposite angles is equal to 180 degrees</note>, that is, <note>angle ABC + angle ADC = 180 degrees</note>, <note>angle BAD + angle BCD = 180 degrees</note>.

<note>Quadrilateral ABCD is inscribed in circle O</note>, and the vertices of the quadrilateral <note>A, B, C, D are all on the circle</note>. According to the <note>Diagonal Supplementary Theorem for Cyclic Quadrilateral</note>, the sum of each pair of opposite angles of quadrilateral ABCD is equal to 180°. Specifically, <note>∠ABC + ∠ADC = 180°</note>; <note>∠BAD + ∠BCD = 180°</note>.


Question: Find x.

Answer: 22.5

1. Let the large triangle be ABC (counterclockwise from the top vertex), the vertical dividing line of the hypotenuse and the long right-angle side be EF, point E is on AC, point F is on AB.

2. From the figure, we get: AF=8, BF=18, AE=10, AC=x, ∠BAC=∠FAE, ∠ABC=∠AFE=90°, according to the similarity theorem (AA), triangle ABC is similar to triangle AFE.

3. According to the definition of similar triangles, △ABC∽△AEF, the ratio of sides is AB/AF = AC/AE.

4. Since AB=AF+BF=18+8=26, substituting the values: 26/8=AC/10, 8AC=26*10=260, 8AC=260/8, AC=32.5.

5. Solving for AC=32.5, find EC(x)=AC-AE=22.5.

6. Through the above reasoning, the final answer is x = 22.5.

Similarity Theorem for Triangles (AA)

Two triangles are similar if two of their corresponding angles are congruent.

In the figure of this problem, triangle ABC and triangle AFE have angle ABC equal to angle AFE, and angle BAC equal to angle FAE, so triangle ABC is similar to triangle AFE.

Definition of Similar Triangles

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

In the figure of this problem, <note>triangle ABC and triangle AFE</note> are similar triangles. According to the definition of similar triangles: <note>angle ABC = angle AFE</note>, <note>angle BAC = angle FAE</note>, angle ACB = angle AEF; <note>AB/AF = BC/FE = AC/AE</note>.

Definition of Perpendicular Lines

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

The lines <note>EF and AF</note> intersect to form the <note>angle ∠AFE which is 90 degrees</note>, therefore according to the <note>definition of perpendicular lines</note>, <note>lines AF and FE</note> are perpendicular to each other.


Question: Find x.

Answer: 4 \sqrt { 6 }

1. Let the right-angle vertex at the lower left be C, AC=z, AB=x, AD=y. It is known that triangles ABC and BCD are two right triangles.

2. In triangles ABC and BCD, angle B is the common angle, angle ACB=angle BDC=90 degrees. According to the similarity theorem of triangles (AA), triangles ABC and BCD are similar.

3. According to the definition of similar triangles, we have: BC/AB = BD/BC, which is x/12 = 8/x. Further simplification gives x^2 = 8*12.

5. Solving for the square root, we get x = 4√6.

Definition of Right Triangle

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

<note>The right angle of the larger right triangle is located at the lower left corner</note>, the smaller right triangle's <note>right angle is located internally at the lower right side</note>. In the larger right triangle, <note>the angle located at the lower left corner is a right angle (90 degrees)</note>, thus this triangle is a right triangle. <note>Sides z and x are the legs, side 12 is the hypotenuse</note>. In the smaller right triangle, <note>the angle located at the lower right side is a right angle (90 degrees)</note>, thus this triangle is a right triangle. <note>Sides y and 8 are the legs, side x is the hypotenuse</note>.

Definition of Similar Triangles

<note>In the figure of this problem, since the larger right triangle and the smaller right triangle are similar</note>, the corresponding sides <note>x and 12</note> and <note>8 and x</note> are proportional, that is <note>x/12 = 8/x</note>.

Similarity Theorem for Triangles (AA)

In the figure of this problem, triangles <note>ABC</note> and <note>BCD</note>, if <note>angle B</note> is equal to <note>angle B</note>, and <note>angle ABC</note> is equal to <note>angle BDC</note>, then triangle <note>ABC is similar to triangle BCD</note>.


Question: As shown in the figure, BD bisects ∠ABC, CD∥AB, if ∠BCD=70°, then the degree of ∠CDB is ()

Answer: 55°

1. Given CD is parallel to AB, according to the definition of corresponding angles and the parallel postulate 2 of parallel lines, ∠BCD and ∠CBA are supplementary, i.e., ∠BCD + ∠CBA = 180°.

2. Given ∠BCD = 70°, substituting into the previous conclusion gives ∠CBA = 180° - 70° = 110°.

3. Given BD bisects ∠ABC, according to the definition of angle bisector, ∠DBC = ∠ABD = 1/2∠CBA. Substituting gives ∠DBC = ∠ABD = 110° / 2 = 55°.

4. According to the definition of alternate interior angles and the parallel postulate 2 of parallel lines, ∠ABD = ∠CDB = 55°.

5. Through the above reasoning, the final answer is ∠CDB = 55°.

平行线; 内错角; 普通三角形

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 ABC is point B<note>, from <note>point B a line BD is drawn<note>, this line divides <note>angle ABC into two equal angles<note>, namely <note>angle ABD and angle DBC are equal<note>. Therefore, <note>line BD is the angle bisector of angle ABC<note>.

Parallel Postulate 2 of Parallel Lines

Original text: <note>CD∥AB</note>, and the line <note>BE</note> intersects them, forming the following geometric relationships: 1. Corresponding angles: <note>none</note>. 2. Alternate interior angles: <note>∠CDB and ∠ABD are equal</note>. 3. Consecutive interior angles: <note>∠BCD and ∠CBA are supplementary, that is, ∠BCD + ∠CBA = 180°</note>. These relationships illustrate that when two parallel lines are intersected by a third line, the corresponding angles are equal, the alternate interior angles are equal, and the consecutive interior angles are supplementary.

Definition of Consecutive Interior Angles

When two lines are intersected by a third line, the angles that lie on the same side of the transversal and inside the two intersected lines are called consecutive interior angles. If the two lines are parallel, then the consecutive interior angles are supplementary. Conversely, if the consecutive interior angles are supplementary, then the two lines are parallel.

<note>In the figure of this problem, two lines xx and xx are intersected by a third line xx, with angles xxx and xxx on the same side of the intersecting line xx, and within the intersected lines xx and xx</note>, so <note>angles xxx and xxx are consecutive interior angles</note>. <note>Consecutive interior angles xxx and xxx are supplementary, that is, angle xxx + angle xxx = 180 degrees</note>.

Definition of Alternate Interior Angles

<note>In the figure of this problem, two parallel lines xxx and xxx are intersected by a line xx, where angle xxx and angle xxx are located between the two parallel lines and on opposite sides of the intersecting line xx. Therefore, angle xxx and angle xxx are alternate interior angles</note>. Alternate interior angles are equal, that is, <note>angle xxx is equal to angle xxx</note>.


Question: As shown in the figure, the lateral surface of a cone is a semicircle with a radius of 10. Find the radius of its base.

Answer: 5

1. Let the radius of the base of the cone be r, and the lateral surface unfolded diagram be a semicircle with a radius of 10.

2. According to the problem statement, the circumference of the base of the cone is equal to the arc length of the semicircle. According to the formula for the circumference of a circle, the circumference of the base of the cone = 2πr, the circumference of the semicircle = 1/2 the circumference of the circle = 1/2 * 10 * 2π = π * 10, thus 2πr = π * 10.

4. Therefore, the radius of the base of the cone is 5.

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>The base of the cone is a circle</note>, its <note>radius is r</note>, <note>circumference is 2πr</note>. The lateral surface of the cone's unfolded diagram is <note>a semicircle</note>, <note>the radius of the semicircle is 10</note>, <note>the circumference of the semicircle is π * 10</note>.

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>A semicircle is formed by a diameter and an arc with a radius of 10</note>. The circumference of the semicircle is (1/2 * 2π * 10) + 10 = π * 10.

Development of a Cone

The development (or net) of a cone is a sector of a circle, where the radius of the sector is the slant height of the cone, and the arc length of the sector is equal to the circumference of the cone's base.

<note>In the figure of this problem</note>, the lateral development of the cone is <note>a semicircle</note>. The radius of the semicircle is <note>the slant height of the cone (10)</note>, and the arc length of the semicircle (<note>π * 10</note>) is equal to <note>the circumference of the base of the cone (2πr)</note>. Therefore, <note>the radius of the semicircle (10)</note> is equal to <note>the slant height of the cone</note>, and the arc length of the semicircle (<note>π * 10</note>) is equal to <note>the circumference of the base circle of the cone (2πr)</note>, where <note>r is the radius of the base circle</note>.


Question: Find x such that the quadrilateral is a parallelogram.

Answer: 13

1. For the given quadrilateral, it can be seen from the figure that the two pairs of opposite sides are 2x - 5 and 3x - 18, as well as 5y and 2y + 12.

2. According to the problem statement, assuming the quadrilateral is a parallelogram, then the two pairs of opposite sides are parallel and equal.

3. Based on the definition of a parallelogram, for one pair of opposite sides, we have 2x - 5 = 3x - 18.

4. Solving the equation 2x - 5 = 3x - 18, by rearranging terms we get 2x - 3x = -18 + 5, which simplifies to -x = -13, hence x = 13.

5. For the other pair of opposite sides, the equation is 5y = 2y + 12.

6. Solving the equation 5y = 2y + 12, we get 5y - 2y = 12, which simplifies to 3y = 12, so y = 4.

7. Verify whether the above values x = 13 and y = 4 satisfy the condition that all opposite sides are equal.

8. Substituting x = 13 into 2x - 5 and 3x - 18, we get 2 * 13 - 5 = 21 and 3 * 13 - 18 = 21. The equality holds.

9. Substituting y = 4 into 5y and 2y + 12, we get 5 * 4 = 20 and 2 * 4 + 12 = 20. The equality holds.

10. After verification, it is found that x = 13 and y = 4 satisfy the given conditions, hence the quadrilateral is a parallelogram.

11. Since the problem requires solving for the value of x, the final answer is x = 13.

Definition of Parallelogram

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

<note>The two pairs of opposite sides of the quadrilateral are 2x - 5 and 3x - 18, as well as 5y and 2y + 12</note>. According to the definition of a parallelogram, <note>the two pairs of opposite sides are parallel and equal</note>, that is, <note>2x - 5 = 3x - 18</note>, <note>5y = 2y + 12</note>.

Definition of Quadrilateral

A quadrilateral is a closed figure formed by four line segments, which are referred to as the sides of the quadrilateral. A quadrilateral has four vertices and four interior angles.

<note>A quadrilateral is composed of four line segments 2x - 5, 3x - 18, 5y, and 2y + 12</note>, these segments are called the sides of the quadrilateral.


Question: Find x.

Answer: 5 \sqrt { 3 }

1. Let the three vertices of the triangle be A, B, and C, it is known that ∠ACB=90°, according to the definition of a right triangle, △ABC is a right triangle.

2. It is known that ∠CAB=60°, ∠ACB=90°, according to the triangle angle sum theorem, ∠ABC=180°-∠ACB-∠BAC=180°-90°-60°=30°.

3. According to the properties of a 30°-60°-90° triangle, the side opposite the 60° angle is √3 times the shorter side, i.e., BC=√3AC.

4. It is known that BC=15, AC=x, substituting the specific values we have 15=√3x, x=5√3.

5. After the above reasoning, the final answer is 5√3.

直角三角形; 正弦; 余弦

Definition of Right Triangle

In the diagram of this problem, in triangle ABC, <note>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</note>, <note>side AB is the hypotenuse</note>.

Triangle Angle Sum Theorem

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

In <note>triangle ABC</note>, <note>angle ABC</note>, <note>angle BAC</note>, and <note>angle ACB</note> are <note>the three interior angles of triangle ABC</note>. According to the <note>Triangle Angle Sum Theorem</note>, <note>angle ABC + angle BAC + angle ACB = 180°</note>.

Properties of 30°-60°-90° Triangle

In a 30°-60°-90° triangle, the side opposite the 30-degree angle (the shorter leg) is half the length of the hypotenuse. The side opposite the 60-degree angle (the longer leg) is √3 times the length of the shorter leg.

In the figure of this problem, <note>30°-60°-90° triangle ABC</note>, <note>angle ABC is 30 degrees</note>, <note>angle BAC is 60 degrees</note>, <note>angle BCA is 90 degrees</note>. <note>Side AB is the hypotenuse</note>, <note>side AC is opposite the 30-degree angle</note>, <note>side BC is opposite the 60-degree angle</note>. According to the properties of a 30°-60°-90° triangle, <note>side AC is half of side AB</note>, <note>side BC is √3 times side AC</note>. That is: <note>AC = 1/2 * AB</note>, <note>BC = AC * √3</note>.


Question: As shown in the figure, AB∥CD, AE bisects ∠CAB and intersects CD at point E. If ∠C=70°, then the degree of ∠AED is ()

Answer: 125°

1. Given AB is parallel to CD, according to ##Parallel Line Axiom 2 and the definition of same-side interior angles##, we get ∠BAC + ∠ACD = 180°.

2. ∵ ∠C = 70°, from step 1 we get ∠BAC = 180° - ∠C = 180° - 70° = 110°.

3. ∵ AE bisects ∠CAB, according to the definition of angle bisector, we get ∠CAE = 0.5×∠BAC = 0.5×110° = 55°.

4. ##According to the exterior angle theorem of triangles, we get ∠AED = ∠ACE + ∠CAE##.

5. ##Since ∠ACE = ∠C (i.e., ∠C = 70°)##, we get ∠AED = ∠C + ∠CAE = 70° + 55° = 125°.

6. Through the above reasoning, the final answer is 125°.

平行线; 内错角; 等腰三角形

Definition of Angle Bisector

<note>The vertex of angle CAB is point A</note>, a line AE is drawn from <note>point A</note>, this line divides <note>angle CAB into two equal angles</note>, namely <note>angle CAE and angle EAB are equal</note>. Therefore, <note>line AE is the angle bisector of angle CAB</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>Given AB∥CD</note>, the line <note>AC intersects these two parallel lines</note>, forming the following geometric relationships: ##1. <note>Corresponding angles are equal</note>. 2. <note>Alternate interior angles are equal</note>. 3. <note>Consecutive interior angles: ∠CAB and ∠ACD are supplementary, i.e., ∠CAB + ∠ACD = 180°</note>##. These relationships illustrate that when <note>two parallel lines AB and CD are intersected by a third line AC</note>, corresponding angles are equal, alternate interior angles are equal, and consecutive interior angles are supplementary.##

In the figure of this problem, <note>two lines AB and CD are intersected by a third line AC</note>, <note>the two angles CAB and ACD are on the same side of the transversal AC</note>, and <note>within the lines AB and CD being intersected</note>, so <note>angles CAB and ACD are consecutive interior angles</note>. <note>Consecutive interior angles CAB and ACD are supplementary</note>, that is, <note>angle CAB + angle ACD = 180 degrees</note>.

Definition of Triangle

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

<note>Triangle ACE is a geometric figure formed by three non-collinear points A, C, E and their connecting line segments AC, AE, CE</note>。<note>Points A, C, E are the three vertices of the triangle</note>，<note>Line segments AC, AE, CE are the three sides of the triangle</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.

<note>Angle AED is an exterior angle of the triangle</note>, <note>angles CAE and ACE are the two non-adjacent interior angles to the exterior angle AED</note>, according to the Exterior Angle Theorem of Triangle, <note>the exterior angle AED is equal to the sum of the two non-adjacent interior angles CAE and ACE</note>, that is, <note>angle AED = angle CAE + angle ACE</note>.

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

The original text: <note>An interior angle of polygon ACE is ∠AEC</note>, the angle ∠AED formed by extending <note>the adjacent sides CE and AE of the interior angle</note> is called <note>the exterior angle of the interior angle ∠AEC</note>.


Question: Find the area of the figure. Round the result to the nearest tenth.

Answer: 50.2

1. The given condition is that the radius of the circle is 4 inches. The area of the circle can be calculated using the formula A = ##πr^2##, where A is the area, r is the radius, and π is the constant pi.

2. According to the given condition r = 4 inches, substitute the radius into the area formula to get A = ##π * (4)^2##.

3. By calculation, we get A = 16π.

4. Using the common approximation π ≈ 3.14 for calculation, A ≈ 16 * 3.14.

5. The result of the calculation is A ≈ 50.24.

6. Round the area to the nearest tenth as required by the problem, the result is 50.2.

7. Based on the above reasoning, the final answer is 50.2.

Definition of Radius

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

The line segment formed from <note>the center of the circle</note> to <note>any point on the circumference</note> is called the <note>radius</note>. The segment marked as <note>4 inches</note> in the figure is the <note>radius</note>.

Area Formula of a Circle

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

<note>The radius of the circle is 4 inches</note>. According to the area formula of a circle, the area A of the circle is equal to pi π multiplied by <note>the square of the radius 4</note>, which is <note>A = π(4)^2</note>.


Question: Find the perimeter of the figure. If necessary, round to the nearest tenth.

Answer: 24

1. First, analyze the geometric figure given in the problem. It is a concave polygon. Clearly given are the lengths of each side: the 3 outer sides each have a length of 5, and the 3 inner sides each have a length of 2.

2. According to the formula for the perimeter of a polygon. The perimeter of the polygon is equal to the sum of the lengths of all its sides. That is, the perimeter P = 5+5+5+2+2+2=21

3. Therefore, the perimeter is 21.

Perimeter Formula for Polygons

The perimeter of a polygon is equal to the sum of the lengths of all its sides. For an n-sided polygon, if the lengths of the sides are \( a_1, a_2, \ldots, a_n \), then the perimeter \( P \) is given by \( P = a_1 + a_2 + \ldots + a_n \).

In the figure of this problem, it is known that <note>each side of the n-sided polygon is 5, 5, 5, 2, 2, 2</note>. According to the <note>Perimeter Formula for Polygons</note>, <note>the perimeter of the n-sided polygon is equal to the sum of the lengths of all its sides</note>, that is, <note>Perimeter P = 5 + 5 + 5 + 2 + 2 + 2</note>.


Question: In the figure above, ABCD is a rectangle. What is the area of triangle AFB?

Answer: 9

1. Given condition: ABCD is a rectangle, the length of side AB is 6, and the length of side BC is 3.

2. According to the definition of a rectangle, the opposite sides of a rectangle are parallel and equal, so AD = BC = 3, AB = DC = 6.

3. According to the definition of a rectangle, each interior angle of rectangle ABCD is a right angle, so ∠DAB = ∠ADC = 90°.

4. Draw a perpendicular line EF through point F, intersecting AB at point E, ∠AEF = 90°.

5. According to the theorem of the sum of interior angles of a quadrilateral, the sum of the interior angles of quadrilateral AEFD is 360°, so ∠DFE = 360° - ∠FDA - ∠DAE - ∠AEF = 360° - 90° - 90° - 90° = 90°. According to the rectangle determination theorem, quadrilateral AEFD is a rectangle.

6. According to the definition of a rectangle, AD = EF = 3, and the height EF of △AFB is 3.

7. The length of the base AB is 6, so the base length of △AFB is 6, and the height is 3.

8. According to the triangle area formula: Area = 1/2 × Base × Height, the area of △AFB = 1/2 × 6 × 3.

9. Simplified calculation: the area of △AFB = 1/2 × 18 = 9.

矩形; 普通三角形; 垂线

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 <note>interior angles ∠DAB, ∠ABC, ∠BCD, and ∠CDA all being right angles (90 degrees)</note>, and <note>sides AB and CD are parallel and equal in length</note>, <note>sides AD and BC are parallel and equal in length</note>. <note>Quadrilateral AEFD is a rectangle</note>, with <note>interior angles ∠DAE, ∠AEF, ∠EFD, and ∠FDA all being right angles (90 degrees)</note>, and <note>sides DF and AE are parallel and equal in length</note>, <note>sides AD and EF are parallel and equal in length</note>.

Area Formula of a Triangle

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

In the figure of this problem, in triangle AFB, side AB is the base, and segment EF is the height. According to the area formula of a triangle, the area of triangle AFB is equal to the base AB multiplied by the height EF and then divided by 2, that is, Area = (6 * 3) / 2.

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 EF perpendicular to the opposite side AB (or its extension) from vertex F</note> is the altitude from vertex F. <note>The line segment FE forms a right angle (90 degrees) with side AB (or its extension)</note>, which indicates that the line segment EF is the perpendicular distance from vertex F to the opposite side AB (or its extension).

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

<note>In quadrilateral AEFD, angle DAE, angle AEF, angle EFD, and angle FDA are the four interior angles of the quadrilateral</note>, according to the Sum of Interior Angles of a Quadrilateral Theorem, <note>the sum of these four interior angles is 360°</note>, i.e., <note>angle DAE + angle AEF + angle EFD + angle FDA = 360°</note>.


Question: Find the area of the kite.

Answer: 336

1. It is known that a quadrilateral with intersecting and perpendicular diagonals is either a rhombus or a kite. According to the figure, the shape is a kite, and its diagonals intersect and are perpendicular to each other.

2. The area formula for a kite is: Area = 1/2 * Diagonal 1 * Diagonal 2.

3. According to the information in the figure, it is known that Diagonal 1 (vertical direction) is 32 cm. Let Diagonal 2 (horizontal direction) be d. By structural symmetry and the characteristics of the kite shown in the figure, d is 21 cm.

4. Substitute the known diagonal lengths into the area formula: Area = 1/2 * 32 cm * 21 cm.

5. Calculate the area: Area = 1/2 * 32 cm * 21 cm = 336 square centimeters.

6. Therefore, through the above reasoning, the final answer is 336 square centimeters.

Definition of Diagonal

A diagonal is a line segment connecting one vertex of a polygon to another vertex that is not adjacent to it.

Original: <note>在本题图中，四边形风筝中</note>，<note>对角线是连接顶点的非相邻顶点的线段</note>。因此，<note>线段分别连接风筝的非相邻顶点的线段即为对角线</note>。

Translation: <note>In the figure of this problem, in the quadrilateral kite</note>, <note>the diagonal is the line segment connecting the non-adjacent vertices</note>. Therefore, <note>the line segments connecting the non-adjacent vertices of the kite are the diagonals</note>.

Area Formula of Kite

The area of a kite is given by half the product of its diagonals.

<note>The two diagonals of the quadrilateral are perpendicular to each other</note>, <note>one of the diagonals bisects the other</note>, then this quadrilateral is a kite, and the area of the kite = 1/2 * 32 * 21.


Question: △ R S T ≅ △ A B C. Find x.

Answer: 10

1. According to the given conditions, ΔRST is congruent to ΔABC (ΔRST ≅ ΔABC).

2. According to the definition of congruent triangles, corresponding angles are equal and corresponding sides are equal, i.e., ∠R = ∠A, ∠S = ∠B, ∠T = ∠C.

3. From the angles given in the figure, we get ∠S = (5x + 20)°, ∠B = (3x + 40)°.

4. Since corresponding angles in congruent triangles are equal, ∠S = ∠B, so (5x + 20)° = (3x + 40)°.

5. Solve the equation: (5x + 20) = (3x + 40).

6. Rearrange the terms: (5x - 3x) = 40 - 20.

8. Divide both sides by 2 to get: x = 10.

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

普通三角形; 对称; 平移

Definition of Congruent Triangles

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

<note>Triangle RST and triangle ABC are congruent triangles</note>, the corresponding sides and corresponding angles of triangle RST are equal to those of triangle ABC, namely: <note>side ST = side BC</note> <note>side RS = side AB</note> <note>side RT = side AC</note>, and the corresponding angles are also equal: <note>angle RST = angle ABC</note> <note>angle SRT = angle BAC</note> <note>angle STR = angle ACB</note>.


Question: In the figure, square ABDC is inscribed in \\odot K. Find the measure of a central angle.

Answer: 90

####

##1##. In square ABCD, F is the intersection point of diagonals AD and BC. ##According to the properties of the diagonals of a square##, these two diagonals are perpendicular and equal in length within the square.

##2. Since the two diagonals are perpendicular and equal in length within the square, all angles formed by the diagonals within square ABCD are right angles, i.e., ∠CFD=∠AFB=∠BFD=∠AFC=90°##.

##3. According to the definition of a central angle, a central angle refers to the angle between the lines connecting two adjacent vertices of a regular polygon to its center. In square ABCD, point C and point D are two adjacent vertices of the square, point F is the center of the square, and ∠CFD is the angle between the lines connecting the two vertices to the center, i.e., the angle between CF and DF##.

##4. Given that ∠CFD=90°, the central angle is 90 degrees##.

##5##. Through the above reasoning, the final answer is 90 degrees.

正方形; 圆; 圆心角; 圆内接四边形; 垂线

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>In quadrilateral ABDC</note>, <note>sides AB, BD, CD, and CA are equal</note>, and <note>angles ∠ABD, ∠BDC, ∠DCA, and ∠CAB are all right angles (90 degrees)</note>, so ABDC is a square.

Properties of Diagonals in a Square

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

In the diagram of this problem, <note>in square ABDC, diagonals AD and BC are segments connecting opposite corners</note>. According to the properties of diagonals in a square, <note>AD and BC are equal</note>, and <note>AD and BC bisect each other perpendicularly</note>, forming <note>four 90-degree angles at their intersection</note>. Therefore, <note>AD = BC</note>, and <note>they are perpendicular to each other at the intersection</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 CFD is a central angle, formed by the lines connecting the center F of the regular polygon with two adjacent vertices C and D</note>. According to the definition of the central angle, <note>angle CFD is the central angle of the regular polygon, and its measure is equal to the central angle of the circumscribed circle corresponding to each side of the regular polygon</note>.


Question: As shown in the figure, in ⊙O, chord AC ∥ radius OB, ∠BOC=50°, then the degree of ∠OBA is ()

Answer: 25°

1. Given that in circle O, chord AC is parallel to radius OB, and ∠BOC = 50°.

2. According to the inscribed angle theorem, the central angle ∠BOC is twice the inscribed angle ∠BAC corresponding to chord AC. Therefore, we can deduce that ∠BOC = 2∠BAC.

3. From ∠BOC = 50° and ∠BOC = 2∠BAC, we get ∠BAC = 25°.

4. Since chord AC is parallel to radius OB, ∠BAC and ∠OBA are alternate interior angles. According to the parallel line axiom 2, we get ∠BAC = ∠OBA.

5. From ∠BAC = 25°, we get ∠OBA = 25°.

6. Through the above reasoning, the final answer is ∠OBA = 65°.

圆; 圆心角; 弧; 平行线; 对顶角

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 circle O, point O is the center</note>, <note>the radius is OA or OB or OC</note>. All points in the figure that are at a distance equal to OA (or OB or OC) from <note>point O</note> are on <note>circle O</note>.

Definition of Chord

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

In the figure of this problem, in circle O, points A and C are any two points on the circle, <note>line segment AC</note> connects these two points, so <note>line segment AC is a chord of circle O</note>.

Definition of Radius

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

Definition of Central Angle

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

In the figure of this problem, in circle O, <note>points B and C are two points on the circle</note>, <note>the center of the circle is point O</note>. <note>The angle ∠BOC formed by the lines OB and OC</note> is called the central angle.

Definition of Inscribed Angle

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

<note>In the figure of this problem, in circle O</note>, <note>the vertex A of angle BAC is on the circumference</note>, <note>the two sides of angle BAC intersect circle O at points B and C respectively</note>. Therefore, <note>angle BAC is an inscribed angle</note>.

Inscribed Angle Theorem

In a circle, an inscribed angle is equal to half of the central angle that subtends the same arc.

In <note>circle O</note>, <note>points A, B, and C are on the circle</note>, <note>the central angle corresponding to arc AC and arc BC is ∠BOC</note>, <note>the inscribed angle is ∠BAC</note>. According to the Inscribed Angle Theorem, <note>∠BAC is equal to half of the central angle ∠BOC corresponding to arc AC</note>, that is, <note>∠BAC = 1/2 ∠BOC</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>Chord AC is parallel to radius OB</note>, according to Parallel Postulate 2 of Parallel Lines, <note>chord AC and radius OB are intersected by line OC</note>, forming the following geometric relationship: <note>alternate interior angles: angle BAC and angle OBA are equal</note>. Therefore, <note>∠BAC = ∠OBA = 25°</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.

In the figure of this problem, <note>two parallel lines AC and OB</note> are intersected by a line <note>OC</note>, where <note>angle BAC and angle OBA are located between the two parallel lines and on opposite sides of the intersecting line OC</note>. Therefore, <note>angle BAC and angle OBA are alternate interior angles</note>. Alternate interior angles are equal, that is, <note>angle BAC is equal to angle OBA</note>.


Question: As shown in the figure, Xiao Dong uses a 3.2m long bamboo pole as a measuring tool to measure the height of the school flagpole. He moves the bamboo pole so that the shadow of the bamboo pole and the top of the flagpole fall exactly at the same point on the ground. At this time, the bamboo pole is 8m away from this point and 22m away from the flagpole. Then the height of the flagpole is ()

Answer: 12m

1. According to the problem statement, add an auxiliary line BE to the triangle such that BE∥CD, and AE is the length of the shadow of the bamboo pole, which is 8 meters. AD is the total length of the shadow of the bamboo pole and the flagpole top, which is 30 meters (22 meters + 8 meters).

2. Because ##angle D = angle AEB = 90 degrees, angle A is the common angle, so triangle AEB is similar to triangle ADC. According to the definition of similar triangles, corresponding sides are proportional.##

3. According to the definition of similar triangles##, the proportional relationship is \rac{AE}{AD} = \rac{BE}{CD}, that is, \rac{8}{30} = \rac{3.2}{CD}.

4. By solving this proportion equation, we get CD = 12 meters.

5. Through the above reasoning, we conclude that the height of the flagpole is 12 meters.

Definition of Similar Triangles

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

<note>Triangle AEB and triangle ADC are similar triangles</note>. According to the definition of similar triangles: <note>∠BAE = ∠CAD</note>, <note>∠AEB = ∠ADC</note>, <note>∠ABE = ∠ACD</note>; <note>AE/AD = BE/CD = AB/AC</note>.

Similarity Theorem for Triangles (AA)

Two triangles are similar if two of their corresponding angles are congruent.

<note>Angle AEB is equal to angle ADC</note>, and <note>angle BAE is equal to angle CAE</note>, so <note>triangle ABE is similar to triangle ACD</note>.


Question: Find the length of \widehat Z Y. Round to two decimal places.

Answer: 7.85

1. According to the problem, the radius of the circle is 6 inches, and the angle YXZ is 75 degrees. We need to calculate the arc length ZY, which is the arc length subtended by the central angle ZXY.

2. According to the conversion formula between degrees and radians, radians = degrees * (π/180) = 75 * (π/180) = 5π/12

3. According to the formula for the arc length of a sector, the arc length L is equal to the central angle θ (expressed in radians) multiplied by the radius r: L = θr. Substituting θ = 5π/12 and r = 6 into the formula, we get L = 5π/2

4. Taking the approximate value of π as 3.14 and substituting it into L = 5π/2, we get L = 7.85

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>Point X is the center of the circle</note>, <note>Point Y is any point on the circle</note>, <note>Line segment XY is the line segment from the center of the circle to any point on the circle, thus line segment XY is the radius of the circle</note>, with a length of 6 inches.

Definition of Central Angle

<note>The vertex of angle YXZ is at the center X of the circle</note>, <note>both sides XY and XZ intersect the circle</note>, so <note>angle YXZ is a central angle</note>, and the angle measures 75 degrees.

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 \( r \): \( L = \theta r \).

<note>In the figure of this problem, in sector ZXY, the central angle ∠ZXY is 75 degrees, and the radius is 6 inches</note>. According to the formula for the length of an arc of a sector, <note>the arc length L is equal to the central angle θ multiplied by the radius r, i.e., L = θr = (5π/12) × 6</note>.

Formula for Conversion between Degrees and Radians

Radians = Degrees × (π/180), Degrees = Radians × (180/π)

<note>ZXY's angle is 75°</note>, according to the formula for conversion between degrees and radians, it converts to <note>radians = degrees * (π/180), which is 75 * (π/180)</note>


Question: Find the perimeter of the parallelogram.

Answer: 78

1. Let the parallelogram be parallelogram ABCD, the given condition is that the length of the base of the parallelogram is 23 feet, and the length of the other side is 16 feet.

2. According to the perimeter of the parallelogram, the perimeter of the parallelogram is equal to twice the sum of its two sets of adjacent sides. From the figure, it is known that the lengths of the two sets of adjacent sides are 23 feet and 16 feet respectively. According to the perimeter formula of the parallelogram: P = 2(a + b), we get: P = 2(23 + 16) = 2 * 39 = 78.

3. After the above reasoning, the final answer is 78 feet.

Definition of Parallelogram

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

In the figure of this problem, <note>quadrilateral ABCD is a parallelogram</note>, <note>side AB is parallel to and equal to side CD</note>, <note>side AD is parallel to and equal to side BC</note>.

Perimeter of a Parallelogram

The perimeter of a parallelogram is equal to twice the sum of the lengths of its two adjacent sides. The formula is: \( P = 2(a + b) \), where \( a \) and \( b \) are the lengths of the two adjacent sides of the parallelogram.

<note>In the figure of this problem, in parallelogram ABCD, sides CD and BC are adjacent sides, and sides AD and AB are also adjacent sides</note>. According to the formula for the perimeter of a parallelogram, <note>the perimeter of a parallelogram is equal to twice the sum of its two pairs of adjacent sides, i.e., perimeter P = 2(BC + CD)</note>.


Question: As shown in the figure, in the right triangle Rt△ABC, ∠C=90°, AC=6, BC=8, find the value of tanA.

Answer: \frac{4}{3}

1. Given in the right triangle ABC, ∠ACB = 90°, AC = 6, BC = 8.

2. According to the definition of the tangent function, in a right triangle, the tangent value of an acute angle is equal to the ratio of the length of the opposite side to the length of the adjacent side.

3. In the right triangle ABC, the opposite side of ∠BAC is BC, and the adjacent side is AC, so tan(∠BAC) = BC / AC.

4. Substituting the given values, we get tan(∠BAC) = 8 / 6.

5. Simplifying the fraction 8 / 6, we get 4 / 3.

6. Through the above reasoning, the final answer is 4 / 3.

Definition of Tangent Function

In a right-angled triangle, the tangent value of an acute angle is equal to the length of the opposite side divided by the length of the adjacent side.

<note>In right triangle ABC</note>, <note>∠ACB = 90°</note>, let <note>∠BAC be the acute angle</note>, <note>the opposite side of ∠BAC is BC</note>, <note>the adjacent side is AC</note>. Therefore, <note>tan(∠BAC) = BC / AC</note>.

Definition of Right Triangle

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

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


Question: Each pair of polygons is similar. Find G F.

Answer: 7.5

1. Given that the two triangles △RST and △EGF are similar triangles, the corresponding angles are equal, and the corresponding sides are proportional.

####

##2##. According to the properties of similar triangles, the corresponding sides of △RST and △EGF satisfy the proportional relationship, i.e., RS/EG = RT/EF = ST/GF.

##3##. According to the given conditions, RT = 15, EF = 11.25, establish the proportional relationship RT/EF = ST/GF, i.e., 15/11.25 = 10/x.

##4##. Solve the proportion equation, 15/11.25 = 10/x, cross-multiplying both sides to get 15 * x = 11.25 * 10.

##5##. Solve the above equation to get x = (11.25 * 10) / 15 = 7.5.

##6##. Through the above reasoning, the final answer is GF = 7.5.

Definition of Similar Triangles

<note>Triangles RST and EGF are similar triangles</note>. According to the definition of similar triangles: <note>∠R = ∠E</note>, <note>∠T = ∠F</note>, <note>∠S = ∠G</note>; <note>RS/EG = ST/GF = RT/EF</note>.


Question: As shown in the figure, the slant height of the conical tent top AB=10cm, the radius of the base BO=5m, the lateral surface area of this conical tent top (excluding seams) is ()

Answer: 50πcm^{2}

1. Given the slant height AB=10cm and the base radius BO=5cm.

2. The formula for the lateral surface area of a cone is: S = πrl, where r is the base radius and l is the slant height.

3. Substituting the given conditions r=5cm and l=10cm, we calculate: S = π * 5cm * 10cm = 50π cm^2.

4. Therefore, the lateral surface area of the conical tent top is 50π cm^2.

圆锥; 圆; 弧; 线段; 扇形

Generatrix

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

<note>A point on the circumference of the base is B, and the vertex is A</note>, <note>the line segment AB is the generatrix</note>. The generatrix is the line segment from a point on the circumference of the base to the vertex in a cone. It is known that <note>AB=10cm</note>.

Cone

A cone is a geometric figure with a circular base and a single vertex. Its surface consists of a curved lateral surface extending from the base to the vertex.

In the figure of this problem, <note>cone OABC</note> has a base that is <note>a circle</note>, <note>the radius of the circle is BO=5cm</note>, <note>the center of the circle is point O</note>. <note>The vertex of the cone is point A</note>, the distance between the vertex <note>A and the center O is the height of the cone, denoted as AO</note>. <note>The lateral surface of the cone is a curved surface</note>, the distance from <note>the vertex A to any point on the circumference (such as B, C) is the slant height, denoted as AB or AC</note>.

Development of a Cone

The development (or net) of a cone is a sector of a circle, where the radius of the sector is the slant height of the cone, and the arc length of the sector is equal to the circumference of the cone's base.

In the diagram of this problem, the development of the cone is a sector. <note>The radius of the sector is the slant height of the cone AB=10cm</note>, <note>the arc length of the sector is the circumference of the base circle of the cone 2πr</note>, where <note>r=BO=5cm</note>. Therefore, <note>the radius of the sector AB is equal to the slant height of the cone</note>, <note>the arc length of the sector is equal to the circumference of the base circle of the cone, which is 2πr</note>, where <note>r is the radius of the base circle</note>.

Lateral Surface Area Formula of a Cone

The formula for the lateral surface area of a cone is: \( S = \pi r l \), where \( r \) is the radius of the base, and \( l \) is the slant height.

Original text: <note>The radius of the cone r=OB=5</note>, <note>the slant height AB=10</note>, according to the lateral surface area formula of the cone, <note>S = πrl=π*5*10</note>.


Question: A square circumscribes a circle. Find the exact circumference of the circle.

Answer: 14 \pi

1. The given condition is that the square is circumscribed around the circle, and the side length of the square is 14####.

2. According to the property of the inscribed circle, the circle is tangent to each side of the polygon, and the distance from the center of the circle to the sides of the polygon is equal. Since the square is circumscribed around the circle, the diameter of the circle is equal to the side length of the square.

3. Therefore, the diameter of the circle is equal to the side length of the square, which is 14####.

4. According to the formula for the circumference of a circle C=πd, where d is the diameter of the circle, the circumference of the circle can be obtained as C = π × 14####.

5. Hence, the exact circumference of the circle is 14π####.

Incircle

An incircle of a polygon is a circle that is tangent to each side of the polygon. The center of this circle is called the incenter, and the distance from the incenter to each side of the polygon is equal.

<note>The circle is tangent to all sides of the square</note>. <note>The center of the circle is equidistant from each side of the square</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.

Original text: <note>The diameter of the circle d is equal to the side length of the square, which is 14 yards</note>. According to <note>the circumference formula of the circle C=πd</note>, the circumference of the circle <note>C=π×14 yards</note>, which is 14π yards.

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>The four sides of the square are equal, each side is 14 yards</note>, and <note>all four interior angles are right angles (90 degrees)</note>. These properties ensure that <note>the side length of the square can be used to calculate the diameter of the circumscribed circle</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.

In the figure of this problem, <note>14 is the diameter</note>, connecting <note>the center O and the two points x, x on the circumference</note>, with a length of <note>2 times the radius, that is, diameter = 2r = 14</note>.


Question:
The area of rectangle ABCD above is 168. What is the length of segment AC?

(A) 7√2
(B) 14
(C) 21
(D) 24
(E) 25

Answer: 25

1. Given the area of rectangle ABCD is 168, ##width BC## = 7. The area formula for a rectangle is length times width, so we use w to represent the rectangle's ##length AB##, thus we have w * 7 = 168.

2. Solving w * 7 = 168, we get w = 168 / 7 = 24, so the rectangle's ##length AB## = 24.

3. ##According to the definition of a rectangle and the definition of a diagonal, angle ABC is 90 degrees, AC is the diagonal of the rectangle, and according to the definition of a right triangle, triangle ABC is a right triangle. In right triangle ABC, according to the Pythagorean theorem, the hypotenuse AC is equal to the square root of the sum of the squares of the two right-angle sides.##

4. ##The right-angle sides of right triangle ABC are BC## and AB, ##given BC=7, AB=24, then## AC = √(##BC^2## + AB^2) = √(24^2 + 7^2).

5. Calculate 24^2 = 576 and 7^2 = 49, then sum 576 + 49 = 625.

6. Therefore, AC = √625 = 25.

7. Through the above reasoning, the final answer is 25.

矩形; 直角三角形; 线段

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 ABCD, <note>side AB</note> and <note>side BC</note> are the length and width of the rectangle, so the area of the rectangle = <note>AB * BC</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 ABCD is a rectangle</note>, with interior angles <note>∠DAB</note>, <note>∠ABC</note>, <note>∠BCD</note>, <note>∠CDA all being right angles (90 degrees)</note>, and sides <note>AB and CD are parallel and equal in length</note>, sides <note>AD and BC are parallel and equal in length</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 this problem's diagram, AC is the diagonal of rectangle ABCD, dividing the rectangle into two right triangles. In <note>right triangle ABC</note>, <note>∠ABC is a right angle (90 degrees)</note>, <note>sides AB and BC</note> are the legs, and <note>side AC is the hypotenuse</note>. Therefore, according to the Pythagorean Theorem, <note>AC² = AB² + BC²</note>. Given the data <note>BC</note> = 7, <note>AB</note> = 24, the diagonal AC = √(24² + 7²) = √(576 + 49) = √625 = 25.

Definition of Diagonal

A diagonal is a line segment connecting one vertex of a polygon to another vertex that is not adjacent to it.

In the figure of this problem, in rectangle <note>ABCD</note>, the vertices <note>A, B, C, D</note>, the diagonal is the line segment connecting vertex <note>A</note> and the non-adjacent vertex <note>C</note>. Therefore, the line segment <note>AC</note> is the diagonal of rectangle <note>ABCD</note>.

Definition of Right Triangle

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


Question: As shown in the figure, quadrilateral ABCD is inscribed in ⊙O. If ∠C=36°, then the degree of ∠A is ()

Answer: 144°

1. Quadrilateral ABCD is inscribed in circle O. According to the ##(corollary 3 of the inscribed angle theorem) the opposite angles of a cyclic quadrilateral are supplementary##, which means the opposite angles of a cyclic quadrilateral are supplementary. Therefore, we have ∠DAB + ∠DCB = 180°.

2. Given ∠DCB = 36°, substituting it into the above equation, we get ∠DAB + 36° = 180°.

3. Through simple algebraic calculation, we find ∠DAB = 180° - 36° = 144°.

圆内接四边形; 圆周角; 对顶角; 邻补角; 圆

Cyclic Quadrilateral

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

<note>The four vertices A, B, C, and D of quadrilateral ABCD are on the same circle O</note>. This circle is called <note>the circumcircle of quadrilateral ABCD</note>. Therefore, <note>quadrilateral ABCD is a cyclic quadrilateral</note>. According to the properties of cyclic quadrilaterals, it can be concluded that <note>the sum of the opposite angles is 180 degrees, i.e., ∠DAB + ∠DCB = 180 degrees, ∠ABC + ∠CDA = 180 degrees</note>.

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

<note>Quadrilateral ABCD is inscribed in circle O</note>, <note>the vertices A, B, C, and D of the quadrilateral are all on the circle</note>. According to the diagonal supplementary theorem for cyclic quadrilateral, <note>the sum of each pair of opposite angles of quadrilateral ABCD is equal to 180°</note>. Specifically, <note>∠DAB + ∠DCB = 180°</note>; <note>∠ABC + ∠ADC = 180°</note>.


Question: Find R S if triangle Q R S is an equilateral triangle.

Answer: 2

1. Given condition: △QRS is an equilateral triangle. According to the properties of an equilateral triangle, we have QR = RS = QS.

2. Based on the properties of an equilateral triangle, we can set up the following equations:

  QR = 4x, RS = 2x + 1, QS = 6x - 1.

  Therefore, 4x = 2x + 1 = 6x - 1.

3. First, solve the first set of equations: 4x = 2x + 1.

  Solving this equation: 4x - 2x = 1 => 2x = 1 => x = 1/2.

4. Using the same value of x, solve the second set of equations to verify consistency:

  6x - 1 = 2x + 1.

  Substituting x = 1/2 we get: 6*(1/2) - 1 = 3 - 1 = 2 => 2*(1/2) + 1 = 1 + 1 = 2.

5. Calculate:

  RS = 2x + 1 = 2*(1/2) + 1 = 1 + 1 = 2.

6. The final answer is 2.

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°).

In the figure of this problem, <note>triangle QRS is an equilateral triangle</note>. <note>The lengths of sides QR, RS, and QS are equal</note>, and <note>the measures of angles QRS, RQS, and QSR are equal, each being 60°</note>. According to the conditions given in the problem, we can set up <note>QR=4x, RS=2x+1, QS=6x-1</note>, so it should satisfy <note>4x = 2x + 1 = 6x - 1</note>.


Question: As shown in the figure, AB is tangent to ⊙O at point B, the extension of AO intersects ⊙O at point C, connect BC, if ∠A=36°, then ∠C equals ()

Answer: 27°

1. Given that line AB is tangent to circle O at point B, according to the property of tangents, we have ∠ABO = 90°.

2. According to the problem statement, ∠A = 36°. Since ∠ABO = 90°, ##in triangle AOB, using the triangle angle sum theorem, ∠A + ∠ABO + ∠BOA = 180°##, we get ∠BOA = 180° - 90° - 36° = 54°.

3. ##Let AO intersect the circle at point D. From the figure, we can see that ∠BOD is the central angle of arc DB, and ∠BCD is the inscribed angle of arc DB. According to the inscribed angle theorem, when the central angle of arc DB is ∠BOD, the corresponding inscribed angle ∠BCD = 1/2 ∠BOD. Thus, ∠BCD = 1/2 * 54° = 27°##.

圆; 切线; 圆周角; 圆心角; 等腰三角形

Definition of Inscribed Angle

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

In the figure of this problem, <note>the vertex C of angle ∠BCD in circle O is on the circumference</note>, <note>the two sides of angle ∠BCD intersect circle O at points B and D respectively</note>. Therefore, <note>angle ∠BCD is an inscribed angle</note>.

Definition of Central Angle

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

In the figure of this problem, in circle O, <note>points B and D are two points on the circle, and the center of the circle is point O</note>. <note>The angle ∠BOD formed by the lines OB and OD is called the central angle</note>.

A tangent line to a circle is perpendicular to the radius that passes through the point of tangency.

<note>Point B is the tangent point of line AB and the circle</note>, <note>segment OB is the radius of the circle</note>. According to the property of the tangent line to a circle, <note>the tangent AB is perpendicular to the radius OB at the tangent point B</note>, that is, <note>∠ABO=90°</note>.

Triangle Angle Sum Theorem

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

In the figure of this problem, in triangle ABO, <note>angle ∠A</note>, <note>angle ∠ABO</note>, and <note>angle ∠BOA</note> are the three interior angles of triangle ABO. According to the Triangle Angle Sum Theorem, <note>angle ∠A + angle ∠ABO + angle ∠BOA = 180°</note>.

Inscribed Angle Theorem

In a circle, an inscribed angle is equal to half of the central angle that subtends the same arc.

<note>In circle O, points B and D are on the circle</note>, the arc BD corresponds to the <note>central angle ∠BOD</note>, and the <note>inscribed angle is ∠BCD</note>. According to the <note>Inscribed Angle Theorem</note>, ∠BCD is equal to half of the <note>central angle ∠BOD corresponding to arc BD</note>, that is, <note>∠BCD = 1/2 ∠BOD</note>.


Question: Find the value of x.

Answer: 74

1. The given condition is that the sum of the exterior angles of a pentagon is 360°.

2. The five exterior angles of the pentagon are (x)°, (x+4)°, 56°, (x+10)°, (x-6)°.

3. According to the sum of the exterior angles of the pentagon being 360°, we have the equation: x + (x+4) + 56 + (x+10) + (x-6) = 360.

4. Combining like terms, we get: 4x + 64 = 360.

5. Rearranging and calculating, we get: 4x = 296.

6. Solving the equation, we get x = 74.

7. Through the above reasoning, we finally obtain the value of x as 74.

五边形; 普通多边形; 三角形的外角

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

In the figure of this problem, the five exterior angles of the pentagon are respectively: <note>x°</note>, <note>(x+4)°</note>, <note>56°</note>, <note>(x+10)°</note>, <note>(x-6)°</note>. Each exterior angle is formed at a vertex by the extension of one adjacent side and the other side.

Exterior Angle Sum Theorem of Polygon

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

Original: <note>The five exterior angles of the pentagon are x°, (x+4)°, 56°, (x+10)°, (x-6)°</note>. According to the Exterior Angle Sum Theorem of Polygon, <note>the sum of these exterior angles is equal to 360°</note>, that is <note>x° + (x+4)° + 56° + (x+10)° + (x-6)° = 360°</note>.