Question: As shown in the figure, AB is a fixed ladder leaning against the wall. The distance from the foot of the ladder B to the foot of the wall C is 1.6m. The distance from a point D on the ladder to the wall is 1.4m, and the length of BD is 0.5m. Then the length of the ladder is ()

Answer: 4m

1. Given that the distance from the foot of the ladder at B to the foot of the wall at C is 1.6 meters, the distance from point D on the ladder to the wall is 1.4 meters, and BD is 0.5 meters long.

2. According to the theorem of proportional segments in parallel lines, since the line DE is parallel to the side BC of triangle ABC and intersects the other two sides AB and AC, the segments it intersects are proportional to the corresponding segments of the original triangle, so AD/AB = DE/BC.

3. Let the length of the ladder be x meters, then AB = x meters, AD = x - 0.5 meters, BC = 1.6 meters, DE = 1.4 meters.

4. From the proportional relationship of similar triangles, we get: (x - 0.5)/x = 1.4/1.6.

5. Solving the above equation, we clear the denominator to get 1.6(x - 0.5) = 1.4x.

6. Expanding and rearranging terms, we get 1.6x - 0.8 = 1.4x.

9. Through the above reasoning, we finally conclude that the length of the ladder is 4 meters.

Definition of Parallel Lines

<note>Line DE and line BC</note> are located in the same plane, and they do not intersect. Therefore, according to the definition of parallel lines, <note>line DE and line BC</note> are parallel lines.

Proportional Segments Theorem

<note>In triangle ABC, line DE is parallel to side BC</note>, and it intersects the other two sides AB and AC at points D and E. Then, according to the Proportional Segments Theorem, we have: <note>AD/AB = DE/BC</note>, that is, the intercepted segments are proportional to the corresponding segments of the original triangle.


Question: Find \cos C.

Answer: \frac { 4 } { 5 }

1. From the figure, ∠BAC=90°, according to the definition of a right triangle, triangle ABC is a right triangle. In the right triangle ABC, AB=3, BC=5.

2. According to the Pythagorean theorem, the sum of the squares of the two legs of a right triangle is equal to the square of the hypotenuse. It is known that in the right triangle ABC, AB^2 + AC^2 = BC^2.

3. Substitute the known data into the Pythagorean theorem, 3^2 + AC^2 = 5^2, which can be simplified to 9 + AC^2 = 25.

4. Solving the equation gives AC^2 = 16, thus AC = 4.

5. According to the cosine function, cos C = length of adjacent side / length of hypotenuse = AC / BC.

6. Substitute the known data, cos C = 4 / 5.

7. Through the above reasoning, the final answer is 4/5.

直角三角形; 余弦; 线段

Definition of Right Triangle

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

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

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 the figure of this problem, in the right triangle ABC, side <note>AC</note> is the adjacent side of <note>∠BCA</note>, and side <note>BC</note> is the hypotenuse. Therefore, according to the definition of the cosine function, the cosine value of <note>∠BCA</note> is equal to the ratio of the adjacent side <note>AC</note> to the hypotenuse <note>BC</note>, that is, <note>cos(∠BCA)</note> = <note>AC / BC</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, in the right triangle ABC, <note>∠BAC is a right angle (90 degrees)</note>, sides <note>AB and AC are the legs</note>, <note>side BC is the hypotenuse</note>, so according to the Pythagorean Theorem, <note>BC² = AB² + AC²</note>.


Question: If \triangle L M N \sim \triangle Q R S, Q R = 35, R S = 37, S Q = 12, and N L = 5, find the perimeter of \triangle L M N.

Answer: 35

1. Given △ L M N and △ Q R S are similar, the corresponding sides of similar triangles are proportional. ##Therefore N L/S Q = L M/Q R = M N/R S##.

2. Substitute the given data N L = 5, S Q = 12, Q R = 35, R S = 37, ##get 5/12 = L M/35 = M N/37##.

3. Solve for L M through the proportional relationship, ##get L M = 5/12 × 35 = 175/12##.

4. Solve for M N through the proportional relationship, ##get M N = 5/12 × 37 = 185/12##.

5. The perimeter of △ L M N is the sum of the three sides, ##i.e., 5 + 175/12 + 185/12##.

6. Calculate the perimeter of the triangle, ##first convert 5 to 60/12, then the sum of the three sides is 60/12 + 175/12 + 185/12##.

7. The sum obtained is ##420/12## = 35.

8. Therefore, the perimeter of △ L M N is 35.

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, since it is known that <note>△LMN is similar to △QRS</note>, the theorem can be directly applied to obtain that <note>NL, SQ, LM, QR, MN, and RS are proportional</note>, that is: <note>NL/SQ = LM/QR = MN/RS</note>. By substituting the known values, we get <note>5/12 = LM/35 = MN/37</note>.

Formula for the Perimeter of a Triangle

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

Original text: <note>The three sides of triangle NLM are NL, LM, and MN respectively</note>. According to the formula for the perimeter of a triangle, the perimeter of the triangle is equal to the sum of the lengths of its three sides, that is, the perimeter P = <note>NL + LM + MN</note>.


Question: Find AB.

Answer: 4.1

1. Given CD=CB, and AC⊥BD, so ∠ACD=∠ACB=90°.

2. According to the definition of a right triangle, because ∠ACD=∠ACB=90°, triangles ABC and ADC are both right triangles. Also, according to the congruence theorem (SAS), since BC=DC, ∠ACD=∠ACB=90°, and the two triangles share AC as the right side, therefore ΔABC≌ΔADC.

3. Since ΔABC≌ΔADC, according to the definition of congruent triangles, side AB is equal to side AD, and the length of side AD in ΔADC is 4.1, so AB=AD=4.1.

4. Therefore, through the above reasoning steps, it is concluded that AB = 4.1.

直角三角形; 垂线; 线段; 中点

Definition of Triangle

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

In the figure of this problem, <note>triangle ABC is a geometric figure composed of three non-collinear points A, B, C and their connecting line segments AC, AB, BC</note>. <note>Points A, B, C are the three vertices of the triangle</note>, <note>line segments AC, AB, BC are the three sides of the triangle</note>. <note>Triangle ADC is a geometric figure composed of three non-collinear points A, D, C and their connecting line segments AC, AD, DC</note>. <note>Points A, D, C are the three vertices of the triangle</note>, <note>line segments AC, AD, DC are the three sides of the triangle</note>.

Definition of Right Triangle

<note>In the figure of this problem, in triangle ABC, angle ACB is a right angle (90 degrees)</note>, therefore <note>triangle ABC is a right triangle</note>. Side AC and side BC are <note>the legs</note>, and side AB is <note>the hypotenuse</note>. <note>In triangle ADC, angle ACD is a right angle (90 degrees)</note>, therefore <note>triangle ADC is a right triangle</note>. Side AC and side DC are <note>the legs</note>, and side AD is <note>the hypotenuse</note>.

Triangular Congruence Theorem (SAS)

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

In the diagram of this problem, triangle ABC and triangle ADC have side BC equal to side DC, side AC equal to side AC, and angle ACB equal to angle ACD. Therefore, according to the Triangular Congruence Theorem (SAS), these two triangles are congruent.

Definition of Congruent Triangles

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

<note>Triangle ABC and triangle ADC are congruent triangles</note>, the corresponding sides and corresponding angles of triangle ABC are equal to those of triangle ADC, namely: <note>side BC = side DC</note> <note>side AB = side AD</note> <note>side AC = side AC</note>, and the corresponding angles are also equal: <note>angle BCA = angle DCA</note> <note>angle BAC = angle DAC</note> <note>angle ABC = angle ADC</note>.


Question: Find m \angle R S T.

Answer: 55

1. According to the image, RS = RT. Based on the definition of an isosceles triangle, triangle RST is an isosceles triangle, and ∠SRT = 70°.

2. In the isosceles triangle RST, according to the properties of an isosceles triangle, the angles opposite the equal sides are equal, so ∠RST = ∠RTS.

3. Let ∠RST = ∠RTS = x. Then, according to the triangle angle sum theorem (i.e., the sum of the three interior angles of a triangle is equal to 180°), we have: x + x + 70° = 180°.

4. Solving this equation, we get: 2x + 70° = 180°, so 2x = 110°, hence x = 55°.

5. Therefore, ∠RST = 55°.

6. Through the above reasoning, the final answer is ∠RST = 55°.

等腰三角形; 普通三角形

Definition of Isosceles Triangle

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

<note>In the figure of this problem, in triangle RST, side RS and side RT are equal</note>, therefore <note>triangle RST is an isosceles triangle</note>.

Triangle Angle Sum Theorem

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

In the figure of the problem, <note>in triangle RST</note>, angles ∠RST, ∠RTS, and ∠SRT are <note>the three interior angles of triangle RST</note>. According to the Triangle Angle Sum Theorem, ∠RST + ∠RTS + ∠SRT = 180°.

Properties of Isosceles Triangle

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

In the <note>isosceles triangle RST</note>, <note>sides RS and RT are equal</note>. Therefore, according to the properties of isosceles triangles, the angles opposite the equal sides are equal, i.e., <note>angle RST = angle RTS</note>.


Question: Find the value of m \angle A in quadrilateral ABCD.

Answer: 135

1. The given condition is that ∠B and ∠C in quadrilateral ABCD are both 90°.

2. According to the theorem of the sum of interior angles of a quadrilateral, we get: ∠A + ∠B + ∠C + ∠D = 360°.

3. Substituting the given conditions, ∠B = 90°, ∠C = 90°, therefore we get: ∠A + ∠D = 360° - 90° - 90° = 180°.

4. Given ∠D = x°, ∠A = 3x°. Substituting into ∠A + ∠D = 180°, we get 3x° + x° = 180°.

5. Solving the above equation, we get 4x° = 180°, therefore x° = 45°.

6. From x° = 45°, we get ∠A = 3x° = 3 * 45° = 135°.

7. Through the above reasoning, the final answer is ∠A = 135°.

平行四边形; 邻补角; 直角三角形

Definition of Quadrilateral

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

<note>Quadrilateral ABCD</note> is composed of <note>four line segments AB, BC, CD, and DA</note>, which are called the sides of the quadrilateral. <note>Quadrilateral ABCD has four vertices, namely point A, point B, point C, and point D</note>, and it has four interior angles, namely <note>angle DAB, angle ABC, angle BCD, and angle CDA</note>.

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

In the figure of this problem, in quadrilateral <note>ABCD</note>, <note>angle DAB, angle ABC, angle BCD, and angle CDA 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>, that is, <note>angle DAB + angle ABC + angle BCD + angle CDA = 360°</note>.


Question: Find E G if G is the incenter of 	riangle A B C.

Answer: 5

1. Given that G is the incenter of triangle ABC, according to the definition of the incenter of a triangle, we know FG = EG.

2. In the right triangle CFG, CG = 13, CF = 12, according to the Pythagorean theorem, we know FG = √(13² - 12²) = 5.

3. Therefore, EG = FG = 5, through the above reasoning, the final answer is EG = 5.

普通三角形; 点; 垂线

Incenter of a Triangle

The incenter of a triangle is the intersection point of the angle bisectors of the three interior angles of the triangle. It is also the center of the triangle's inscribed circle (incircle).

In the figure of this problem, in triangle ABC, <note>point G is the incenter of the triangle</note>. The incenter of a triangle is formed by <note>the intersection of the three angle bisectors of the triangle</note>. Specifically, <note>the bisector of angle BAC</note>, <note>the bisector of angle ABC</note>, and <note>the bisector of angle ACB intersect at point G</note>. <note>Point G is the center of the inscribed circle of the triangle</note>, and it is equidistant from the three sides of the triangle.

Definition of Right Triangle

In <note>triangle CGF</note>, <note>∠GFC is a right angle (90 degrees)</note>, therefore <note>triangle CGF is a right triangle</note>. <note>Side CF and side GF are the legs</note>, <note>side CG is the hypotenuse</note>.

Pythagorean Theorem

<note>In the right triangle CGF, ∠GFC is a right angle (90 degrees)</note>, <note>side CF and side GF are the legs</note>, <note>side CG is the hypotenuse</note>, so according to the Pythagorean Theorem, <note>CG² = CF² + GF²</note>.


Question: As shown in the figure, the slant height of the cone is 5cm, and the height is 4cm. Then the base area of the cone is ()

Answer: 9πcm^{2}

1. Let the three vertices of the triangle formed by the dashed lines in the figure be A, B, and C.

2. From the figure, we can see that AB is perpendicular to BC, so ∠ABC is 90 degrees. According to the definition of a right triangle, triangle ABC is a right triangle. In the right triangle ABC, its hypotenuse is the slant height of the cone, and the legs are the height of the cone and the radius of the base.

3. Given that the length of the slant height AC of the cone is 5 cm, and the length of the height AB is 4 cm. According to the Pythagorean theorem, let the radius of the base circle be r, then we have: 5² = r² + 4².

4. Calculating, we get: r = √(25 - 16) = √9 = 3 (cm).

5. From the figure, we can see that the base area of the cone is the area of the base circle. According to the formula for the area of a circle = π * r².

6. Substituting the value of the radius r, we get: the base area of the cone = π * 3² = 9π (square cm).

7. Through the above reasoning, the final answer is 9π (square cm).

圆锥; 直角三角形; 线段

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.

<note>The base of the cone is the circular bottom</note>, <note>the radius of the circle is 3 cm</note>, <note>the center of the circle is the center of the base</note>. <note>The vertex of the cone is the point at the top</note>, <note>the distance between the vertex and the center of the circle is the height of the cone</note>, denoted as 4 cm. <note>The lateral surface of the cone is a curved surface</note>, from the vertex to <note>any point on the circumference is the slant height</note>, denoted as 5 cm.

Area Formula of a Circle

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

<note>Area Formula of a Circle</note>, the area A of the circle is equal to <note>pi π multiplied by the square of the radius r</note>, which is A = πr^2. From the problem, we calculate <note>radius r = 3 cm</note>, thus the <note>base area</note> is π * 3^2 = 9π square centimeters.

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 radius of the base circle is the line segment from the center of the base circle to any point on the circumference, and the radius is calculated to be 3 centimeters.

Definition of Right Triangle

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

In the diagram of this problem, <note>the generatrix of the cone, the vertical height line, and the radius of the base circle form a right triangle</note>. <note>The generatrix of the cone is the hypotenuse</note>, <note>and the vertical height line and the radius of the base circle are the legs</note>.

Pythagorean Theorem

In the figure of this problem, <note>the hypotenuse of the right triangle is 5 cm long</note>, <note>the legs are 4 cm long and the radius of the base circle r</note>. According to the Pythagorean Theorem, <note>5² = r² + 4²</note>, therefore <note>r = √(5² - 4²) = 3 cm</note>.


Question: Find x.

Answer: \sqrt { 7 }

1. Given that one of the legs of a right triangle is 3√2, the hypotenuse is 5, and the other leg is represented by x. According to the Pythagorean theorem (i.e., in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the two legs), we get the equation (3√2)^2 + x^2 = 5^2.

2. Calculate (3√2)^2, which is 9 * 2 = 18.

3. Substitute the calculation result into the Pythagorean theorem, obtaining the equation 18 + x^2 = 25.

4. Move 18 to the right side of the equation, getting x^2 = 25 - 18.

5. Calculate 25 - 18 = 7, thus obtaining x^2 = 7.

6. Solve for x from x^2 = 7, x = √7.

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

Definition of Right Triangle

<note>One angle in the triangle is a right angle (90 degrees)</note>, therefore the triangle is a right triangle. <note>The sides with lengths 3√2 and x are the legs, and the side with length 5 is the hypotenuse.</note>

Pythagorean Theorem

<note>In a right triangle</note>, the square of the hypotenuse is equal to the sum of the squares of the two legs. Given that <note>one leg is 3√2</note>, <note>the hypotenuse is 5</note>, and <note>the other leg is represented by x</note>. According to the <note>Pythagorean Theorem</note>, ##<note>5^2</note> = <note>(3√2)^2</note> + <note>x^2</note>##, i.e., <note>25 = 18 + x^2</note>.


Question: As shown in the figure, △ABC is inscribed in ⊙O. If ∠OAB=26°, then the measure of ∠C is ()

Answer: 64°

1. Connect OB.

2. In △OAB, it is known that OA = OB, ##according to the definition of an isosceles triangle, △OAB is an isosceles triangle; and according to the properties of an isosceles triangle##, it follows that ∠OAB = ∠OBA.

3. It is also known that ∠OAB = 26°, therefore we can deduce that ∠OBA = 26°.

4. ##According to the triangle angle sum theorem, we get ∠OAB + ∠OBA + ∠AOB = 180°, that is, ∠AOB = 180° - ∠OAB - ∠OBA = 180° - 26° - 26° = 128°.##

5. ##According to the definition of a central angle, ∠AOB is a central angle; according to the definition of an inscribed angle, ∠ACB is an inscribed angle. Arc AB corresponds to ∠ACB and ∠AOB. According to the inscribed angle theorem, we get ∠ACB = 1/2 * ∠AOB = 1/2 * 128° = 64°.##

6. Therefore, through the above reasoning, we finally deduce that the measure of ∠C is 64°.

圆; 圆周角; 普通三角形

Definition of Isosceles Triangle

In the triangle OAB, sides OA and OB are equal, therefore the triangle OAB is an isosceles triangle.

Properties of Isosceles Triangle

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

Triangle Angle Sum Theorem

<note>Angle OAB</note>, <note>Angle OBA</note>, and <note>Angle AOB</note> are the three interior angles of triangle OAB. According to the Triangle Angle Sum Theorem, <note>Angle OAB + Angle OBA + Angle AOB = 180°</note>.

Definition of Inscribed Angle

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

In the problem diagram, <note>the vertex C of angle C is on the circumference of circle O</note>, <note>the two sides of ∠ACB intersect circle O at points A and B respectively</note>. Therefore, <note>∠ACB 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 <note>circle O</note>, <note>point A and point B are two points on the circle, and the center of the circle is point O</note>. The angle ∠AOB formed by the lines <note>OA and OB</note> is called the central 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 diagram of this problem, in circle O, points A, B, and C are on the circle, <note>the central angle corresponding to arc ACB and arc AB is ∠AOB</note>, <note>the inscribed angle is ∠ACB</note>. According to the Inscribed Angle Theorem, <note>∠ACB is equal to half of the central angle ∠AOB corresponding to arc AB, that is, ∠ACB = 1/2 ∠AOB</note>.


Question: If F G H J is a kite, find m \angle G F J.

Answer: 80

1. Given that quadrilateral FGHJ is a kite, ##according to common knowledge, a kite is a symmetrical figure, and based on symmetry, we can deduce ∠F=∠H##.

2. ##According to the theorem of the sum of interior angles of a quadrilateral, the sum of the interior angles of quadrilateral FGHJ is 360°, therefore ∠F+∠H=360°-∠G-∠J=360°-128°-72°=160°##.

3. ##Since ∠F=∠H, we have ∠F=∠H=80°##.

####

##4##. Through the above reasoning, the final answer is ##∠GFJ = 80°##.

菱形; 对称; 等腰三角形; 邻补角

Symmetry

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

<note>Quadrilateral FGHJ has symmetry</note>. Specifically, <note>quadrilateral FGHJ is symmetric about line GJ</note>, <note>line GJ is the axis of symmetry of quadrilateral FGHJ</note>. Each part of quadrilateral FGHJ has a symmetric corresponding part on the other side of the axis of symmetry, <note>thus ∠F=∠H</note>.

In <note>quadrilateral FGHJ</note>, <note>angle GFH</note>, <note>angle GHJ</note>, <note>angle HJF</note>, and <note>angle FGH</note> are the four interior angles of the quadrilateral. According to the Sum of Interior Angles of a Quadrilateral Theorem, the sum of these four interior angles is 360°, that is, <note>angle GFH</note> + <note>angle GHJ</note> + <note>angle HJF</note> + <note>angle FGH</note> = 360°.


Question: Find x.

Answer: 17

1. Let the triangle in the figure be ABC, it is known that AD is perpendicular to BC, ∠ADB = ∠ADC = 90°, BC = 16, AD = 15, BD = DC = 1/2BC = 8.

2. According to the definition of a right triangle, triangle ADC is a right triangle, and the hypotenuse AC = x.

3. According to the Pythagorean theorem, 15^2 + 8^2 = x^2.

5. Based on the above reasoning, we get x = 17.

Definition of Right Triangle

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

Pythagorean Theorem

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

<note>In the right triangle ADC, angle ADC is a right angle (90 degrees)</note>, sides AD and DC are the legs, <note>side AC is the hypotenuse</note>, so according to the Pythagorean Theorem, <note>AC² = AD² + DC²</note>.


Question: Find x. A = 148 m^2.

Answer: 18.5

1. Given the area of the parallelogram A = 148 square meters, ##from the figure## the length of the base is 8 meters.

2. According to the area formula of the parallelogram, the area A is equal to the base length multiplied by the height, i.e., A = base length × height.

3. Substitute the given conditions into the formula, we get 148 = 8 × height.

4. Solve the equation 8 × height = 148, we get height = 148 ÷ 8.

5. Calculate to get height = 18.5.

6. Therefore, the answer is x = 18.5 meters.

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>the quadrilateral is a parallelogram</note>, and its <note>opposite sides are parallel and equal</note>. <note>The length of the base is 8 meters</note>.

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).

In the diagram of this problem, the altitude of the parallelogram is the line segment perpendicular to the <note>base</note> from the <note>top right vertex</note>, and its <note>vertical distance</note> is x meters.

Area Formula of a Parallelogram

The area of a parallelogram is calculated by multiplying the length of its base by the corresponding height, which can be expressed as \( A = \text{base length} \times \text{height} \).

<note>In the figure of this problem, the area of the parallelogram A = 148 square meters</note>, <note>the length of the base is 8 meters</note>, <note>the corresponding height is the vertical distance from the base to the opposite side, denoted as x meters</note>. Therefore, according to the area formula of the parallelogram, we get 148 = 8 × x, by solving the equation, <note>we find x = 18.5 meters</note>.


Question: As shown in the figure, line AB∥CD, Rt△DEF is placed as shown, ∠EDF=90°, if ∠1+∠F=70°, then the degree of ∠2 is ()

Answer: 20°

1. Given that line AB ∥ CD, according to the definition of same-side interior angles, ∠ABD and ∠BDC form a pair of same-side interior angles. According to the parallel axiom 2, ∠ABD and ∠BDC are supplementary, i.e., ∠ABD + ∠BDC = 180°.

2. Given that ∠1 + ∠F = 70°, let the point corresponding to ∠1 be point G. In △GFB, ∠ABD is its exterior angle. According to the exterior angle theorem of triangles, ∠ABD = ∠1 + ∠F. Therefore, it can be concluded that ∠ABD = 70°.

3. According to step 1, ∠ABD + ∠BDC = 180°. Substituting the given condition, we get 70° + ∠BDC = 180°, thus solving for ∠BDC = 110°.

4. Given that ∠EDF = 90°, ∠BDC = ∠2 + ∠EDF = 110°, it can be concluded that ∠2 = ∠BDC - ∠EDF = 110° - 90°.

5. After calculation, it is concluded that ∠2 = 20°.

平行线; 直角三角形; 内错角; 三角形的外角

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>Two lines AB and CD</note> are intersected by <note>a third line DF</note>, <note>∠ABD and ∠BDC</note> are on the same side of <note>the intersecting line DF</note> and within <note>the intersected lines AB and CD</note>, so <note>∠ABD and ∠BDC are consecutive interior angles</note>.

Parallel Postulate 2 of Parallel Lines

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

<note>Line AB ∥ CD</note>, line <note>BD is the transversal</note>, forming the following geometric relationship: <note>Consecutive interior angles: ∠ABD and ∠BDC are supplementary</note>, that is, <note>∠ABD + ∠BDC = 180 degrees</note>.

Exterior Angle Theorem of Triangle

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

In the diagram of this problem, in triangle GFB, <note>angle ABD is an exterior angle of the triangle</note>, <note>angle BGF and angle F are the two non-adjacent interior angles to exterior angle ABD</note>, according to the Exterior Angle Theorem of Triangle, <note>exterior angle ABD is equal to the sum of the two non-adjacent interior angles ∠BGF and ∠F</note>, that is, <note>∠ABD = ∠1 + ∠F</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.

The interior angle of the small triangle where angle 1 is located is ∠ACF, and the extended adjacent sides of this interior angle, <note>FB and AB forming the angle ∠ABD</note>, are called <note>the exterior angle of the interior angle ∠ABF</note>.


Question: Quadrilateral E F G H is a rectangle. If m \angle H G E = 13, find m \angle F G E.

Answer: 77

1. Given quadrilateral EFGH is a rectangle, ##EG and FH are the diagonals of rectangle EFGH##.

2. ##According to the definition of a rectangle, ∠GHE=90°, and according to the definition of a right triangle, triangle GHE is a right triangle##.

3. ##In right triangle GHE, given ∠HGE = 13°, ∠GHE=90°, according to the triangle angle sum theorem, ∠HGE+∠GHE+∠HEG=13°+90°+∠HEG=180°, i.e., ∠HEG=180°-13°-90°=77°##.

4. ##According to the definition of a rectangle, EH∥FG, and according to the parallel line axiom 2, alternate interior angles are equal, then ∠HEG= ∠FGE=77°##.

####

##5. ##Through the above reasoning, the final answer is: ##∠FGE##= 77°.

矩形; 对顶角; 邻补角

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 diagram of this problem, <note>quadrilateral EFGH is a rectangle</note>, <note>its interior angles ∠HEF, ∠EFG, ∠FGH, ∠GHE are all right angles (90 degrees)</note>, and <note>side EF is parallel and equal in length to side HG</note>, <note>side EH is parallel and equal in length to side FG</note>####

Triangle Angle Sum Theorem

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

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

Definition of Right Triangle

In the diagram of this problem, in triangle GHE, <note>angle GHE is a right angle (90 degrees)</note>, so triangle GHE is a right triangle. <note>Side EH and side HG are the legs</note>, <note>side EG is the hypotenuse</note>.


Question: Find the area of the trapezoid.

Answer: 132

1. Given that the upper base of the trapezoid is 16 ft, the lower base is 6 ft, and the height of the trapezoid is 12 ft.

2. The formula for calculating the area of the trapezoid is: ##Area = 1/2 × (Upper base + Lower base) × Height##.

3. Substitute the given conditions into the area formula: ##Area = 1/2 × (16 ft + 6 ft) × 12 ft##.

4. Calculate: ##Area = 1/2 × 22 ft × 12 ft##.

5. Finally, calculate the result: Area = 132 square feet.

6. Through the above reasoning, the final answer is that the area is 132 square feet.

Definition of Trapezoid

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

Original: <note>This quadrilateral is a trapezoid because it has only one pair of parallel sides</note>. In the figure of this problem, <note>the upper base of the trapezoid is 16 feet, the lower base is 6 feet, and the other two sides are not parallel</note>. Therefore, according to <note>the definition of trapezoid</note>, this quadrilateral is a trapezoid because it has only one pair of parallel sides.

Definition of Altitude

The line segment <note>from the vertex of the lower base perpendicular to the upper base (or its extension)</note> is the altitude of that vertex. <note>The lower base and the upper base (or its extension) form a right angle (90 degrees)</note>, which indicates that the line segment is the <note>vertical distance from the vertex of the lower base to the upper base (or its extension)</note>.

Trapezoid Area Formula

Given a trapezoid with the lengths of its two parallel sides as \(a\) and \(b\), and the height between these sides as \(h\), the area of the trapezoid is given by \(\frac{(a + b) \cdot h}{2}\).

<note>The upper base of the trapezoid is 16 feet, the lower base is 6 feet, and the height is 12 feet</note>. According to the trapezoid area formula, the area of the trapezoid is (upper base + lower base) * height / 2, <note>area = 1/2 × (16 feet + 6 feet) × 12 feet = 132 square feet</note>.


Question: As shown in the figure, ⊙O is the circumcircle of quadrilateral ABCD. If ∠O = 110°, then the degree of ∠C is ()

Answer: 125°

1. Given ∠O=110°, according to the inscribed angle theorem, ##we get ∠BAD=1/2∠O=55°##.

2. According to the problem statement, quadrilateral ABCD is a cyclic quadrilateral, so its opposite angles are supplementary, i.e., ∠A+∠C=180°.

3. Therefore, we get ∠BAD + ∠BCD = 180°.

4. Substituting the given conditions and the result from step 1, we get 55° + ∠BCD = 180°.

5. Thus, we get ∠BCD = 180° - 55° = 125°.

6. Through the above reasoning, we finally get the answer as 125°.

圆; 圆内接四边形; 圆心角; 对顶角; 邻补角

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>.

Inscribed Angle Theorem

In the figure of this problem, in circle ⊙O, points A, B, C, and D are on the circle, <note>the central angle corresponding to arc BD is ∠BOD</note>=110°, and the inscribed angle is ∠BAD. According to the Inscribed Angle Theorem, ∠BAD is equal to <note>half of the central angle ∠BOD corresponding to the arc BD</note>, that is, <note>∠BAD = 1/2 ∠BOD</note>= 55°.

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

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

In this problem diagram, <note>quadrilateral ABCD is a cyclic quadrilateral</note>, so 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>∠BAD + ∠BCD = 180°</note>, that is, <note>55° + ∠C = 180°</note>. Thus, <note>∠C = 125°</note>.

Definition of Central Angle

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


Question: Find x in the given parallelogram.

Answer: 38

1. Given that quadrilateral JKLM is a parallelogram, according to the ##properties of parallelogram theorem##, the opposite interior angles are equal, i.e., ##∠JML = ∠LKJ and ∠MJK = ∠KLM##.

2. ##∠MJK= 75°, its opposite angle ∠KLM is (2x-1)°, so (2x-1)°=75°, solving: x=38##.

####

##3##. Through the above reasoning steps, the final solution is x = 38.

平行四边形; 对顶角; 内错角; 同旁内角

Definition of Parallelogram

<note>Side JM is parallel and equal to side KL</note>, <note>side JK is parallel and equal to side ML</note>.

Properties of Parallelogram Theorem

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

<note>In parallelogram JKLM</note>, <note>the opposite angles ∠JML and ∠LKJ are equal, and the opposite angles ∠MJK and ∠KLM are equal</note>; <note>side JM and side KL are equal, and side JK and side ML are equal</note>; <note>the diagonals JL and MK bisect each other</note>, that is, <note>the intersection point divides diagonal JL into two equal segments, and the intersection point divides diagonal MK into two equal segments</note>.


Question: Find y.

Answer: 6

1. Let the three vertices of the triangle be A, B, and C. According to the right angle symbol in the figure, angle C is 90 degrees. Based on the definition of a right triangle, triangle ABC is a right triangle.

2. Given that angle C is 90°, angle B is 60°, according to the triangle sum theorem, angle C + angle B + angle A = 90° + 60° + angle A = 180°, thus angle A = 180° - 90° - 60° = 30°.

3. Given that the hypotenuse AB is 12, according to the properties of a 30°-60°-90° triangle, the shorter leg BC (the side opposite the 30° angle) is half of the hypotenuse AB.

4. Therefore, in this right triangle ABC, the length of the shorter leg opposite the 30° angle, BC = AB/2 = 12/2 = 6.

5. From the previous step, we know BC is the shorter leg corresponding to the 30° angle, which is y, thus the length of y is 6.

6. Therefore, based on the above reasoning steps, the final answer is y = 6.

Definition of Right Triangle

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

<note>In the figure of this problem, there is a 90-degree angle in the triangle</note>, so the triangle is a right triangle. <note>Side y and side x are the legs</note>, and the hypotenuse is ##12## units long.

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

Triangle Angle Sum Theorem

In the right triangle <note>ABC</note>, the <note>60° angle, 90° angle, and 30° angle are the three interior angles of the triangle</note>. According to the Triangle Angle Sum Theorem, <note>60° + 90° + 30° = 180°</note>.


Question: Find m \angle 1.

Answer: 53

1. From the figure, it can be observed that three lines intersect at a point, forming a straight angle with ∠1, a 37° angle, and a right angle.

2. According to the definition of a straight angle, we get ∠1 + 37° + 90° = 180°.

3. Solving for ∠1, we get ∠1 = 90° - 37° = 53°.

4. Through the above reasoning, the final answer is that the measure of ∠1 is 53°.

Definition of Straight Angle

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

In the figure of this problem, <note>the ray rotates around the endpoint to form a straight line with the initial side, forming a straight angle</note>. According to the <note>definition of a straight angle</note>, the measure of a straight angle is <note>180 degrees</note>.


Question: Using parallelogram W X Y Z, find m \angle X Y Z.

Answer: 105

1. Given that quadrilateral WXYZ is a parallelogram, according to the ##properties of parallelograms theorem##, the opposite angles of a parallelogram are equal, thus ##∠XWZ = ∠XYZ##.

2. According to the information given in the problem, ##∠XWZ## = 105°.

3. Therefore, from the conclusion in step 1, we get ##∠XYZ## = 105°.

4. Hence, through the above reasoning, the final answer is ##∠XYZ## = 105°.

Definition of Parallelogram

Original text: <note>In the figure of this problem, quadrilateral WXYZ is a parallelogram</note>, <note>side WX is parallel and equal to side YZ</note>, <note>side WZ is parallel and equal to side XY</note>.

Properties of Parallelogram Theorem

In the figure of this problem, in <note>parallelogram WXYZ</note>, <note>the opposite angles ∠XWZ and ∠XYZ are equal</note>; <note>the opposite angles ∠WXY and ∠WZY are equal</note>; <note>sides WX and YZ are equal</note>, <note>sides XY and WZ are equal</note>; <note>the diagonals WY and XZ bisect each other</note>.


Question: As shown in the figure, m \angle 1 = 123. Find the degree of \angle 14.

Answer: 57

1. Let the horizontal lines be AB and CD, and the vertical lines be EF and GH. According to the problem, AB∥CD, EF∥GH.

2. According to the parallel postulate 2 and the definition of corresponding angles, we get ∠1=∠3=∠11=123°.

3. ∠11 and ∠14 are on the same line and are adjacent supplementary angles. According to the definition of adjacent supplementary angles 'the sum of two adjacent angles formed by a straight line is 180°', we get ∠14 + ∠11 = 180°.

4. Currently, we know ∠11 = 123°, using the equation ∠14 + 123° = 180°, we can calculate ∠14 = 180° - 123° = 57°.

5. Through the above reasoning, the final answer is ∠14 = 57°.

Definition of Parallel Lines

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

In the diagram of this problem, <note>line AB and line CD are in the same plane and they do not intersect</note>, so according to the definition of parallel lines, <note>line AB and line CD are parallel lines</note>. <note>Line EF and line GH are in the same plane and they do not intersect</note>, so according to the definition of parallel lines, <note>line EF and line GH are parallel lines</note>.

Parallel Postulate 2 of Parallel Lines

##In the diagram of this problem, <note>two parallel lines EF and GH are intersected by a third line CD</note>, forming the following geometric relationship:## Corresponding angles: <note>angle 1 and angle 3 are equal</note>. <note>Two parallel lines AB and CD are intersected by a third line GH</note>, forming the following geometric relationship: Corresponding angles: <note>angle 11 and angle 3 are equal</note>. These relationships illustrate that when <note>two parallel lines are intersected by a third line</note>, corresponding angles are equal##.

Definition of Linear Pair of Angles

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

In the figure of this problem, <note>∠11 and ∠14 have a common side (line)</note>, their <note>other sides are respectively (lines) and (lines) that are extensions in opposite directions</note>, so <note>∠11 and ∠14 are adjacent supplementary angles</note>. Therefore, according to the <note>Adjacent Supplementary Angles Theorem</note>, <note>∠14 + ∠11 = 180°</note>. Since <note>∠11 = 123°</note>, therefore <note>∠14 = 180° - 123° = 57°</note>.


Question: Given that as shown in the figure, AB is the diameter of ⊙O, CD is the chord of ⊙O, ∠CDB=40°, then the degree of ∠CBA is ()

Answer: 50°

1. According to the problem, we know AB is the diameter of ⊙O. Connect AC to form triangle ACB.

2. According to (Corollary 2 of the Inscribed Angle Theorem), the inscribed angle subtended by the diameter is a right angle, ∠ACB=90°.

3. The problem states that ∠CDB=40°. According to the definition of inscribed angles, ∠CDB and ∠CAB are two inscribed angles subtended by the same arc CB. According to Corollary 1 of the Inscribed Angle Theorem, we have ∠CAB=∠CDB=40°.

4. In △ACB, since ∠ACB=90° and ∠CAB=40°, according to the Triangle Sum Theorem, we have ∠ACB+∠CAB+∠CBA=180°. Substituting the values, we get 90°+40°+∠CBA=180°. Calculating, we find ∠CBA=180°-90°-40°=50°.

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

圆; 圆周角; 弦; 直角三角形; 中点

Definition of Inscribed Angle

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

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

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

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

<note>In circle O, the inscribed angle ∠ACB subtended by the diameter AB is a right angle (90 degrees)</note>.

In a circle, any two inscribed angles that subtend the same arc are equal.

In the figure of this problem, <note>in circle O</note>, <note>the inscribed angles ∠CDB and ∠CAB corresponding to the same arc CB are equal</note>, that is, <note>∠CDB = ∠CAB</note>.

Triangle Angle Sum Theorem

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


Question: As shown in the figure, AB is parallel to CD. If ∠B=20°, then ∠C is ()

Answer: 20°

1. Given AB is parallel to CD and ∠B=20°.

2. According to the parallel lines ##Parallel Postulate 2, alternate interior angles are equal, we get ∠B=∠C=20°##.

3. ##Therefore, the value of ∠C is 20°##.

####

Definition of Parallel Lines

In the figure of this problem, <note>line AB</note> and <note>line CD</note> lie in the same plane and do not intersect, so according to the definition of parallel lines, line AB and line CD are parallel lines.

Parallel Postulate 2 of Parallel Lines

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

<note>Two parallel lines AB and CD</note> are intersected by <note>a third line BC</note>, forming the following geometric relationship: <note>alternate interior angles: angle B and angle C are equal</note>.

Definition of Alternate Interior Angles

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

The two parallel lines <note>AB and CD</note> are intersected by <note>a line BC</note>, where <note>angle B</note> and <note>angle C</note> are located between the two parallel lines and on opposite sides of the intersecting line BC. Therefore, <note>angle B and angle C are alternate interior angles</note>. Alternate interior angles are equal, that is, <note>angle B is equal to angle C</note>.


Question: If c = 5, find a.

Answer: 2.5

1. Given that triangle ABC is a right triangle, ∠ACB is a right angle, according to the definition of the sine function: In a right triangle, the sine value of an acute angle is equal to the ratio of the length of the side opposite the angle to the length of the hypotenuse.

2. From the figure, ∠CAB = 30°, side c = 5, then sin30° = a/c.

3. Substitute the given conditions, sin30° = 1/2, therefore a/5 = 1/2.

4. Solve a/5 = 1/2, yielding a = 5 × 1/2 = 2.5.

5. Through the above reasoning, the final answer is a = 2.5.

直角三角形; 正弦; 余弦

Definition of Sine Function

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

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


Question: Given: As shown in the figure, AB is the diameter of circle O, CD is a chord, connect AD and AC, ∠CAB=55°, then ∠D=()

Answer: 35°

1. Given that AB is the diameter of circle O, according to ##(Corollary 2 of the Inscribed Angle Theorem) the inscribed angle subtended by the diameter is a right angle##, we obtain ∠ACB = 90°.

2. Given ∠CAB = 55°, using the triangle angle sum theorem, we obtain ∠CBA = 90° - ∠CAB = 90° - 55° = 35°.

3. According to ##Corollary 1 of the Inscribed Angle Theorem##, ∠ADC = ∠CBA.

4. From step 2, we know ∠CBA = 35°, therefore ∠ADC = 35°.

5. Through the above reasoning, the final answer is 35°.

圆; 圆周角; 直角三角形; 弦; 线段

Definition of Diameter

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

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

Definition of Chord

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

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

Definition of Inscribed Angle

In the figure of this problem, in circle O, <note>angle ABC</note> has its <note>vertex B</note> on the circumference, and the two sides of <note>angle ABC</note> intersect circle O at points C and <note>point A</note>. Therefore, <note>angle ABC</note> is an inscribed angle. Similarly, <note>angle ADC</note> has its vertex D on the circumference, and the two sides of <note>angle ADC</note> intersect circle O at points C and <note>point A</note>. Therefore, <note>angle ADC</note> is an inscribed angle. Similarly, <note>angle ACB</note> has its vertex C on the circumference, and the two sides of <note>angle ACB</note> intersect circle O at points B and point A. Therefore, <note>angle ACB</note> is an inscribed angle.

Triangle Angle Sum Theorem

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

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

In the figure of this problem, in circle O, the inscribed angles <note>∠ABC</note> and <note>∠ADC</note> corresponding to <note>arc AB</note> are equal. According to Corollary 1 of the Inscribed Angle Theorem, <note>the inscribed angles ∠ABC and ∠ADC corresponding to the same arc AB are equal</note>, that is, <note>∠ADC = ∠ABC</note>.

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


Question: For trapezoid ABCD, S and T are the midpoints of the two legs. If AB = x + 4, CD = 3x + 2, and ST = 9, find AB.

Answer: 7

1. Given that trapezoid ABCD, where S and T are the midpoints of AC and BD respectively, AB = x + 4, CD = 3x + 2, ##midline## ST = 9.

2. According to the midline theorem of trapezoid, ##the midline of a trapezoid is half the sum of the lengths of the two parallel sides. For trapezoid ABCD, the midline is ST, thus the formula is:## ST = (AB + CD) / 2.

3. Substitute the given conditions into the midline formula: 9 = ((x + 4) + (3x + 2)) / 2.

4. Solve the equation: (x + 4) + (3x + 2) = 18, simplify to get: 4x + 6 = 18.

5. Rearrange to solve for x: 4x = 12.

6. Divide both sides by 4, to get: x = 3.

7. Substitute x = 3 into AB = x + 4, to get AB = 3 + 4 = 7.

8. Through the above reasoning, the final answer is AB = 7.

中点; 平行线; 梯形; 线段

Definition of Trapezoid

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

<note>Side AB and side CD are parallel</note>, while <note>side AC and side BD are not parallel</note>. Therefore, according to the <note>definition of trapezoid</note>, quadrilateral ABCD is a trapezoid because it <note>has exactly one pair of parallel sides</note>.

Midpoint of a Line Segment

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

In the figure of this problem, <note>the midpoint of line segment AC is point S</note>. According to the definition of the midpoint of a line segment, <note>point S divides line segment AC into two equal parts, that is, the lengths of line segments AS and SC are equal</note>. That is, <note>AS = SC</note>. In the figure of this problem, <note>the midpoint of line segment BD is point T</note>. According to the definition of the midpoint of a line segment, <note>point T divides line segment BD into two equal parts, that is, the lengths of line segments BT and TD are equal</note>. That is, <note>BT = TD</note>.

Median Line Theorem of Trapezoid

The median line of a trapezoid is the line segment that connects the midpoints of the non-parallel sides. This line segment is parallel to the bases (the parallel sides of the trapezoid) and its length is equal to half the sum of the lengths of the two bases.

In the figure of this problem, in the trapezoid <note>ABCD</note>, side <note>AB</note> and side <note>CD</note> are the two bases of the trapezoid, point <note>S</note> and point <note>T</note> are the midpoints of the two legs of the trapezoid, and segment <note>ST</note> is the median line connecting the midpoints of the legs. According to the <note>Median Line Theorem of Trapezoid</note>, segment <note>ST</note> is parallel to side <note>AB</note> and side <note>CD</note>, and the length of segment <note>ST</note> is equal to half the sum of the lengths of side <note>AB</note> and side <note>CD</note>, that is, <note>ST = (AB + CD) / 2</note>.


Question: Each pair of polygons is similar. Find AC.

Answer: 7.6

1. From the figure, we know ∠ABC=∠CDE=90 degrees. According to the definition of vertical angles, we get ∠ACB=∠DCE. Based on the theorem of similar triangles (AA), we can conclude that △ACB∽△EDC.

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2. According to the definition of similar triangles, we have AC/CE=CB/CD.

3. Since AC=x+7, CE=12-x, CB=4, CD=6, substituting into the equation, we get (x+7)/(12-x) = 4/6 = 2/3.

4. Performing cross multiplication, we get: 3(x + 7) = 2(12 - x).

5. Expanding the equation, we get: 3x + 21 = 24 - 2x.

6. Rearranging both sides, we get 5x + 21 = 24.

7. Rearranging, we get 5x = 24-21=3.

8. Dividing both sides of the equation by 5, we get x = 3/5.

9. Substituting into AC = x+7, we get AC = (3/5)+7 = 3/5 + 35/5 = 38/5 = 7.6.

10. Through the above reasoning, we finally get the answer as 7.6.

Definition of Similar Triangles

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

The original text: <note>Triangles △ACB and △EDC are similar</note>, <note>the ratio of the corresponding sides is AC/EC=CB/CD</note>.

Definition of Vertical Angles

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

<note>Two intersecting lines AE and BD intersect at point C</note>, forming four angles: <note>angle DCE</note>, <note>angle ACB</note>, <note>angle ACD</note>, and <note>angle BCE</note>. According to the definition of vertical angles, <note>angle ACB and angle DCE are vertical angles</note>, <note>angle ACD and angle BCE are vertical angles</note>. Since the angles of vertical angles are equal, <note>angle ACB = angle DCE</note>, <note>angle ACD = angle BCE</note>.

Similarity Theorem for Triangles (AA)

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

Given <note>∠ACB=∠ECD</note> and <note>∠ABC=∠CDE=90°</note>, therefore <note>△ACB～△EDC</note>.


Question: As shown in the figure, a∥b, ∠1=158°, ∠2=42°, ∠4=50°. Then ∠3=()

Answer: 70°

1. Pass through the vertex of ∠2 and make AB parallel to a, pass through the vertex of ∠3 and make CD parallel to b, let the vertex of ∠2 be E, the vertex of ∠3 be F, the vertex of ∠1 be G, and the vertex of ∠4 be H.

2. Since line a is parallel to line b, line AB is parallel to line a, and line CD is parallel to line b, according to the transitivity of parallel lines, we can obtain a∥b∥AB∥CD.

3. According to the parallel axiom 2 of parallel lines and the definition of same-side interior angles, ∠1 and ∠GEB are same-side interior angles, so ∠1 + ∠GEB = 180°. Given ∠1 = 158°, so ∠GEB = 180° - ∠1 = 180° - 158° = 22°.

4. According to the parallel axiom 2 of parallel lines and the definition of alternate interior angles, because b∥AB∥CD, ∠BEF and ∠EFC are alternate interior angles, ∠CFH and ∠4 are alternate interior angles, that is, ∠BEF = ∠EFC, ∠CFH = ∠4.

5. From the figure, we can see ∠3 = ∠EFC + ∠CFH, ∠BEF = ∠EFC, ∠CFH = ∠4, so ∠3 = ∠BEF + ∠4.

6. From the figure, we can see ∠2 = ∠GEB + ∠BEF = 42°, it has been calculated that ∠GEB = 22°, so ∠BEF = ∠2 - ∠GEB = 42° - 22° = 20°. Also, because ∠3 = ∠BEF + ∠4, ∠4 = 50°, so ∠3 = 20° + 50° = 70°.

平行线; 同旁内角; 内错角; 对顶角

Definition of Parallel Lines

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

<note>Line a and line b lie in the same plane and they do not intersect</note>, so according to the definition of parallel lines, <note>line a and line b are parallel lines</note>. <note>Line AB and line a lie in the same plane and they do not intersect</note>, so according to the definition of parallel lines, <note>line AB and line a are parallel lines</note>. <note>Line CD and line b lie in the same plane and they do not intersect</note>, so according to the definition of parallel lines, <note>line CD and line b are parallel lines</note>.

Parallel Postulate 2 of Parallel Lines

Original text: <note>Two parallel lines a and ##AB##</note> are intersected by a third line, forming the following geometric relationships: 1. Corresponding angles: ##None. 2. Alternate interior angles: None. 3. Consecutive interior angles: <note>∠1 and ∠GEB are supplementary, i.e., ∠1 + ∠GEB = 180 degrees</note>. These relationships indicate that when two parallel lines are intersected by a third line, <note>consecutive interior angles are supplementary</note>. <note>Two parallel lines AB and CD</note> are intersected by a third line, forming the following geometric relationships: 1. Corresponding angles: None. 2. Alternate interior angles: <note>∠BEF and ∠EFC are equal</note>. 3. None. These relationships indicate that when two parallel lines are intersected by a third line, <note>alternate interior angles are equal</note>. <note>Two parallel lines CD and b</note> are intersected by a third line, forming the following geometric relationships: 1. Corresponding angles: None. 2. Alternate interior angles: <note>∠CFH and ∠4 are equal</note>. 3. None. These relationships indicate that when two parallel lines are intersected by a third line, <note>alternate interior angles are equal</note>.

Definition of Alternate Interior Angles

<note>Line AB is parallel to line CD, and line EF is the transversal</note>, where <note>∠BEF and ∠EFC are located between the two parallel lines and on opposite sides of the transversal EF</note>, thus <note>∠BEF and ∠EFC are alternate interior angles</note>. <note>Alternate interior angles are equal, i.e., ∠BEF=∠EFC</note>. <note>Line CD is parallel to line b, and line FH is the transversal</note>, where <note>∠CFH and ∠4 are located between the two parallel lines and on opposite sides of the transversal FH</note>, thus <note>∠CFH and ∠4 are alternate interior angles</note>. <note>Alternate interior angles are equal, i.e., ∠CFH=∠4</note>.

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>Line a is parallel to line AB, intersected by a third line</note>, ∠1 and ∠GEB are on the same side of the transversal and inside the intersected lines a and AB, so <note>∠1 and ∠GEB are consecutive interior angles</note>. Consecutive interior angles ∠1 and ∠GEB are supplementary, that is, <note>∠1 + ∠GEB = 180 degrees</note>.

Transitivity of Parallel Lines

If two lines are each parallel to a third line, then those two lines are parallel to each other.

<note>Line AB and line b are respectively parallel to line a</note>. According to the transitivity of parallel lines, <note>line AB is parallel to line a</note>, <note>line b is parallel to line z</note>, <note>line x and line a are mutually parallel</note>. Therefore, <note>line AB is parallel to line b</note>. Similarly, we can obtain <note>a∥b∥AB∥CD</note>.


Question: Find x. Assume any line segment that appears to be tangent is tangent.

Answer: 10

1. Let angle P be the exterior angle of circle O. From the figure, we know angle P = 3x°, the degree of arc AB = (4x + 50)°, and the degree of arc CD = 30°. According to the exterior angle theorem of the circle, we get angle P = (degree of arc AB - degree of arc CD) / 2, i.e., 3x° = [(4x + 50)° - 30°] / 2.

3. Multiplying both sides by 2, we get: 4x + 20 = 6x.

4. Rearranging terms, we get 2x = 20, thus simply obtaining x = 10.

5. Through the above reasoning, the final answer is x = 10.

External Angle Theorem of a Circle

The measure of an exterior angle of a circle is equal to one-half the difference of the measures of the intercepted arcs.

<note>Angle APB is the external angle of the circle</note>, <note>Arc AB and Arc CD are the two arcs subtended by angle APB</note>. According to the External Angle Theorem of a Circle, <note>angle APB is equal to half the difference of the degrees of arc AB and arc CD</note>, that is, <note>angle APB = (degrees of arc AB - degrees of arc CD) / 2</note>.


Question: As shown in the figure, points A, B, and C are all on ⊙O. When ∠OBC=40°, the degree of ∠A is ()

Answer: 50°

1. Given that point A, point B, and point C are all on circle O, and ∠OBC=40°. According to the definition of the inscribed angle, ∠A is the inscribed angle of circle O; according to the definition of the central angle, ∠BOC is the central angle of circle O.

2. According to the definition of a circle, OB = OC. In △BOC, according to the definition of an isosceles triangle, △BOC is an isosceles triangle; further, according to the properties of an isosceles triangle, the base angles opposite the equal sides are equal, thus ∠OBC = ∠OCB=40°.

3. According to the triangle angle sum theorem, the sum of the three interior angles of a triangle is 180°, so in △BOC we have ∠BOC + ∠OBC + ∠OCB = 180°, thus ∠BOC=180°-∠OBC-∠OCB=180°-40°-40°=100°.

4. According to the inscribed angle theorem, the inscribed angle is half of the central angle it subtends, so ∠BAC=∠BOC/2.

5. Substituting the known values, we get ∠BAC=100°/2=50°.

6. Through the above reasoning, the final answer is ∠A=50°.

圆; 圆周角; 圆心角; 弧; 弦

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, and the center of the circle is point O</note>. The angle ∠BOC formed by the lines <note>OB and OC</note> is called the central angle.

Definition of Radius

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

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

Definition of Circle

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

In the figure of this problem, <note>circle O</note>, <note>point O is the center of the circle</note>, <note>the radii are OB and OC</note>. All <note>points equidistant from point O to OB and OC</note> are on <note>circle O</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 the diagram of this problem, in circle O, points A, B, and C are on the circle</note>, the central angle corresponding to arc BC and arc BAC is <note>∠BOC</note>, and the inscribed angle is <note>∠BAC</note>. According to the Inscribed Angle Theorem, <note>∠BAC is equal to half of the central angle ∠BOC corresponding to arc BC</note>, that is, <note>∠BAC = 1/2 ∠BOC</note>.

Definition of Isosceles Triangle

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

In the figure of this problem, in triangle BOC, <note>side OB and side OC are equal</note>, therefore <note>triangle BOC is an isosceles triangle</note>.

Properties of Isosceles Triangle

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

In the isosceles triangle <note>BOC</note>, <note>sides OB and OC are equal</note>. Therefore, according to the properties of isosceles triangles, the angles opposite the equal sides are equal, that is, <note>∠OBC = ∠OCB</note>.

Triangle Angle Sum Theorem

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

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

Definition of Inscribed Angle

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

In the figure of this problem, in circle O, <note>the vertex of angle BAC (point A) 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>.


Question: Find m \angle B.

Answer: 61

1. Let the center of the circle be O, connect AO and OC, it is known that the degree of arc AC is 122°, ∠AOC is the central angle, ∠ABC is the inscribed angle.

2. According to the property of the central angle, the degree of the central angle is equal to the degree of its corresponding arc, we can get ∠AOC=122°.

3. According to the inscribed angle theorem, the inscribed angle is equal to half of the central angle corresponding to its arc, ∠ABC=(1/2)*∠AOC=(1/2)*122°=61°.

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

Inscribed Angle Theorem

<note>∠AOC</note> = 122° as the central angle. According to the Inscribed Angle Theorem, the inscribed angle is half of the central angle. Therefore, the inscribed angle subtended by the arc intercepted by <note>∠AOC</note> is <note>∠ABC</note> = 122°/2 = 61°.

Properties of Central Angles

The measure of a central angle is equal to the measure of the arc that it intercepts.

<note>Given that the central angle ∠AOC corresponds to the arc AC</note>. According to the properties of central angles, <note>the degree measure of the central angle is equal to the degree measure of the corresponding arc</note>, that is, <note>the degree measure of ∠AOC = the degree measure of arc AC</note>.

Definition of Central Angle

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

Definition of Arc

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

In the figure of this problem, there are two points A and C on circle O, <note>arc AC</note> is a segment of the curve connecting these two points. According to the <note>definition of arc</note>, <note>arc AC is a segment of the curve between two points A and C on the circle</note>.


Question: Solve for X Z.

Answer: 34

1. Given ZW = WY, WX = WX, ∠ZWX = ∠YWX = 90°, so according to the theorem of congruent triangles (SAS), we know that triangle ZWX and triangle YWX are congruent.

2. From the definition of congruent triangles, we know ZX = YX.

3. Given ZX equals 5y - 6, YX = 18 + 2y, set up the equation 5y - 6 = 18 + 2y.

4. Rearrange the equation: 5y - 2y = 18 + 6.

直角三角形; 等腰三角形

Definition of Congruent Triangles

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

<note>In the diagram of this problem, triangle WYX and triangle WZX are congruent triangles</note>, the corresponding sides and corresponding angles of triangle WYX are equal to those of triangle WZX, namely: <note>side WY = side WZ</note>, <note>side WX = side WX</note>, <note>side YX = side ZX</note>, and at the same time, <note>the corresponding angles are also equal</note>: <note>angle WXY = angle WXZ</note>, <note>angle YWX = angle ZWX</note>, <note>angle WYX = angle WZX</note>.

Triangular Congruence Theorem (SAS)

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

In <note>triangle WZX and triangle WYX</note>, <note>side ZW</note> is equal to <note>side WY</note>, <note>side WX</note> is equal to <note>side WX</note>, and <note>angle ZWX</note> is equal to <note>angle YWX</note>. Therefore, according to the Triangular Congruence Theorem (SAS), these two triangles are congruent.


Question: Find x. Round to the nearest tenth.

Answer: 18.8

1. Let the vertices of the triangle in the figure be points A, B, and C, with BC as the base, and the height from vertex A to base BC be AD. Given: AB=AC=32, AD=y, BD=x, ∠ACB=54°. Therefore, according to the definition and properties of an isosceles triangle, triangle ABC is an isosceles triangle, and ∠ACB=∠ABC=54°. Since AD is the height on BC, ∠ADB=∠ADC=90°. According to the definition of a right triangle, triangles ABD and ACD are both right triangles, so the sine function definition can be used. Sine function definition: In a right triangle, the sine function is defined as the ratio of the length of the side opposite the angle to the hypotenuse.

2. In this problem, it is known that the base angle ∠ABD of triangle ABC is 54°, AD length is y, BD length is x. Therefore, sin(54°) = opposite side (y) / hypotenuse (32).

3. In the figure, it is easy to see that the opposite side is y, which refers to the side opposite the angle 54°.

4. Apply the sine function definition and substitute the known values: sin(54°) ≈ 0.809. Thus, 0.809 = y / 32.

5. Through algebraic operations, we get y = 32 * 0.809 ≈ 25.888.

6. According to the Pythagorean theorem: In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the two legs. Substitute x, y, and the hypotenuse.

7. Now, introduce y and the known sides into the formula: x^2 + 25.888^2 = 32^2. Substitute and calculate: x^2 ≈ 32^2 - 25.888^2.

9. Taking the square root gives the approximate value of x: x ≈ √353.534 ≈ 18.8.

10. Through correct reasoning and algebraic operations, the final answer is x ≈ 18.8.

等腰三角形; 直角三角形; 正弦

Definition of Isosceles Triangle

In the figure of this problem, in triangle ABC, <note>AB=AC=32</note>, therefore <note>triangle ABC is an isosceles triangle</note>.

Properties of Isosceles Triangle

In the figure of this problem, in <note>isosceles triangle ABC</note>, <note>side AB and side AC are equal</note>. Therefore, according to the properties of an isosceles triangle, the angles opposite the equal sides are equal, that is, <note>angle ABC = angle ACB</note>. (Or: In the figure of this problem, in <note>isosceles triangle ABC</note>, <note>angle ABC and angle ACB are equal</note>. Therefore, according to the properties of an isosceles triangle, the sides opposite the equal angles are equal, that is, <note>side AB = side AC</note>.)

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).

The segment AD <note>from vertex A perpendicular to the opposite side BC (or its extension)</note> is the altitude from vertex A. The segment AD forms a right angle (90 degrees) <note>with side BC (or its extension)</note>, indicating that segment AD is <note>the perpendicular distance from vertex A to the opposite side BC (or its extension)</note>.

Definition of Right Triangle

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

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

Definition of Sine Function

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

In <note>right triangle ABD</note>, <note>angle ABD=54° is an acute angle</note>, <note>AD=y is the opposite side of angle 54°</note>, <note>AB=32 is the hypotenuse</note>. According to the definition of the sine function, the sine value of angle 54° is equal to the ratio of the opposite side y to the hypotenuse 32, that is, sin(54°) = y / 32.

Pythagorean Theorem

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

In the right triangle ABD, the hypotenuse is 32, and the two legs are x and y respectively. So according to the Pythagorean Theorem, <note>32^2 = x^2 + y^2</note>.


Question: Find x. Round to two decimal places.

Answer: 27.44

1. Given that the condition is a right triangle, and one side length is 24 with an angle of 61°, we are required to find the length of the hypotenuse x. According to the definition of the sine function in right triangles, we know: sin(θ) = opposite/hypotenuse. Therefore, in this problem, we can use sin(61°) = 24/x.

2. Transform the above equation to x = 24/sin(61°) to calculate the length of the hypotenuse x.

3. Use a calculator to find the value of sin(61°), which gives sin(61°) ≈ 0.8746.

4. Substitute sin(61°) ≈ 0.8746 into x = 24/sin(61°), and calculate to get x ≈ 24/0.8746 ≈ 27.44.

5. Following the above reasoning steps, the answer is x ≈ 27.44. Rounding to two decimal places, the final result is 27.44.

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, the <note>triangle</note> is composed of <note>three non-collinear points and their connecting line segments</note>. In this <note>triangle</note>, the length of the <note>leg</note> is 24, the length of the <note>hypotenuse</note> is x, and the <note>angle is 61°</note>.

Definition of Right Triangle

In the figure of this problem, <note>one angle in the triangle is a right angle (90 degrees)</note>, so this triangle is a right triangle. <note>The length of one leg of the right triangle is 24</note>, <note>the hypotenuse is x</note>.

Definition of Sine Function

<note>Angle 61° is an acute angle</note>, <note>side 24 is the opposite side of angle 61°</note>, <note>side x is the hypotenuse</note>. According to the definition of the sine function, the sine value of angle 61° is equal to the ratio of the opposite side 24 to the hypotenuse x, that is, <note>sin(61°) = 24 / x</note>.


Question: In \odot O, E C and A B are diameters, and \angle B O D \cong \angle D O E \cong \angle E O F \cong \angle F O A. Find m \widehat A D.

Answer: 135

1. Given the condition that in circle O, EC and AB are diameters, ∠ BOD≌∠ DOE≌∠ EOF≌∠ FOA.

2. Since angles BOD, DOE, EOF, and FOA are equal, and these 4 angles form a straight angle, the sum of these four angles is equal to 180°.

3. Therefore, the sum of these four angles is 180°, let each angle be x, according to the equation 4x = 180°, we can solve for x = 45°.

4. The central angle corresponding to arc AD is the sum of angles DOE, EOF, and FOA, so ∠AOD = 3 * 45° = 135°, therefore arc AD = 135°.

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5. Based on the previous reasoning, the final answer is arc AD = 135°.

圆; 圆心角; 弧; 弦; 直线

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>EC and AB are diameters</note>, connecting <note>the center O</note> and <note>points E, C, and A, B on the circumference</note>, with a length of <note>2 times the radius</note>, i.e., <note>EC = 2r</note> and <note>AB = 2r</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 diagram of this problem, <note>circle O</note>, <note>point B and point E</note> are two points on the circle, <note>the center of the circle is point O</note>. The angle ∠BOE formed by <note>line segments OB and OE</note> is called a central angle. Similarly, the angle ∠EOF formed by <note>point E and point F</note>, the angle ∠FOA formed by <note>point F and point A</note>, and the angle ∠AOD formed by <note>point A and point D</note> are also central angles.

Definition of Straight Angle

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

<note>Ray OA rotates around endpoint O to form a straight line with the initial side</note>, forming a <note>straight angle AOB</note>. According to the <note>definition of straight angle</note>, <note>the measure of a straight angle is 180 degrees</note>, i.e., <note>angle AOB = 180 degrees</note>.


Question: As shown in the figure, when the width of the water surface AB in the circular bridge hole is 8 meters, the arc ACB is exactly a semicircle. When the water surface rises by 1 meter, the width of the water surface A′B′ in the bridge hole is ()

Answer: 2√{15}米

1. Let the center of the circle be O, draw the chord A′B′ with the chordal distance OE, and connect OA′.

2. According to the problem statement, OE⊥A'B', ∠OEA'=∠OEB'=90°, then the triangle OA′E is a right triangle. According to the definition of the radius, the radius is the line segment from the center of the circle to any point on the circle. Given that point A' is on the circle, the distance from point A' to the center O is the radius, which is half of the diameter. Given that the diameter is 8, so OA' = 4. Thus, in the right triangle OA′E, OA′ = 4 meters, OE = 1 meter.

3. According to the Pythagorean theorem, in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the two legs. Therefore, in the right triangle OA′E, OA′^2 = OE^2 + A′E^2, that is, 4^2 = 1^2 + A′E^2, thus A′E = √15 meters.

4. According to the perpendicular bisector theorem, in a circle, the chord perpendicular to the diameter is bisected by the diameter. Given that OE is the chordal distance, and the chordal distance is on the diameter of the circle, so OE also perpendicularly bisects the chord A'B'. Therefore, A′B′ = 2A′E = 2√15 meters.

5. Through the above reasoning, the final answer is 2√15 meters.

圆; 弦; 弧; 垂线; 直角三角形

Definition of Radius

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

<note>In the figure of this problem, in circle O, point O is the center of the circle, point A is any point on the circle, and the line segment ##OA'## is a line segment from the center to any point on the circle</note>, therefore <note>the line segment ##OA'## is the radius of the circle</note>.

Chord Central Distance

The perpendicular distance from the center of a circle to a chord is referred to as the chord central distance.

<note>In circle O, point O is the center of the circle</note>, line segment <note>A'B' is a chord of the circle</note>, line segment <note>OE is the perpendicular segment from the center O to the chord A'B'</note>. According to the definition of chord central distance, <note>the length of segment OE is the vertical distance from the center to the chord A'B', referred to as the chord central distance</note>.

Definition of Right Triangle

<note>In the figure of this problem, in triangle OA′E, angle ∠OA′E is a right angle (90 degrees)</note>, therefore <note>triangle OA′E is a right triangle</note>. <note>Sides OA′ and OE are the legs</note>, <note>side A′E is the hypotenuse</note>.

Pythagorean Theorem

<note>In the right triangle OA′E, ∠OA′E is a right angle (90 degrees)</note>, sides OE and A′E are the legs, and side OA′ is the hypotenuse, so according to the Pythagorean Theorem, <note>OA′^2 = OE^2 + A′E^2</note>, that is <note>4^2 = 1^2 + A′E^2</note>, thus we get <note>A′E = √15 meters</note>.

Perpendicular Diameter Theorem

If a diameter is perpendicular to a chord, then that diameter bisects the chord and also bisects the two arcs subtended by that chord.

In the figure of this problem, in circle O, <note>diameter OE is perpendicular to chord A′B′</note>, then according to the Perpendicular Diameter Theorem, <note>diameter OE bisects chord A′B′</note>, that is, <note>A′E = E′B′</note>, and <note>diameter OE bisects the arcs subtended by chord A′B′</note>, that is, <note>arc A′C = arc CB′</note>.


Question: In \odot T, Z V = 1, and T W = 13. Find X Y.

Answer: 10

1. According to the problem conditions, ##circle T is a circle, and ZV and TV## are part of the radius of this circle.

2. Knowing ZV = 1 and TW = 13, we need to find the length of segment XY.

3. From the figure, it is known that TV ⊥ ZV, and TV is part of the line of the diameter. This means ##angle XVT is a right angle##.

4. ##According to the definition of a right triangle, triangle XVT is a right triangle. Given XT and ZT are radii of circle T, ZV = 1, TW = 13, so XT = 13, and VT = 13 - 1 = 12##.

5. ##According to the Pythagorean theorem, in the right triangle XVT, XT^2 = XV^2 + VT^2. Given XT = 13 and VT = 12, so XV^2 = XT^2 - VT^2 = 169 - 144 = 25, and taking the square root: XV = 5##.

6. ##According to the perpendicular bisector theorem, the radius ZT perpendicularly bisects the chord XY, so XY equals twice XV, i.e., XY = 2XV, thus XY = 2 * 5 = 10##.

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>in circle T</note>, <note>point T is the center of the circle</note>. All <note>points at a distance equal to the radius from point T are on circle T</note>.

Definition of Radius

Original text: <note>Point T is the center of the circle</note>, <note>Point W is any point on the circle</note>, <note>Line segment TW is the line segment from the center of the circle to any point on the circle</note>, therefore <note>Line segment TW is the radius of the circle</note>. Similarly, <note>in circle T, point T is the center of the circle</note>, <note>Point X is any point on the circle</note>, <note>Line segment TX is the line segment from the center of the circle to any point on the circle</note>, therefore <note>Line segment TX is the radius of the circle</note>. Similarly, <note>in circle T, point T is the center of the circle</note>, <note>Point Z is any point on the circle</note>, <note>Line segment TZ is the line segment from the center of the circle to any point on the circle</note>, therefore <note>Line segment TZ is the radius of the circle</note>.

Definition of Chord

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

Original: <note>In circle T, points X and Y are any two points on the circle, and line segment XY connects these two points</note>, so <note>line segment XY is a chord of circle T</note>.

Definition of Triangle

<note>Triangle XVT</note> is a geometric figure composed of three <note>non-collinear points X, V, T</note> and their <note>connecting line segments XT, XV, VT</note>. <note>Points X, V, T are the three vertices of the triangle</note>, and <note>line segments XT, XV, VT are the three sides of the triangle</note>.

Definition of Right Triangle

<note>In the figure of this problem, in triangle XVT, angle XVT is a right angle (90 degrees)</note>, therefore triangle XVT is a right triangle. <note>Side XV and side VT are the legs, side XT is the hypotenuse.</note>

Pythagorean Theorem

<note>In the right triangle XVT</note>, <note>∠XVT is a right angle (90 degrees)</note>, the sides <note>XV and TV are the legs</note>, and the side <note>XT is the hypotenuse</note>, so according to the Pythagorean Theorem, <note>XT^2=XV^2+VT^2</note>.

Perpendicular Diameter Theorem

<note>In circle T, radius ZT is perpendicular to chord XY</note>, then according to the Perpendicular Diameter Theorem, <note>radius ZT bisects chord XY, that is, XV=VY</note>, and <note>radius ZT bisects the arcs subtended by chord XY, that is, arc ZX=arc ZY</note>.


Question: Find the area of the parallelogram. If necessary, round to the nearest tenth.

Answer: 1440

1. Given that the base length of the parallelogram is 40 inches and the height is 36 inches. To calculate the area of the parallelogram, we can use the area formula, which states that the area of a parallelogram is equal to the base times the height.

2. According to the parallelogram area formula: Area = Base × Height, in this problem the base length and height are 40 inches and 36 inches respectively.

3. Calculation: Area = 40 inches × 36 inches = 1440 square inches.

4. Since the problem requires the result to be rounded to one decimal place, the final result remains unchanged.

5. Through the above reasoning, the final answer is 1440.0 square inches.

平行四边形; 平行线; 垂线

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, the quadrilateral satisfies <note>two pairs of opposite sides are parallel and equal</note>, namely <note>the top side is parallel and equal to the bottom side</note>, and <note>the left side is parallel and equal to the right side</note>. Therefore, this is <note>a parallelogram</note>.

Area Formula of a Parallelogram

The area of a parallelogram is calculated by multiplying the length of its base by the corresponding height, which can be expressed as \( A = \text{base length} \times \text{height} \).

<note>The length of the base of the parallelogram is 40 inches</note>, <note>the corresponding height is the vertical distance from the base to the opposite side, denoted as 36 inches</note>. Therefore, according to the area formula of a parallelogram, the area of the parallelogram is equal to <note>the base length of 40 inches multiplied by the corresponding height of 36 inches</note>, i.e., A = 40 inches × 36 inches = 1440 square inches.

Height of a Parallelogram

The height (or altitude) of a parallelogram is the perpendicular distance from a vertex on one side to the line containing the opposite side.

<note>In the figure of this problem, the upper and lower bases of the parallelogram are parallel.</note> <note>The vertical distance from the top left vertex to the lower base is the height of the parallelogram.</note> Specifically, <note>draw a line segment perpendicular to the base from the top left vertex</note>, the length of this line segment (36 inches) is the height of the parallelogram.


Question: A square is inscribed in a circle with an area of 18 \\pi square units. Find the side length of the square.

Answer: 6

1. Given that the area of the circle is 18π square units. According to the formula for the area of a circle A = πr^2, we can deduce that the square of the radius r is 18.

2. ####Thus, the radius r = √18.

3. ##Let the square ABCD be inscribed in the circle O. According to the definition of a square, ∠A=∠B=∠C=∠D=90°, AB=BC=CD=DA. According to (the corollary of the inscribed angle theorem 2) the angle subtended by the diameter is a right angle, the diagonals AC and BD of the square are the diameters of the circle##.

4. Therefore, the diameter of the circle ##AC=BD=2r##= 2√18 = 2√(9*2) = 6√2.

5. ##In △ABD, ∠A=90°, so according to the definition of a right triangle, △ABD is a right triangle. According to the Pythagorean theorem, AB^2 + AD^2 = BD^2, since AB = AD, we get 2AB^2 = (6√2)^2##.

6. Thus, we can deduce the side length of the square ##AB= 6##.

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

Area Formula of a Circle

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

In the figure of this problem, <note>the area of the circle is 18π square units</note>. According to <note>the area formula of a circle</note>, <note>the area A of the circle is equal to π multiplied by the square of the radius r, i.e., A = πr^2</note>. <note>Given that A = 18π</note>, we can solve for <note>πr^2 = 18π</note>, and then calculate <note>the radius r = √18</note>.

Definition of Square

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

In the quadrilateral <note>ABCD</note>, <note>sides AB, BC, CD, and DA are equal</note>, and <note>angles ABC, BCD, CDA, and DAB are all right angles (90 degrees)</note>, so <note>ABCD is a square</note>.

Definition of Diameter

<note>Diameter of a circle</note> is a <note>line segment</note> that passes through the <note>center of the circle</note> and has <note>both endpoints on the circle</note>, with a <note>length of 2 times the radius</note>, i.e., <note>diameter d = 2r</note>. Calculated as <note>d = 2√18</note>.

Definition of Inscribed Angle

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

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

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

<note>In circle O, the vertex of angle BAD (point A) 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 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 the square <note>ABCD</note>, the vertices <note>A</note>, <note>B</note>, <note>C</note>, and <note>D</note>, the diagonal is the line segment connecting vertex <note>A(B)</note> and the non-adjacent vertex <note>C(D)</note>. Therefore, the line segments <note>AC</note> and <note>BD</note> are the diagonals of the square <note>ABCD</note>.

Definition of Right Triangle

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

In the figure of this problem, in triangle ABD, <note>angle BAD is a right angle (90 degrees)</note>, therefore <note>triangle ABD is a right triangle</note>. <note>Side AD and side AB are the legs</note>, <note>side BD is the hypotenuse</note>.

Pythagorean Theorem

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

<note>In the figure of this problem, in the right triangle ABD, angle BAD is a right angle (90 degrees)</note>, the sides <note>AB and AD are the legs</note>, the side <note>BD is the hypotenuse</note>, so according to the Pythagorean Theorem, <note>BD² = AB² + AD²</note>.


Question: AB is the diameter, AC = 8 inches, BC = 15 inches. Find the radius of the circle.

Answer: 8.5

1. Given that AB is the diameter, according to the definition of diameter, AB is a diameter of the circle, and a diameter of a circle is the longest line segment passing through the center of the circle.

2. Given AC = 8 inches, BC = 15 inches. To find the radius of the circle, we can first use this information to calculate the length of AB.

3. According to the corollary of the inscribed angle theorem, the inscribed angle subtended by the diameter is a right angle. Therefore, the inscribed angle ∠ACB subtended by the diameter AB is 90°, so triangle ABC is a right triangle.

4. According to the Pythagorean theorem, we have: AB squared = AC squared + BC squared, i.e., AB^2 = AC^2 + BC^2.

5. Substituting the given values: AB^2 = 8^2 + 15^2 = 64 + 225 = 289.

6. Therefore, the length of AB is: AB = √(289) = 17 inches.

7. The radius is half of the diameter, so the radius of the circle is: 17 / 2 = 8.5 inches.

Definition of Diameter

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

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

In the figure of this problem, <note>在本题图中，圆中，直径AB</note> subtends a <note>圆周角∠ACB是直角（90度）</note>.

Pythagorean Theorem

<note>In right triangle ABC</note>, <note>∠ACB is a right angle (90 degrees)</note>, sides AC and BC are the legs, side AB is the hypotenuse, so according to the Pythagorean Theorem, <note>AB^2 = AC^2 + BC^2</note>.

Definition of Inscribed Angle

In the figure of this problem, in circle O, <note>the vertex of angle ACB (point C) 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.

Definition of Radius

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


Question: In the figure, \triangle A B C \sim \triangle D E F. B G is the median of \triangle A B C, E H is the median of \triangle D E F. If B C = 30, B G = 15, and E F = 15, find E H.

Answer: 7.5

1. Given that △ABC ～△DEF, according to the definition of similar triangles, AB/DE = BC/EF = AC/DF, ∠C=∠F.

2. Since BG is the median of △ABC and EH is the median of △DEF, according to the definition of the median of a triangle, G is the midpoint of AC, H is the midpoint of DF. According to the midpoint of a line segment, AG=GC=?AC, DH=HF=?DF, then GC/HF=?AC/?DF=AC/DF=BC/EF.

3. Since ∠C=∠F, GC/HF=BC/EF, according to the SAS similarity theorem, △BCG is similar to △EFH. According to the definition of similar triangles, BG/EH=BC/EF=GC/HF.

4. Given BC = 30, EF = 15, BG=15, therefore 15/EH = 30/15 = 2.

5. Solving the equation gives EH = 15/2 = 7.5.

6. The above reasoning concludes that the length of EH is 7.5.

普通三角形; 线段; 中点; 位似

Definition of Similar Triangles

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

<note>△ABC and △DEF are similar triangles</note>. According to the definition of similar triangles: <note>∠A = ∠D</note>, <note>∠B = ∠E</note>, <note>∠C = ∠F</note>; <note>AB/DE = BC/EF = AC/DF</note>. <note>Triangle BCG is similar to triangle EFH</note>, according to the definition of similar triangles: <note>BG/EH=BC/EF=GC/HF</note>.

Definition of Median of a Triangle

A median of a triangle is a line segment drawn from one vertex of the triangle to the midpoint of the opposite side.

In the figure of this problem, in triangle ABC, <note>vertex B</note> is a vertex of the triangle, <note>opposite side AC</note> is the side opposite to vertex B. <note>Point G</note> is the midpoint of side AC, <note>segment BG</note> is the segment from vertex B to the midpoint G of opposite side AC, therefore BG is a median of triangle ABC. In triangle DEF, <note>vertex E</note> is a vertex of the triangle, <note>opposite side DF</note> is the side opposite to vertex E. <note>Point H</note> is the midpoint of side DF, <note>segment EH</note> is the segment from vertex E to the midpoint H of opposite side DF, therefore EH is a median of triangle DEF.

Similarity Theorem of Triangles (SAS)

Two triangles are similar if they have two pairs of corresponding sides in the same ratio, and the included angle between these pairs of sides is equal.

Original: <note>在本题图中，在三角形BCG和三角形EFH中</note>，<note>GC/HF=BC/EF</note>，<note>且∠C=∠F</note>，则<note>三角形BCG与三角形EFH相似</note>。

Translation: <note>In the figure of this problem, in triangles BCG and EFH</note>, <note>GC/HF = BC/EF</note>, <note>and ∠C = ∠F</note>, then <note>triangles BCG and EFH are similar</note>.

Midpoint of a Line Segment

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

<note>G is the midpoint of AC</note>, <note>H is the midpoint of DF</note>, according to the definition of the midpoint of a line segment, <note>GC=AG=?AC</note>, <note>HF=DH=?DF##</note>.