Question: In rhombus L M P Q, m \angle Q L M = 2 x^{2} - 10, m \angle Q P M = 8 x, and M P = 10. Find the perimeter of L M P Q.

Answer: 40

1. According to the definition of a rhombus, in the rhombus LMPQ, all sides are equal. Given the condition MP = 10, it follows that LM = MP = PQ = QL = 10.

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2. According to the formula for the perimeter of a rhombus, the perimeter of the rhombus = 4 × side length. Given that the side length is 10, the perimeter is 4 × 10 = 40.

3. Through the above reasoning, the final answer is 40.

菱形; 邻补角; 平行四边形

Definition of Rhombus

In the figure of this problem, all sides <note>LM</note>, <note>MP</note>, <note>PQ</note>, and <note>QL</note> of the quadrilateral <note>LMPQ</note> are equal, so the quadrilateral <note>LMPQ</note> is a rhombus. Additionally, the diagonals <note>LQ</note> and <note>MP</note> of the quadrilateral <note>LMPQ</note> are perpendicular bisectors of each other, meaning the diagonals <note>LQ</note> and <note>MP</note> intersect at point <note>R</note>, and <note>angle LRM is a right angle (90 degrees)</note>, and <note>LR = PR</note> and <note>MR = QR</note>.

Perimeter Formula for Rhombus

The perimeter of a rhombus is equal to four times the length of one of its sides, i.e., \(C = 4 \cdot a\).

In the figure of this problem, <note>in the rhombus LMPQ, side MP is a side of the rhombus</note>, according to the <note>perimeter of a rhombus is equal to four times the length of its side, i.e., C = 4MP</note>.


Question: Find the area of the figure. Round the result to one decimal place.

Answer: 30.2

1. The given condition is that the diameter of the circle is 6.2 cm. According to the definition of the radius, the radius is half of the diameter, which means the radius of the circle is 3.1 cm.

2. According to the formula for the area of a circle, \(A = πr^2\), where \(A\) represents the area and \(r\) represents the radius.

3. The radius of the circle \(r = 3.1 cm\). Substitute the radius into the area formula, i.e., \(A = π(3.1)^2\).

4. Calculate the square of \(3.1\), obtaining \(3.1^2 = 9.61\).

5. Substitute \(9.61\) into the area formula, obtaining \(A = π × 9.61\).

6. Estimate the value of π as \(3.14\), then \(A ≈ 3.14 × 9.61 = 30.1674\).

7. Round the calculated area to one decimal place, obtaining the approximate area of the circle as 30.2 square cm.

8. Through the above reasoning, the final answer is 30.2 square cm.

Definition of Diameter

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

In the figure of this problem, <note>the red line segment is the diameter</note>, connecting <note>the center of the circle and two points on the circumference</note>, with a length equal to 2 times the radius, that is, <note>diameter = 6.2 cm</note>.

Definition of Radius

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

<note>The center of the circle is the black dot</note>, <note>the line segment between any point on the circumference and the center of the circle is the radius</note>, and its length is half of the diameter of the circle, which is 3.1 centimeters.

Area Formula of a Circle

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

<note>The diameter of the circle is 6.2 cm</note>, according to the properties of the circle, <note>the radius is half of the diameter, so the radius r of the circle is 3.1 cm</note>. According to the area formula of a circle, <note>the area A of the circle is equal to π times the square of the radius 3.1 cm, that is \$A = π(3.1)^2\$</note>, that is, <note>\$A = π × 9.61\$</note>, substituting the approximate value of π as 3.14, we get <note>\$A ≈ 3.14 × 9.61 = 30.1674\$ square cm</note>.


Question: m \angle 2 = 2 x, m \angle 3 = x. Find m \angle 3.

Answer: 30

1. According to the definition of a right triangle, ∠GHJ=90° in △GHJ, so △GHJ is a right triangle, and ∠2 and ∠3 are two interior angles of △GFJ. Therefore, based on the complementary property of acute angles in a right triangle, we get: ∠2+∠3=90°.

2. Given ∠2 = 2x and ∠3 = x, substituting these values we get: 2x + x = 90°.

3. By calculation, we get: 3x = 90°, solving for x gives x = 30°.

4. Therefore, ∠3 = x = 30°.

5. Through the above reasoning, the final answer is 30°.

圆周角; 直角三角形; 弧

Definition of Right Triangle

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

In <note>triangle GHJ</note>, <note>angle GHJ is a right angle (90 degrees)</note>, therefore <note>triangle GHJ is a right triangle</note>. <note>Side GH and side HJ are the legs</note>, <note>side GJ is the hypotenuse</note>.

Complementary Property of Acute Angles in Right Triangle

In a right triangle, the sum of the two acute angles, other than the right angle, is 90°.

In the <note>right triangle GHJ</note>, <note>angle GHJ is a right angle (90 degrees)</note>, <note>angle HGJ (angle 2) and angle GJH (angle 3) are the two acute angles other than the right angle</note>. According to the complementary property of acute angles in a right triangle, the sum of angle HGJ (angle 2) and angle GJH (angle 3) is 90 degrees, that is, <note>angle HGJ (angle 2) + angle GJH (angle 3) = 90°</note>.


Question: Find x.

Answer: 14

1. Given that the triangle is a right triangle, the shorter leg is 7, the angle between the hypotenuse of length x and the leg of length y is 30°, and we need to find the length of the hypotenuse x.

2. According to the definition of sine in trigonometry, the sine of an angle is equal to the ratio of the opposite side to the hypotenuse. In this problem, it is sin30°, so we have sin30° = opposite side / hypotenuse.

3. Knowing sin30° = 1/2 and using the definition of sine: sin∠ = opposite side / hypotenuse, we get 7/x = 1/2.

4. From 7/x = 1/2, we solve for x: x = 7 × 2 = 14.

5. Through the above reasoning, we finally get the answer as 14.

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 a right triangle, the angle of 30° is an acute angle</note>, <note>side 7 is the opposite side of the angle 30°</note>, <note>side x is the hypotenuse</note>. According to the definition of the sine function, <note>the sine value of the angle 30° is equal to the ratio of the opposite side 7 to the hypotenuse x</note>, that is, <note>sin(30°) = 7 / x</note>.

Definition of Right Triangle

<note>In a triangle, one of the interior angles is 90 degrees</note>, therefore the triangle is a right triangle. <note>Sides 7 and side y are the legs</note>, <note>side x is the hypotenuse</note>.


Question: Find m \angle W.

Answer: 76

1. Given that the figure is a pentagon UVWYZ, with its interior angles being ∠U, ∠V, ∠W, ∠Z, ∠Y, and these angles are expressed as (x-8)°, (3x-11)°, (x+8)°, (2x+7)° and x° respectively.

2. According to the interior angle sum theorem of polygons, the formula for the sum of the interior angles of a polygon is (n-2)×180°, where n is the number of sides. For a pentagon, n=5, so the sum of its interior angles is (5-2)×180°=540°.

3. Using the given expressions for the five interior angles, set up the equation: (x-8)° + (3x-11)° + (x+8)° + (2x+7)° + x°=540°.

4. Combine like terms to get the sum of the coefficients of x as 8x and the sum of the constant terms as -4, thus the equation becomes: 8x - 4 = 540.

5. Solve the equation 8x - 4 = 540 by first adding 4 to both sides to get: 8x = 544.

6. Then divide both sides by 8 to get: x = 68.

7. Substitute x = 68 into the expression for ∠W = (x + 8)°, to get ∠W = (68 + 8)° = 76°.

8. Through the above reasoning, the final answer is ∠W=76°.

Definition of Interior Angles of a Polygon

The interior angles of a polygon are the angles formed between each pair of adjacent sides within the polygon, specifically referring to the angles inside the polygon.

In the figure of this problem, <note>the interior angles of pentagon UVWYZ are ∠U, ∠V, ∠W, ∠Y, and ∠Z</note>, with the corresponding expressions being <note>(x-8)°</note>, <note>(3x-11)°</note>, <note>(x+8)°</note>, <note>x°</note>, and <note>(2x+7)°</note>. These interior angles are <note>the angles formed between two adjacent sides and are within pentagon UVWYZ</note>.

Polygon Interior Angle Sum Theorem

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

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

Polygon

A polygon is a closed figure in a plane formed by a finite number of line segments joined sequentially where each segment intersects exactly two other segments at its endpoints.

In the figure of this problem, <note>polygon UVWYZ is a closed figure on a plane</note>. It is formed by several line segments <note>UV, VW, WY, YZ, ZU</note> connected in sequence, and each two line segments intersect only at <note>endpoints U, V, W, Y, Z</note>. Each vertex of the polygon is a common endpoint of two line segments, and these segments form a closed path without intersections.


Question: Find x, if AE = 3, AB = 2, BC = 6, and ED = 2x - 3.

Answer: 6

1. From the figure, we can see that BE∥CD. According to the theorem of proportional segments formed by parallel lines, we get: AE/AD=AB/AC. Given AB = 2, AE=3, BC=6, ED=2x-3. Since points A, E, D are on the same line, AD=AE+DE=3+2x-3=2x; similarly, since points A, B, C are on the same line, AC=AB+BC=2+6=8.

2. Substitute into the proportion: 3/2x = 2/8.

3. Simplify to get: 3/2x = 1/4.

4. Cross-multiply this proportion to get 2x = 3*4.

5. Divide both sides by 2 to get x = 6.

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

线段; 普通三角形; 内错角

Proportional Segments Theorem

If a line is parallel to one side of a triangle and intersects the other two sides or their extensions, then it divides the segments it intersects proportionally in relation to the corresponding segments of the original triangle.

In <note>triangle ADC</note>, <note>line EB is parallel to side DC</note>, and it intersects the other two sides <note>AD and AC (or their extensions) at points E and B</note>. Then, according to the <note>Proportional Segments Theorem</note>, we have: <note>segment AE/segment AD = segment AB/segment AC</note>. That is, the intercepted segments are proportional to the corresponding segments of the original triangle.


Question: Use rectangle L M N P and parallelogram L K M J to solve the problem. If L N = 10, L J = 2 x + 1, P J = 3 x - 1, find x.

Answer: 2

1. Given rectangle L M N P, ##according to the properties of the diagonals of a rectangle, the diagonals are equal and bisect each other, so LN = PM, and the intersection point J of diagonals LN and PM bisects the diagonals LN and PM, i.e., LJ = NJ = 1/2 * LN##.

2. ##Given LN = 10, LJ = 2x + 1, substitute the given value: 2x + 1 = 1/2 * 10##.

3. ##Solve the equation: 2x + 1 = 5, finally solve: x = 2##.

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##4##. Through the above reasoning, the final answer is x = 2.

矩形; 平行四边形; 线段

Definition of Rectangle

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

<note>Quadrilateral L M N P is a rectangle</note>, with its interior angles <note>∠LMN</note>, <note>∠MNP</note>, <note>∠NPL</note>, <note>∠PLM</note> all being right angles (90 degrees), and <note>side LM is parallel and equal in length to side NP</note>, <note>side MN is parallel and equal in length to side LP</note>.

Property of Diagonals in a Rectangle

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

In this problem, in rectangle L M N P, <note>side L M is parallel and equal to side P N</note>, <note>side L P is parallel and equal to side M N</note>. <note>Diagonals L N and P M are equal and bisect each other</note>, that is, <note>the intersection point J of diagonals L N and P M is the midpoint of both diagonals</note>. Therefore, <note>segment L J is equal to segment J N</note>, <note>segment P J is equal to segment J M</note>.


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

Answer: 169.7

1. Given that the base length of the parallelogram is 14 mm, ##let the vertices of the parallelogram (starting from the top left point in a clockwise direction) be labeled as ABCD. According to the properties of parallelograms, the lengths of the top and bottom sides are both 7 mm. Draw the height h from the top right vertex to the base.##

2. Based on the information in the figure, it can be seen that the angle between ##side BC## and ##base CD## is 60°. From ##vertex B, draw a perpendicular line to the base CD, and let the foot of the perpendicular be point E. This perpendicular divides the base into CE = 7 mm and DE = 14 - 7 = 7 mm##.

3. ##According to the definition of a right triangle, the right triangle BCE is formed##, and we can use the ##tangent function to calculate the height h##.

4. ##Here, tan(60°) = BE/CE = h/7 = √3, therefore h = 7 * √3.##

5. ##Simplifying, the height h = 7 * 1.732 = 12.12 mm##.

6. The problem requires finding the area of the parallelogram. The area formula for a parallelogram is 'base * height', where the base is 14 mm and the height is 12.12 mm. Therefore, the area is 14 * 12.12.

7. Calculating the area, 14 * 12.12 = ##169.68 mm^2##, and rounding to one decimal place, it becomes 169.7.

8. Based on the above discussion, we arrive at the final answer: The approximate area of the parallelogram is ##169.7 mm^2##.

平行四边形; 垂线; 正弦

Definition of Parallelogram

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

<note>In the figure of this problem, the two pairs of opposite sides of this shape are parallel and equal</note>, the length of the base is 14 mm, and <note>the top side is parallel to the base</note>.

Properties of Parallelogram Theorem

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

In the parallelogram ABCD, the angles BAD and BCD are equal, the angles ABC and ADC are equal; the sides AB and CD are equal, the sides AD and BC are equal; the diagonals AC and BD bisect each other, that is, the intersection point divides the diagonal AC into two equal segments, and divides the diagonal BD into two equal segments.

Definition of Perpendicular Lines

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

Vertex <note>B</note> draws a perpendicular line to <note>base CD</note>, forming a <note>right angle</note> with the base. Therefore, according to the definition of perpendicular lines, this <note>perpendicular line is perpendicular to the base</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 figure of this problem, the line segment BE from <note>vertex B perpendicular to the opposite side CD (or its extension)</note> is <note>the altitude from vertex B</note>. <note>The line segment BE forms a right angle (90 degrees) with the side CD (or its extension)</note>, which indicates that <note>the line segment BE is the perpendicular distance from vertex B to the opposite side CD (or its extension)</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 14 mm</note>, <note>the corresponding height is the vertical distance from the base to the opposite side, noted as 12.12 mm</note>. Therefore, according to the area formula of a parallelogram, the area of the parallelogram is equal to <note>the length of the base 14 mm</note> multiplied by <note>the corresponding height 12.12 mm</note>, that is, A = 14 mm × 12.12 mm.

Definition of Right Triangle

Vertex <note>B</note> forms a perpendicular line with the base, creating a right triangle. One of the internal angles of this right triangle is <note>90 degrees</note>, thus it is a right triangle. The length of the perpendicular line is <note>the longer leg of the right triangle BE</note> (12.12 mm), and the length of the shorter leg of the right triangle <note>CE</note> is 7 mm.

Definition of Tangent Function

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

<note>In the figure of this problem, in the right triangle BCE, angle C is an acute angle, side BE is the opposite leg of angle C, and side CE is the adjacent leg of angle C</note>, so the tangent value of angle C is equal to the length of side BE divided by the length of side CE, <note>i.e., tan(C) = BE / CE</note>.


Question: Find \widehat W N if \triangle I W N is an equilateral triangle and W N = 5.

Answer: \frac { 5 } { 3 } \pi

1. Given that triangle I W N is an equilateral triangle and W N = 5, according to the definition of an equilateral triangle: in an equilateral triangle, all three interior angles are equal and each is 60 degrees, therefore ∠IWN=∠INW=∠WIN=60°. Conclusion: ∠WIN=60°.

2. According to the conversion formula between degrees and radians, convert the angle of arc WN to radians: 60*(π/180)=π/3. Further substituting into the arc length formula of the sector, we get the arc length L of arc WN: θr=π/3*5=5π/3.

3. Through the above reasoning, the final answer is 5π/3.

等边三角形; 圆; 中点

Definition of Equilateral Triangle

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

<note>Triangle IWN is an equilateral triangle</note>. <note>Sides IW, IN, and WN are of equal length</note>, and <note>angles IWN, WIN, and INW are equal, each measuring 60°</note>.

Formula for the Length of an Arc of a Sector

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

In <note>扇形WIN</note>, <note>圆心角WIN为π/3（以弧度制表示）</note>, <note>半径为5</note>. According to the formula for the length of an arc of a sector, the arc length L is equal to <note>圆心角θ乘以半径r</note>, that is, <note>L = θ * r = π/3 * 5</note>.

Formula for Conversion between Degrees and Radians

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

<note>In circle I, the angle of WN is 60°</note>, according to the formula for conversion between degrees and radians, the conversion to radians = degrees * (π/180), which means <note>60° * (π/180) = π/3</note>.


Question: Quadrilateral D E F G is a rectangle. If m \angle E F D = 2 x - 3 and m \angle D F G = x + 12, find m \angle E F D.

Answer: 51

1. Given that quadrilateral DEFG is a rectangle, thus ∠EFD and ∠DFG are two adjacent angles forming a right angle. According to the definition of a rectangle, ∠EFD + ∠DFG = 90°.

2. According to the conditions in the problem, ∠EFD = 2x - 3 and ∠DFG = x + 12.

3. Substituting the expressions for the angles into the equation for the right angle, we get (2x - 3) + (x + 12) = 90.

4. Expanding and simplifying the expression, we obtain 3x + 9 = 90.

5. Subtracting 9 from both sides, we get 3x = 81.

6. Dividing both sides by 3, we get x = 27.

7. Substituting x = 27 into ∠EFD = 2x - 3, we get ∠EFD = 2(27) - 3 = 54 - 3 = 51.

8. Through the above reasoning, the final answer is ∠EFD = 51°.

矩形; 邻补角; 对顶角

Definition of Rectangle

In the figure of this problem, the quadrilateral DEFG is a rectangle, whose <note>interior angles ∠DEF, ∠EFG, ∠FGD, and ∠GDE are all right angles (90 degrees)</note>, and <note>side DE is parallel and equal in length to side FG</note>, <note>side DG is parallel and equal in length to side EF</note>.


Question: Find the area of the shaded region. Keep the result to one decimal place.

Answer: 10.7

1. Given that the length of the rectangle is 10 cm, ##let the center of the left circle be O1 and the center of the right circle be O2, the four vertices of the rectangle be A, B, C, and D, and it is also known that the diameter of the circle is equal to the width of the rectangle##, which is also 5 cm, so the rectangle contains two circles with a diameter of 5 cm.

2. The diameter of the circle is 5 cm, therefore the radius is 2.5 cm. Using the ##formula for the area of a circle,## A = π##r^2##, the area of the circle is π*(2.5)^2 = 6.25π square cm.

3. Since there are two identical circles, their total area is 2*6.25π = 12.5π square cm.

4. ##According to the formula for the area of a rectangle,## the area of the rectangle is length times width, which is 10 cm * 5 cm = 50 square cm.

5. The area of the shaded region is equal to the area of the rectangle minus the area of the two circles, which is 50 - 12.5π.

6. Using an approximate value, taking π ≈ 3.1416, we get 12.5π ≈ 12.5 * 3.1416 = 39.27.

7. Calculate the area of the shaded region: 50 - 39.27 ##=## 10.73.

8. The result rounded to one decimal place is 10.7.

Definition of Circle

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

<note>The center of the left circle is O1</note>, <note>The center of the right circle is O2</note>, both with a radius of 2.5 cm. In the figure, all <note>points that are 2.5 cm away from O1 are on the left circle</note>, and all <note>points that are 2.5 cm away from O2 are on the right circle</note>.

Definition of Diameter

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

<note>The diameter of both circles is 5 centimeters</note>, connecting <note>the center of the circle and two points on the circumference</note>, the length is <note>2 times the radius</note>, that is, <note>diameter = 2 * radius = 5 centimeters</note>.

Definition of Radius

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

In the diagram of this problem, <note>each circle has a radius of 2.5 centimeters</note>. This means that the distance from <note>the center of each circle to any point on the circumference is 2.5 centimeters</note>.

Formula for the Area of a Rectangle

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

<note>The length of the rectangle is 10 cm</note>, <note>the width is 5 cm</note>, so <note>the area of the rectangle = 10 cm * 5 cm</note> = 50 square cm.

Area Formula of a Circle

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

<note>Each circle has a radius of 2.5 cm</note>. According to the area formula of a circle, <note>the area A of a circle is equal to π times the square of the radius 2.5</note>, that is, <note>A = π * (2.5)^2 = 6.25π square cm</note>.

Definition of Rectangle

<note>Quadrilateral ABCD is a rectangle</note>, <note>its interior angles ∠ABC, ∠BCD, ∠CDA, ∠DAB are all right angles (90 degrees)</note>, <note>side AB is parallel to side DC and they are of equal length</note>, <note>side BC is parallel to side DA and they are of equal length</note>.


Question: Find x.

Answer: 8

1. Given that one acute angle in a right triangle is 60°, according to the triangle sum theorem, the other acute angle is 30°.

2. According to the properties of a 30°-60°-90° triangle, the hypotenuse is twice the length of the shorter leg, and the ratio of the shorter leg to the longer leg is 1:√3.

3. The problem states that the longer leg is 8√3. According to the properties of a 30°-60°-90° right triangle, let the shorter leg be a, then the longer leg = a√3. Therefore, a√3 = 8√3.

4. Solving the equation a√3 = 8√3, we get a = 8.

5. Since the shorter leg is x, x equals 8.

直角三角形; 正弦; 余弦

Definition of Right Triangle

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

<note>In a triangle, one angle is a right angle (90 degrees)</note>, therefore the triangle is a right triangle. <note>Side x and side 8√3 are the legs</note>, <note>side y is the hypotenuse</note>.

Triangle Angle Sum Theorem

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

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

Properties of 30°-60°-90° Triangle

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

In the diagram of this problem, in the 30°-60°-90° triangle, the angle of 60° is 30 degrees, the angle of 30° is 60 degrees, and the right angle is 90 degrees. <note>Side y is the hypotenuse</note>, side x is the side opposite the 30-degree angle, <note>side 8√3</note> is the side opposite the 60-degree angle. According to the properties of the 30°-60°-90° triangle, side x is equal to half of <note>side y</note>, <note>side 8√3</note> is equal to x times √3. That is: <note>x = 1/2 * y</note>, <note>8√3 = x * √3</note>.


Question: Find z.

Answer: 3 \sqrt { 19 }

1. Given the lengths of two sides of a right triangle, which are 5 and 14.

2. According to the Pythagorean theorem, 14^2 = 5^2 + z^2.

4. Through the above reasoning, the final answer is z = 3√19.

直角三角形; 对顶角; 直线

Definition of Right Triangle

<note>In the figure of this problem, one interior angle of the triangle is 90 degrees</note>, so the triangle is a right triangle. <note>The sides with lengths 5 and z are the legs</note>, <note>the side with length 14 is the hypotenuse</note>.

Hypotenuse of a Right Triangle

In a right triangle, the hypotenuse is the side opposite the right angle, and it is also the longest side among the three sides of the triangle.

<note>Side 14 is the hypotenuse of right triangle ABC</note>.

Pythagorean Theorem

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

In the figure of this problem, in the right triangle, <note>side 5 and side z are the legs</note>, <note>side 14 is the hypotenuse</note>, so according to the Pythagorean Theorem, <note>14^2 = 5^2 + z^2</note>.


Question: Given: As shown in the figure, in ⊙O, diameter AB is perpendicular to chord CD at point E. If AB=10 and CD=6, then the length of BE is ()

Answer: 1

1. Given that AB is the diameter of ⊙O, AB=10, ##according to the definition of diameter##, then the radius OA=OB=10÷2=5.

2. According to the problem statement, AB is perpendicular to chord CD, with the foot of the perpendicular being E, ##according to the perpendicular bisector theorem, the diameter AB bisects chord CD##, so E is the midpoint of CD.

3. Therefore, CE=CD÷2=6÷2=3.

4. Connect ##OC, since AB is perpendicular to CD, forming a right angle at E, i.e., ∠OEC is 90 degrees, according to the definition of a right triangle, triangle △OCE is a right triangle. In the right triangle △OCE, according to the Pythagorean theorem, OE=√(OC²-CE²)=√(5²-3²)=√(25-9)=√16=4##.

5. ##Given OB=5, OE=4, and since OB=BE+OE=BE+4=5, therefore BE=1##.

6. Through the above reasoning, we finally conclude that BE=1.

圆; 垂线; 直角三角形; 弧; 弦

Definition of Diameter

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

Definition of Radius

<note>In the figure of this problem, in circle O, point O is the center of the circle, points A and B are any points on the circle, and segments OA and OB are segments from the center to any point on the circle</note>, therefore <note>segments OA and OB are the radii of the circle</note>, both with a length of 5.

Pythagorean Theorem

In the right triangle <note>△OCE</note>, <note>∠OEC is a right angle (90 degrees)</note>, <note>sides OE and CE are the legs</note>, <note>side OC is the hypotenuse</note>, so according to the Pythagorean Theorem, <note>OC² = OE² + CE²</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 <note>circle O</note>, <note>diameter AB is perpendicular to chord CD</note>, with E as the foot of the perpendicular. According to the Perpendicular Diameter Theorem, <note>diameter AB bisects chord CD</note>, so <note>CE=ED=3</note>, and <note>diameter AB bisects the arcs subtended by chord CD</note>, so <note>arc AC = arc AD</note>.

Definition of Right Triangle

In triangle <note>△OCE</note>, <note>∠OEC is a right angle (90 degrees)</note>, thus <note>triangle OCE is a right triangle</note>. Side <note>OE</note> and side <note>CE</note> are the legs, and side <note>OC</note> is the hypotenuse.


Question: m \angle 2 = 2 x, m \angle 3 = x. Find m \angle 2.

Answer: 60

1. Given conditions are ∠ 2 = 2x, ∠ 3 = x.

2. Observing the figure, it can be seen that in △GHJ, ∠GHJ is a right angle (90 degrees).

3. In triangle GHJ, according to the triangle angle sum theorem, ∠GHJ + ∠HJG + ∠JGH = 180 degrees.

4. Substituting the given ∠GHJ = 90 degrees, we get: 90° + ∠HJG + ∠JGH = 180°.

5. Therefore, according to the equation, we get: ∠HJG + ∠JGH = 180° - 90° = 90°.

6. According to the problem statement, in △GJH, ∠HJG = ∠ 3 = x, ∠JGH = ∠ 2 = 2x.

7. Substituting ∠HJG = x and ∠JGH = 2x into the equation, solving the equation: x + 2x = 90 => 3x = 90.

8. Solving for x, we get x = 30. Therefore, ∠2 = 2 * 30 = 60.

9. Through the above reasoning, the final answer is 60.

圆内接四边形; 垂线; 直角三角形; 圆周角

Definition of Triangle

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

<note>Triangle △GHJ</note> is a geometric figure formed by three non-collinear points <note>G, H, J</note> and their connecting line segments <note>GH, HJ, JG</note>. <note>Points G, H, J</note> are the three vertices of the triangle, and <note>line segments GH, HJ, JG</note> are the three sides of the triangle.

Triangle Angle Sum Theorem

<note>Angle GHJ, angle HJG, and angle JGH are the three interior angles of triangle GHJ</note>. According to the Triangle Angle Sum Theorem, <note>angle GHJ + angle HJG + angle JGH = 180°</note>.


Question: The perimeter of this polygon is 60 cm. Find the length of each side of the polygon.

Answer: 12

1. Given that the perimeter of the pentagon is 60 cm, and according to the markings on the sides in the figure, ##the five sides of the pentagon are equal, according to the definition of a regular polygon, the pentagon is a regular pentagon##.

####

##2. According to the formula for the perimeter of a regular polygon, let the length of each side of the pentagon be x, then the perimeter of the regular pentagon is 5x.##

##3. ##According to the given condition, the perimeter is 60 cm, so 5x = 60.

##4.## Solving the equation 5x = 60, we get x = 60 / 5 = 12.

##5. ##Therefore, the length of each side is 12 cm.

Definition of Regular Polygon

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

In the figure of this problem, <note>all sides of the pentagon are of equal length</note>, therefore, the pentagon is a <note>regular pentagon</note>.

Formula for the Perimeter of a Regular Polygon

The perimeter of a regular polygon is equal to the length of one side multiplied by the number of sides.

In the figure of this problem, in the <note>regular pentagon</note>, let <note>each side length be x</note>, <note>the number of sides of the regular polygon is 5</note>, so the perimeter of this regular pentagon is <note>perimeter = 5x</note>.


Question: Find Q P.

Answer: 3

1. Given that segment QN ∥ PO, in △MPO, according to the theorem of parallel lines dividing segments proportionally, we have MQ/PQ=MN/NO.

2. Substitute the given values: 5/x=6/(18/5)

3. Solve the equation to get x=3

4. Through the above reasoning, the final answer is QP = 3.

等腰三角形; 平行线; 线段

Proportional Segments Theorem

If a line is parallel to one side of a triangle and intersects the other two sides or their extensions, then it divides the segments it intersects proportionally in relation to the corresponding segments of the original triangle.

In the figure of this problem, in triangle MPO, <note>line QN is parallel to side PO</note>, and <note>it intersects the other two sides MP and MO at points Q and N</note>. Then, according to the Proportional Segments Theorem, we have:
\[
\frac{<note>segment MQ</note>}{<note>segment PQ</note>} = \frac{<note>segment MN</note>}{<note>segment NO</note>}
\]
That is, the intercepted segments are proportional to the corresponding segments of the original triangle.


Question:
y - x =
Requirements:
1. Ensure all mathematical symbols are not translated
2. Do not translate figure labels (e.g., ABCD)
3. Output format: directly return the translated English problem without any additional content (no need to start with 'Question:')

Answer: 10

1. According to the known theorem of the sum of the interior angles of a triangle, the sum of the interior angles of any triangle is 180 degrees.

2. In the upper triangle in the figure, there is one angle of 80 degrees and one angle of 50 degrees, ##the third angle is x##, calculated according to the theorem of the sum of the interior angles of a triangle: x = 180 - 80 - 50 = 50 degrees.

3. Next, look at the lower triangle in the figure, where one angle is 70 degrees, ##the other angle is 50 degrees according to the definition of vertical angles, the third angle is y°##, so it can be calculated by the theorem of the sum of the interior angles of a triangle: y = 180 - 70 - 50 = 60 degrees.

4. The next step is to calculate y - x, through the upper ##and## lower triangles known x = 50 and y = 60, calculate the equation: y - x = 60 - 50 = 10 degrees.

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

Triangle Angle Sum Theorem

<note>In the upper triangle, there is one angle of 80 degrees and one angle of 50 degrees</note>, let the <note>third angle be x</note>, it can be calculated using the Triangle Angle Sum Theorem: <note>x = 180 - 80 - 50 = 50 degrees</note>. Similarly, <note>in the lower triangle, there is one angle of 70 degrees</note>, <note>another angle is 50 degrees according to the definition of vertical angles</note>, <note>the third angle is y</note>, thus it can be calculated using the Triangle Angle Sum Theorem: <note>y = 180 - 70 - 50 = 60 degrees</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 intersect at the center point</note>. According to the definition of vertical angles, their angles are equal. In the previous triangle, <note>one angle is 50 degrees</note>, so in the next triangle, <note>the opposite angle is also 50 degrees</note>.


Question: Find the area of \parallelogram J K L M.

Answer: 24

1. According to the problem statement, ##the given parallelogram JKLM has a base length of ML=4 cm##. Let the height be JJ', i.e., JJ'=6 cm.

2. It is known that the area formula for a parallelogram is: base length multiplied by height, i.e., Area = base * height.

3. From the given conditions, the length of ##base ML## is 4 cm and the length of height JJ' is 6 cm. Substituting these values into the formula, we get: Area = 4 cm * 6 cm.

4. After calculation, the area of parallelogram JKLM is found to be 24 square centimeters.

平行四边形; 垂线; 线段

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

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.

In the figure of this problem, in the parallelogram JKLM, <note>side JK and side ML are parallel</note>. <note>The perpendicular distance from vertex J to the opposite side ML is the height of the parallelogram</note>. Specifically, from <note>vertex J, draw a line segment perpendicular to side ML, the length of this line segment is the height of the 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 \( A = \text{base length} \times \text{height} \).

In the figure of this problem, in <note>parallelogram JKLM</note>, <note>side ML is the base</note>, and the corresponding height is <note>the perpendicular distance from the base ML to the opposite side JK, denoted as JJ'</note>. Therefore, according to the area formula of a parallelogram, the area of parallelogram JKLM is equal to <note>the length of the base ML multiplied by the corresponding height JJ'</note>, that is, <note>A = ML × JJ'</note>.


Question: J K L M is a rectangle. M L P R is a rhombus. \\angle J M K \\cong \\angle R M P, m \\angle J M K = 55, and m \\angle M R P = 70. Find m \\angle K M L.

Answer: 35

1. Given that JKLM is a rectangle, according to the definition of a rectangle, ∠JML is a right angle, i.e., ∠JML = 90°.

2. According to the problem, ∠JMK = 55°.

3. Therefore, ∠KML = ∠JML - ∠JMK = 90° - 55° = 35°.

Definition of Rectangle

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

<note>Quadrilateral JKLM is a rectangle</note>, with <note>interior angles ∠JML, ∠MLK, ∠LKJ, ∠KJM all being right angles (90 degrees)</note>, and <note>sides JK and ML are parallel and equal in length</note>, <note>sides JM and KL are parallel and equal in length</note>.


Question: Rectangle ABCD shown in the figure above has width w and length l that is twice the width. If w = 2, what is the area of the rectangle?

Answer: 8

1. Given that the width of rectangle ABCD is w = 2, according to the definition of a rectangle, the opposite sides are equal and parallel, thus the width ##AB## = ##CD## = w = 2.

2. The problem states that the length l of the rectangle is twice the width w, therefore, according to the given condition, ##BC## = ##AD## = l = 2 * w = 2 * 2 = 4.

##3. The area of rectangle ABCD is calculated using the formula length times width, i.e., S_rectangle_ABCD = AB * BC.##

4. Substituting the specific values, we get ##S_rectangle_ABCD = AB * BC = 4 * 2 = 8##.

5. After the above calculation process, the final area of the rectangle is 8.

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

Formula for the Area of a Rectangle

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

<note>Side AD</note> and <note>Side AB</note> are the length and width of the rectangle, so the area of the rectangle = AB * <note>Side AC</note>.


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

Answer: 31

1. Given that the quadrilateral is a parallelogram, according to the properties of a parallelogram, the opposite angles in a parallelogram are equal. Therefore, angle (6y - 57)° = angle (3y + 36)°.

2. Solving the equation, we get: 3y = 93.

4. Based on the calculation result, we conclude that y = 31.

平行四边形; 同旁内角; 邻补角; 平行线

Definition of Parallelogram

<note>In the figure of this problem, the quadrilateral is a parallelogram</note>, <note>the top side is parallel and equal to the bottom side</note>, <note>the left side is parallel and equal to the right side</note>.

Properties of Parallelogram Theorem

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

In a <note>parallelogram</note>, the opposite angles <note>(6y - 57)° and (3y + 36)° are equal.</note>


Question: In the figure, m \angle 3 = 43. Find the measure of \angle 11.

Answer: 43

1. Given that ∠3 = 43°. According to the information in the figure, since line c ∥ d and b is the transversal, ∠3 and ∠11 are corresponding angles (based on the definition of corresponding angles). According to the parallel postulate 2, ∠3 = ∠11 (corresponding angles are equal).

2. Therefore, ∠11 = 43°.

3. Through the above reasoning, the final answer is 43°.

Definition of Corresponding Angles

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

In the figure of this problem, <note>two parallel lines c and d are intersected by a line b</note>, where <note>angle 3 and angle 11 are on the same side of the intersecting line b and on the same side of the intersected lines c and d</note>, therefore <note>angle 3 and angle 11 are corresponding angles</note>. Corresponding angles are equal, that is, <note>angle 3 is equal to angle 11</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.

Original text: <note>Two parallel lines c and d are intersected by a third line b</note>, forming the following geometric relationship: <note>Corresponding angles: angle 3 and angle 11 are equal</note>. These relationships indicate that when two parallel lines are intersected by a third line, corresponding angles are equal, alternate interior angles are equal, and same-side interior angles are supplementary.


Question: As shown in the figure, one of the angles between the two diagonals of the rectangle is 60°, AC + BD = 20cm, then the length of AB is ()

Answer: 5cm

1. Given that quadrilateral ABCD is a rectangle, according to the property of diagonals of a rectangle, we have AC=BD, and for the intersection point O of the diagonals, we have OA=OC=OD=OB.

2. From the problem statement, we get AC+BD=20. Since AC=BD, we have 2AC=20, thus AC=10. Similarly, BD also equals 10.

3. Because the relationship between OA and AC is OA=AC/2, we have OA=10/2=5.

4. Similarly, the relationship between OB and BD is OB=BD/2, thus we also get OB=10/2=5.

5. Given that angle AOB equals 60 degrees, and OA=5, OB=5.

6. According to the definition of an isosceles triangle, OA=OB=5, so △AOB is an isosceles triangle. Given that one interior angle is 60° in an isosceles triangle, it is an equilateral triangle. Since ∠AOB=60 degrees, △AOB is an equilateral triangle.

7. According to the property of an equilateral triangle, all sides of △AOB are equal, thus AB=OA=5.

矩形; 线段; 直角三角形; 余弦; 平移

Definition of Rectangle

In the figure of this problem, <note>quadrilateral ABCD is a rectangle</note>, its interior angles <note>∠DAB</note>, <note>∠ABC</note>, <note>∠BCD</note>, <note>∠CDA</note> are all <note>right angles (90 degrees)</note>, and <note>side AB is parallel and equal in length to side CD</note>, <note>side AD is parallel and equal in length to side BC</note>.

Equilateral Triangle Identification Theorem (60-Degree Angle in an Isosceles Triangle)

An isosceles triangle with one interior angle measuring 60 degrees is an equilateral triangle.

<note>△AOB is an isosceles triangle</note>, <note>OA=OB</note>, and there is an internal angle of 60°, i.e., <note>∠AOB=60°</note>. According to the Equilateral Triangle Identification Theorem, if an isosceles triangle has an internal angle of 60°, then the lengths of its three sides are equal, and all three internal angles are 60°. Therefore, it can be determined that <note>△AOB is an equilateral triangle</note>.

Definition of Equilateral Triangle

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

<note>Triangle AOB is an equilateral triangle</note>. <note>Sides OA, OB, and AB are of equal length</note>, and <note>angles AOB, OAB, and OBA are equal in measure, each being 60°</note>.

Property of Diagonals in a Rectangle

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

In the rectangle <note>ABCD</note>, the sides <note>AB and CD are parallel and equal</note>, and the sides <note>AD and BC are parallel and equal</note>. The diagonals <note>AC and BD are equal and bisect each other</note>, that is, the intersection point <note>O of diagonals AC and BD is the midpoint of both diagonals</note>. Therefore, <note>segment OA is equal to segment OC</note>, and <note>segment OB is equal to segment OD</note>.

Definition of Isosceles Triangle

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

In the original text: <note>In triangle AOB, OA=OB</note>, therefore <note>triangle AOB is an isosceles triangle</note>.


Question: Equilateral pentagon P Q R S T is inscribed in \odot U. Find m \widehat Q R.

Answer: 72

1. Given the condition: the equilateral pentagon PQRST is inscribed in the circle with center at point U. According to the definition of an equilateral pentagon, the sides of the pentagon are equal, i.e., PQ = QR = RS = ST = TP.

2. The sum of the interior angles of the pentagon is (n - 2)180 degrees, where n is the number of sides. Therefore, for a pentagon, the sum of the interior angles is (5 - 2)180 = 540 degrees.

3. Since the pentagon PQRST is an equilateral pentagon, each of its interior angles is equal. Therefore, each interior angle is 540 / 5 = 108 degrees.

4. According to the property of the central angle, the degree of the central angle is equal to the degree of the arc it subtends. Then, according to the inscribed angle theorem, the degree of the inscribed angle is half the degree of the central angle it subtends. We can derive the formula: arc PS = central angle corresponding to PS = 2 * ∠PTS = 2 * 108° = 216°.

5. Since PQRST is a regular pentagon, the arc PS is equally divided by points PQRS into 3 arcs PQ, arc QR, arc RS. We derive the formula arc QR = 216° / 3 = 72°.

6. Therefore, arc QR is equal to 72 degrees.

7. After the above reasoning, the final answer is 72 degrees.

正多边形; 圆; 圆周角

Polygon Interior Angle Sum Theorem

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

<note>PQRST is a polygon with 5 sides, where 5 represents the number of sides of the polygon</note>. According to the Polygon Interior Angle Sum Theorem, the sum of the interior angles of this polygon is equal to <note>(5-2) × 180°</note> = <note>540°</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, <note>with point U as the center of the circle</note>, the vertex T of angle PTS is <note>on the circumference of the circle</note>, and the two sides of angle PTS <note>intersect the circle U at points P and S</note>. Therefore, <note>angle PTS</note> is an inscribed angle.

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 <note>P and S</note> on the circle with <note>center at point U</note>, and <note>arc PS</note> is a segment of the curve connecting these two points. According to the definition of an arc, <note>arc PS is a segment of the curve between two points P and S on the circle</note>. In the figure of this problem, there are two points <note>P and Q</note> on the circle with <note>center at point U</note>, and <note>arc PQ</note> is a segment of the curve connecting these two points. According to the definition of an arc, <note>arc PQ is a segment of the curve between two points P and Q on the circle</note>. In the figure of this problem, there are two points <note>Q and R</note> on the circle with <note>center at point U</note>, and <note>arc QR</note> is a segment of the curve connecting these two points. According to the definition of an arc, <note>arc QR is a segment of the curve between two points Q and R on the circle</note>. In the figure of this problem, there are two points <note>R and S</note> on the circle with <note>center at point U</note>, and <note>arc RS</note> is a segment of

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 <note>circle with center U</note>, <note>points P and S are two points on the circle</note>, <note>the center of the circle is point U</note>. The angle ∠PUS formed by the lines PU and US is called the central angle.

Properties of Central Angles

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

<note>In the figure of this problem, in circle U, points P, T, and S are on the circle, the central angle corresponding to the major arc PS is ∠PUS, and the inscribed angle is ∠PTS.</note> According to the inscribed angle theorem, <note>∠PTS is equal to half of the central angle ∠PUS corresponding to the arc PS, that is, ∠PTS = 1/2 ∠PUS.</note>


Question: As shown in the figure, in the square ABCD, E is a point on DC, F is a point on the extension of BC, ∠BEC=70°, and △BCE≌△DCF. Connect EF, then the measure of ∠EFD is ()

Answer: 25°

1. Given that quadrilateral ABCD is a square, according to the properties of a square, we obtain ∠BCE=90° and ∠DCF=90°.

2. From the given conditions, ∠BEC=70° and △BCE≌△DCF, according to ##the definition of congruent triangles, two triangles are congruent if their corresponding sides are equal and their corresponding angles are equal.## Thus, we have ∠DFC=70°.

3. Auxiliary line: connect EF.

4. ##From the given conditions, ∠BEC=70° and △BCE≌△DCF, according to the properties of congruent triangles, corresponding sides are equal and corresponding angles are equal,## therefore CE=CF.

####

##5. In the isosceles right triangle △ECF, since its two base angles are equal, we can deduce that ∠EFC=45°.##

##6##. ∠EFD is obtained by the difference between ∠DFC and ∠EFC, ∠EFD=∠DFC-∠EFC=70°-45°=25°.

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

正方形; 等腰三角形; 直角三角形; 邻补角; 三角形的外角

Definition of Square

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

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

Definition of Congruent Triangles

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

In the diagram of this problem, <note>△BCE≌△DCF</note>, <note>the corresponding sides and angles of △BCD are equal to those of △DCF</note>, that is: <note>side BC = side DC</note>, <note>side BE = side DF</note>, <note>side CE = side CF</note>, and <note>∠BCE = ∠DCF</note>, <note>∠BEC = ∠DFC</note>, <note>∠CBE = ∠CDF</note>.

Definition of Isosceles Right Triangle

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

<note>Triangle ECF is an isosceles right triangle</note>, in which <note>∠ECF is a right angle (90 degrees)</note>, <note>sides CE and CF are equal right-angled sides</note>.

Triangle Angle Sum Theorem

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

<note>Angles ECF, CEF, and EFC</note> are the three interior angles of triangle ECF. According to the Triangle Angle Sum Theorem, <note>angle ECF + angle CEF + angle EFC = 180°</note>.

Properties of Isosceles Triangle

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

<note>In the isosceles triangle ECF, sides CE and CF are equal</note>. Therefore, according to the properties of an isosceles triangle, <note>the angles opposite the equal sides are equal</note>, that is, <note>angle ECF = angle EFC</note>.


Question: Use a Pythagorean triple to find x.

Answer: 70

1. From the given figure, it is known that one angle in the triangle is marked with a right angle sign. According to the definition of a right triangle, this triangle is a right triangle. Let this triangle be ABC, where ∠ACB is the right angle, the length of the shorter leg BC is 24, and the length of the hypotenuse AB is 74.

2. According to the Pythagorean theorem, which states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the two legs, i.e., c^2 = a^2 + b^2. In this problem, it is known that a = 24, b = x, and c = 74.

3. Substitute a = 24 and c = 74 into the Pythagorean theorem formula, resulting in 74^2 = 24^2 + x^2.

4. Calculate 74^2 = 5476 and 24^2 = 576.

5. Substitute the values into the equation, 5476 = 576 + x^2, thus x^2 = 5476 - 576.

6. Calculate x^2 = 4900.

7. Take the square root of x^2 to get x = √4900.

8. Calculate √4900 to get x = 70.

9. Based on the above calculations, we have determined that the value of x is 70.

直角三角形; 线段; 直线

Definition of Triangle

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

<note>Triangle ABC is formed by three non-collinear points A, B, C</note> and their <note>connecting line segments AB, BC, CA</note> to create a geometric figure. <note>Points A, B, C are the three vertices of the triangle</note>, <note>line segments AB, BC, CA are the three sides of the triangle</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, let triangle ABC, where <note>∠ACB is a right angle (90 degrees)</note>, therefore <note>triangle ABC is a right triangle</note>. <note>Side AC and side BC are the legs</note>, <note>side AB 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 a right triangle, the legs are 24 and x, and the hypotenuse is 74</note>, so according to the Pythagorean Theorem, <note>74^2 = 24^2 + x^2</note>.


Question: Express the ratio of \\sin P as a decimal, accurate to two decimal places.

Answer: 0.88

1. Given ∠QRP=90°, according to the definition of a right triangle, △QRP is a right triangle.

2. From the figure, it can be seen that the hypotenuse of ΔPQR is PQ, with a length of 17; the legs RP are 8 and RQ are 15. Based on this, we can confirm that the markings in the figure are correct.

3. The required sin∠P is actually the sine value of ∠P in the right triangle ΔPQR. According to the definition, the sine value is the ratio of the opposite side to the hypotenuse, which is RQ/PQ.

4. Based on the previous analysis, RQ is 15 and PQ is 17. Therefore, sin∠P = RQ/PQ = 15/17.

5. Converting the fraction 15/17 to a decimal, we get 0.882352941.

6. Rounding the decimal result to two decimal places, we get 0.88.

7. Through the above reasoning, the final answer is 0.88.

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 the <note>right triangle PQR</note>, <note>angle ∠P is an acute angle</note>, <note>side RQ is the opposite side of angle ∠P</note>, and <note>side PQ is the hypotenuse</note>. According to the definition of the sine function, the <note>sine value of angle ∠P</note> is equal to the <note>ratio of the opposite side RQ to the hypotenuse PQ</note>, that is, <note>sin(∠P) = RQ / PQ = 15 / 17</note>.

Definition of Right Triangle

In the figure of this problem, in triangle PQR, <note>angle QRP is a right angle (90 degrees)</note>, therefore <note>triangle PQR is a right triangle</note>. <note>Side PR and side RQ are the legs</note>, <note>side PQ is the hypotenuse</note>.


Question: Find x. Round the angle value to the nearest degree.

Answer: 98

1. Given that this is a triangle containing angles and side lengths, and based on the figure without a marked right angle, it can be concluded that the triangle is an arbitrary angle triangle. ##Let the angle between the sides with lengths 29 and 61 be ∠B, the angle corresponding to x° be ∠C, and the third angle be ∠A. ∠A corresponds to side a with length 29, ∠B corresponds to side b with length 50, and ∠C corresponds to side c with length 61##.

2. According to the cosine theorem, c^2 = a^2 + b^2 - 2ab*cos(γ), where a, b, and c are the three sides of the triangle, and γ is the angle between sides a and b.

3. For the given side lengths in the triangle, 61 (c), 50 (b), and 29 (a), use the cosine theorem to find the cosine value corresponding to angle x: 61^2 = 29^2 + 50^2 - 2 * 29 * 50 * cos(x).

4. Calculate 61^2 = 3721, 29^2 = 841, 50^2 = 2500.

5. Substitute all calculations into the formula, yielding: 3721 = 841 + 2500 - 2900 * cos(x).

6. Simplify the calculation: 3721 = 3341 - 2900 * cos(##x°##), thus obtaining 2900 * cos(##x°##) = 3341 - 3721 = -380.

7. Based on the above result, find cos(##x°##) = -380 / 2900.

8. Calculate cos(x) to get: cos(##x°##) ≈ -0.13103448.

9. Using a trigonometric table or a calculator to find the inverse cosine value, we get ##x°## ≈ 97.546 degrees.

10. Round the obtained angle to the nearest integer degree, so ##x°## ≈ 98 degrees.

11. Through the above reasoning, the final answer is that ##x°## is approximately 98 degrees.

直角三角形; 正弦; 余弦; 正切

Cosine Theorem

The cosine function is a trigonometric function defined in terms of 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.

<note>In the figure of this problem, the three sides of the triangle are 29, 50, and 61 respectively</note>, <note>angle x is the angle between side 29 and side 50</note>. According to the Cosine Theorem, <note>the square of side 61 is equal to the sum of the squares of the other two sides 29 and 50 minus the product of these two sides and the cosine of the angle x between them multiplied by two</note>, that is, <note>61^2 = 29^2 + 50^2 - 2 * 29 * 50 * cos(x)</note>.

Definition of Triangle

In the diagram of this problem, <note>triangle ABC</note> is a geometric figure formed by <note>three non-collinear points A, B, C</note> and their connecting <note>line segments AB, BC, CA</note>. <note>Points A, B, C</note> are the three vertices of the triangle, and <note>line segments AB, BC, CA</note> are the three sides of the triangle.


Question: A rectangle is inscribed in a circle. Find the exact circumference of the circle.

Answer: 10 \pi

1. Given a rectangle inscribed in a circle, ##according to the inscribed quadrilateral, the four vertices of the rectangle are on the circle. Let the rectangle be rectangle ABCD (from the leftmost point clockwise), and the center of the circle be point O. According to the definition of the diameter, the diameter is a line segment passing through the center of the circle with both ends on the circle, and according to the properties of the diagonals of the rectangle, the diagonals are equal and bisect each other in the rectangle, so the distances from the intersection point of the diagonals to the four vertices are equal. Therefore, the intersection point of the diagonals of the rectangle is the center of the circle. According to (corollary 2 of the inscribed angle theorem), the inscribed angle subtended by the diameter is a right angle, it can be concluded that:## the diagonals of the rectangle are the diameters of the circle.

2. The length of the rectangle is 8 inches, and the width is 6 inches. ##According to the definition of a rectangle and the definition of a right triangle##, the diagonal can be considered as the hypotenuse of a right triangle, with the two legs being 8 inches and 6 inches respectively.

3. By ##Pythagorean theorem: In the right triangle ABD##, the square of the hypotenuse is equal to the sum of the squares of the two legs. The Pythagorean theorem is expressed as: ##BD^2 = AB^2 + AD^2##.

4. In this problem, ##AB = 8 inches, AD = 6 inches##, so: ##BD^2## = 8^2 + 6^2.

5. According to the calculation: ##BD^2## = 64 + 36 = 100, i.e., ##hypotenuse BD## = √100 = 10 inches.

6. Therefore, the diagonal of the rectangle is 10 inches, which is the diameter of the circle.

7. ##According to the formula for the circumference of a circle##, by the circumference formula C = πd (where d is the diameter).

8. So the circumference of the circle in this problem is: C = π × 10 = 10π inches.

9. After the above reasoning, the final answer is: 10π inches.

圆; 矩形; 线段; 弧

Definition of Rectangle

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

<note>The length of the rectangle is 8 inches</note>, <note>the width is 6 inches</note>, so it is a quadrilateral with <note>opposite sides equal</note> and <note>internal angles are right angles</note>. The rectangle's <note>opposite sides are 8 inches and 6 inches respectively</note>, and <note>each internal angle is a right angle (90 degrees)</note>.

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

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

<note>In the diagram of this problem, in circle O, the inscribed angles ABC and ADC subtended by the diameter AC are both right angles (90 degrees)</note>. (Or <note>in the diagram of this problem, the inscribed angles ABC and ADC are both 90 degrees, so the chord AC they subtend is a diameter</note>.) <note>In circle O, the inscribed angles BAD and BCD subtended by the diameter BD are both right angles (90 degrees)</note>. (Or <note>in the diagram of this problem, the inscribed angles BAD and BCD are both 90 degrees, so the chord BD they subtend is a diameter</note>.)

Definition of Circle

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

<note>In the figure of this problem, in circle O, the radius is d/2=10/2=5</note>. In the figure, <note>all points that are at a distance of 5 from point O are on circle O</note>.

Cyclic Quadrilateral

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

In the geometric figure provided in the problem, the four vertices of the rectangle are all on the same circle, <note>this circle is called the circumscribed circle of the rectangle</note>. Therefore, the rectangle is a cyclic quadrilateral. According to the properties of cyclic quadrilaterals, <note>it can be concluded that the sum of the opposite angles is equal to 180 degrees</note>.

Definition of Diameter

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

<note>The diameter is the diagonal of the rectangle</note>, because <note>the rectangle is inscribed in the circle</note>, in which <note>the diagonal passes through the center of the circle and connects two opposite vertices on the circumference</note>. <note>The length of the rectangle's diagonal is 10 inches</note>, that is, <note>the diameter of the circle is 10 inches</note>.

Definition of Right Triangle

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 AB and side AD are the legs</note>, <note>side BD is the hypotenuse</note>.

Pythagorean Theorem

<note>The diagonal of the rectangle</note> divides the rectangle into two right-angled triangles, where the rectangle's <note>length is 8 inches and width is 6 inches</note>. According to the Pythagorean Theorem, the square of the <note>hypotenuse (i.e., the diagonal)</note> is equal to the sum of the squares of the two right-angled sides, i.e., <note>BD^2 = 8^2 + 6^2</note> = 64 + 36 = 100, therefore the <note>hypotenuse BD</note> = √100 = 10 inches.

Circumference Formula of Circle

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

<note>A rectangle is inscribed in a circle, and the diagonal of the rectangle is the diameter of the circle</note>. The length of the rectangle is 8 inches, the width is 6 inches, <note>the diagonal is 10 inches, which means the diameter of the circle is 10 inches</note>. According to the circumference formula of a circle, the circumference C is equal to π times the diameter d, <note>that is, C=πd</note>. Therefore, <note>the circumference of the circle is 10π inches</note>.

Property of Diagonals in a Rectangle

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

In the <note>rectangle ABCD</note>, sides <note>AB and CD are parallel and equal</note>, and sides <note>AD and BC are parallel and equal</note>. The <note>diagonals AC and BD are equal and bisect each other</note>, meaning the intersection point <note>O of diagonals AC and BD is the midpoint of both diagonals</note>. Therefore, segment <note>OA is equal to segment OC</note>, and segment <note>OB is equal to segment OD</note>.


Question: Find the area of the shaded region. Round to the nearest tenth.

Answer: 58.9

1. According to the figure, three small semicircles are arranged in a straight line, and their diameters sum to 15. Assuming the diameter of each small semicircle is d, then 3d = 15, solving for d = 5.

2. Based on the given conditions, the diameter of each small semicircle is 5, so the radius of each small semicircle r = 2.5.

3. Thus, the area of each small semicircle A = π * ##r?## / 2, where r = 2.5, so A = π * ##(2.5)?## / 2.

4. Calculating, the area of each small semicircle A = (π * 6.25) / 2 ≈ 9.821.

5. The total area of the three small semicircles is 3 * A ≈ 3 * 9.821 = 29.463.

6. According to the figure, the diameter of the large semicircle is also 15, so the radius of the large semicircle R = 7.5.

7. Using the geometric formula, the area of the large semicircle is ##B = π * R? / 2, where R = 7.5, so B = π * (7.5)? / 2##.

8. Calculating, the area of the large semicircle ##B = (π * 56.25) / 2 ≈ 88.358##.

9. Finally, the area of the shaded region is the area of the large semicircle minus the total area of the three small semicircles, i.e., the area of the shaded region = ##B## - 3 * A ≈ 88.358 - 29.463 = 58.895.

10. Rounding to the nearest tenth, the area of the shaded region is 58.9.

Definition of Semicircle

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

<note>Each small semicircle is composed of a diameter (d=5) and its corresponding arc</note>; <note>The large semicircle is composed of a diameter (15) and its corresponding arc</note>.

Definition of Radius

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

<note>The radius of the small semicircle is 2.5</note>, which means the length of the line segment from the center of the small semicircle to any point on the circle is 2.5; <note>The radius of the large semicircle is 7.5</note>, which means the length of the line segment from the center of the large semicircle to any point on the circle is 7.5.

Definition of Diameter

The diameter of the small semicircle is 5, the diameter of the large semicircle is 15. The diameter of the small semicircle connects the center and two points on the circumference, with a length of 2 times the radius, that is, the diameter of the small semicircle is 2 * 2.5 = 5. The diameter of the large semicircle connects the center and two points on the circumference, with a length of 2 times the radius, that is, the diameter of the large semicircle is 2 * 7.5 = 15.

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>In the diagram of this problem, the radii of the three small semicircles and one large semicircle are 2.5 and 7.5 respectively</note>. According to the area formula of a circle, the area A of a circle is equal to the product of pi (π) and the square of the radius. <note>The area of each small semicircle is A = π * ##(2.5) ?## / 2</note>, <note>the area of the large semicircle is ##B = π * (7.5) ?## / 2</note>.


Question: Find x.

Answer: 90

1. Given four angles as shown in the diagram, according to the given conditions, the four angles are equal and each is x°.

2. ##According to the definition of vertical angles##, vertical angles formed by two intersecting lines are equal. In this problem, since the given angles are all x°, this is verified.

3. According to the ##definition of adjacent supplementary angles##, the sum of the ##adjacent supplementary angles## formed by a straight line is 180°.

####

##4. Then the adjacent two angles form: ##x° + x° = 180°.

##5##. Combining like terms gives ##2x°## = 180°.

##6##. Dividing both sides of the equation by 2, we get x° = 90°.

##7##. Through the above reasoning, the final answer is 90°.

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</note> form four angles: <note>upper left angle</note>, <note>upper right angle</note>, <note>lower left angle</note>, and <note>lower right angle</note>. According to the definition of vertical angles, <note>the upper left angle and the lower right angle are vertical angles</note>, <note>the upper right angle and the lower left angle are vertical angles</note>. Since vertical angles are equal, <note>upper left angle = lower right angle = x°</note>, <note>upper right angle = lower left angle = x°</note>.

Definition of Linear Pair of Angles

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

The two adjacent angles <note>x° and x° on the horizontal line</note> have a common side <note>(horizontal line)</note>, and their other sides <note>(vertical line) are extensions in opposite directions</note>, so these two angles are adjacent supplementary angles.


Question: Find the value of x for equilateral triangle R S T, given R S = x + 9, S T = 2 x, and R T = 3 x - 9.

Answer: 9

1. Given conditions are RS = x + 9, ST = 2x, and RT = 3x - 9.

2. According to the definition of an equilateral triangle, the three sides of triangle RST are equal, so RS = ST = RT, i.e., x + 9 = 2x = 3x - 9.

3. First, solve the equation x + 9 = 2x: move x to one side of the equation to get 9 = 2x - x, so 9 = x.

4. Next, verify by substituting x = 9 into the equation x + 9 = 3x - 9 to get 9 + 9 = 3 * 9 - 9, simplifying to 18 = 18, thus x = 9 is correct.

5. Through the above reasoning, the final answer is x = 9.

Definition of Equilateral Triangle

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

<note>In the diagram of this problem</note>, triangle RST is an equilateral triangle. The lengths of <note>side RS</note>, <note>side ST</note>, and <note>side RT</note> are equal, and the measures of <note>∠RST, ∠SRT, and ∠STR</note> are equal, each being 60°.


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

Answer: 64

1. Given conditions: One pair of adjacent sides of a parallelogram are 20 cm and 12 cm respectively, and the included angle is 60 degrees. ##According to the formula for the perimeter of a parallelogram, we get: Perimeter=2*(20+12)=64 cm##.

####

##2##. Through reasoning, we calculate that the perimeter of the parallelogram is 64 cm.

平行四边形; 平行线; 普通三角形

Definition of Parallelogram

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

<note>The quadrilateral in the figure of this problem is a parallelogram</note>, <note>with both pairs of opposite sides parallel and equal</note>.

Perimeter of a Parallelogram

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

<note>The two sets of adjacent sides of the parallelogram are 20 cm and 12 cm respectively</note>, therefore the perimeter P = 2(20 + 12) = 64 cm.


Question: \overrightarrow Q P and \overrightarrow Q R are opposite rays, and \overrightarrow Q T bisects \angle S Q R. If m \angle S Q R = 6 x + 8 and m \angle T Q R = 4 x - 14, find m \angle S Q T.

Answer: 58

1. Given \\overrightarrow{QP} and \\overrightarrow{QR} are opposite rays, according to the definition of rays, \\angle PQR is a straight line, therefore ##\\angle PQR## is a straight angle, and the degree of ##\\angle PQR## is 180 degrees.

2. Given \\overrightarrow{QT} bisects \\angle SQR, according to the definition of angle bisector, \\angle SQT = \\angle TQR. Thus, we can conclude \\angle SQT = \\angle TQR = \\frac{1}{2} \\angle SQR.

3. According to the problem statement, \\angle SQR = 6x + 8 and \\angle TQR = 4x - 14. Combining with the definition of bisector, we get \\angle TQR = \\frac{1}{2} \\angle SQR = \\frac{1}{2}(6x + 8).

4. Let \\frac{1}{2}(6x + 8) = 4x - 14, solve the equation:

   \\frac{1}{2}(6x + 8) = 3x + 4,

   3x + 4 = 4x - 14,

   18 = x.

5. By substituting x = 18 into \\angle SQR = 6x + 8, we get \\angle SQR = 6(18) + 8 = 116.

6. Since \\overrightarrow{QT} is the bisector of \\angle SQR, \\angle SQT = \\frac{1}{2}(116) = 58.

7. Therefore, through the steps, we can conclude \\angle SQT = 58.

Definition of Angle Bisector

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

The vertex of <note>angle \( \angle SQR \)</note> is point Q. A line <note>QT</note> is drawn from point Q, which divides <note>angle \( \angle SQR \) into two equal angles</note>, that is, <note>\( \angle SQT \) and \( \angle TQR \) are equal</note>. Therefore, the line <note>QT is the angle bisector of \( \angle SQR \)</note>. In algebraic terms: <note>\( \angle SQT = \angle TQR = \frac{1}{2} \angle SQR \)</note>.

Definition of Straight Angle

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

The ray QP rotates around the endpoint Q to form a straight line with the initial side, forming <note>平角PQR</note>. According to the definition of a straight angle, the measure of a straight angle is 180 degrees, i.e., <note>角PQR=180度</note>.


Question: If R L = 5, R T = 9, and W S = 6, find R W.

Answer: 7.5

1. Correctly understand the problem: Given segment RL = 5, RT = 9, WS = 6, it is required to find the length of RW.

2. ##Given segment LW is parallel to one side TS of triangle RTS, according to the theorem of parallel lines dividing segments proportionally, we get RL/LT = RW/WS.##

3. ##Since segment LT = RT - RL, substituting the given conditions we get LT = 9 - 5 = 4.##

4. ##Substitute the corresponding values into RL/LT = RW/WS, we get 5/4 = RW/6.##

5. ##Cross-multiplying the equation: 4RW = 30.##

6. ##Divide both sides by 4, we get RW = 30/4 = 7.5.##

####

##7##. Through the above steps, the final length of RW is 7.5.

普通四边形; 平行线; 内错角; 三角形的外角

Definition of Line Segment

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

In the figure of this problem, <note>line segment RL</note>, <note>line segment RT</note>, <note>line segment LT</note>, <note>line segment WS</note>, and <note>line segment RW</note> are line segments. <note>Line segment RL</note> has two endpoints R and L, and every point on <note>line segment RL</note> lies between endpoint R and endpoint L. <note>Line segment RT</note> has two endpoints R and T, and every point on <note>line segment RT</note> lies between endpoint R and endpoint T. <note>Line segment LT</note> has two endpoints L and T, and every point on <note>line segment LT</note> lies between endpoint L and endpoint T. <note>Line segment WS</note> has two endpoints W and S, and every point on <note>line segment WS</note> lies between endpoint W and endpoint S. <note>Line segment RW</note> has two endpoints R and W, and every point on <note>line segment RW</note> lies between endpoint R and endpoint W.

Proportional Segments Theorem

If a line is parallel to one side of a triangle and intersects the other two sides or their extensions, then it divides the segments it intersects proportionally in relation to the corresponding segments of the original triangle.

In the figure of this problem, in triangle RTS, the line LW is parallel to <note>side TS</note> and intersects its other two sides RT and <note>RS</note> at points L and W. Then, according to the Proportional Segments Theorem, we have: <note>RL/LT = RW/WS</note>. That is, the segments intercepted are proportional to the corresponding segments of the original triangle.

Definition of Triangle

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

<note>Triangle RTS is a geometric figure composed of three non-collinear points R, T, S and their connecting line segments RT, RS, TS</note>。<note>Points R, T, S are the three vertices of the triangle</note>，<note>Line segments RT, RS, TS are the three sides of the triangle</note>。

Definition of Parallel Lines

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

The lines <note>LW and TS</note> lie in the same plane and do not intersect, so according to the definition of parallel lines, <note>LW and TS</note> are parallel lines.


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

Answer: 4

1. Assume the quadrilateral in the figure is parallelogram ABCD, given the interior angles ∠BAC = 25x°, ∠ACB = 10y°, ∠ACD = 100°, ∠DAC = 40°.

2. For quadrilateral ABCD to be a parallelogram, according to the definition of a parallelogram, the opposite sides of a parallelogram are parallel and equal. By the parallel axiom 2 and the definition of alternate interior angles, alternate interior angles are equal, and ∠BAC and ∠ACD are alternate interior angles, so 25x° = 100°.

3. Solve the equation 25x = 100, multiply both sides of the equation by 1/25, to get x = 4.

4. Therefore, when x = 4, the quadrilateral is a parallelogram.

平行四边形; 内错角; 邻补角

Definition of Parallelogram

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

Definition of Parallel Lines

In the figure of this problem, <note>line AB and line CD are located in the same plane</note>, and <note>they have no intersection points</note>, so according to the definition of parallel lines, <note>line AB and line CD are parallel lines</note>.

Parallel Postulate 2 of Parallel Lines

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

<note>Two parallel lines AB and CD</note> are intersected by <note>a third line BC</note>, forming the following geometric relationships: 1. Corresponding angles: None. 2. Alternate interior angles: <note>∠BAC and ∠ACD are equal</note>. 3. Same-side interior angles: None. These relationships illustrate that when two parallel lines are intersected by a third line, the alternate interior angles are equal.

Definition of Alternate Interior Angles

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

In the figure of this problem, <note>two parallel lines AB and CD are intersected by a line BC</note>, where <note>∠BAC and ∠ACD are between the two parallel lines</note>, and <note>on opposite sides of the intersecting line BC</note>, therefore <note>∠BAC and ∠ACD are alternate interior angles</note>. <note>Alternate interior angles are equal, i.e., ∠BAC is equal to ∠ACD</note>.


Question: Triangle L M N is an equilateral triangle, and M P bisects L N. Find y.

Answer: 18

1. Triangle LMN is an equilateral triangle. According to the definition of an equilateral triangle, all angles are equal, i.e., \\(
angle L =
angle M =
angle N = 60°\\).

2. Since MP bisects LN, according to the theorem that the altitude, median, and angle bisector coincide in an isosceles triangle (an equilateral triangle is a special isosceles triangle): MP is the altitude, median, and angle bisector of triangle LMN, so \\(
angle LMP =
angle NMP\\), and each is equal to \\(rac{
angle LMN}{2} = 30°\\).

3. In \\(	riangle LMP\\), applying the triangle angle sum theorem: \\(
angle L +
angle LPM +
angle LMP = 180°\\), substituting the known values we get \\(
angle L + 5y° + 30° = 180°\\).

4. Solving \\(
angle L + 5y + 90 = 180\\), we get: \\(
angle L + 5y = 90\\).

5. Dividing both sides by 5, \\(
angle L + y = 18\\). Determine the value of y.

6. The final answer is \\(
angle L + y = 18\\).

等边三角形; 线段; 中点

Definition of Equilateral Triangle

<note>In the figure of this problem, triangle LMN is an equilateral triangle</note>. <note>The lengths of sides LM, MN, and LN are equal</note>, and <note>the measures of angles LMN, MLN, and MNL are equal, each being 60°</note>.

Coincidence Theorem of Altitude, Median, and Angle Bisector in Isosceles Triangle

In an isosceles triangle, the angle bisector of the vertex angle not only bisects the vertex angle but also bisects the base and is perpendicular to the base.

In <note>isosceles triangle LMN</note>, the vertex angle is <note>∠LMN</note>, and the base is <note>side LN</note>. The angle bisector of the vertex angle <note>MP</note> not only bisects the vertex angle <note>∠LMN</note> but also bisects the base <note>LN</note>, making <note>PL = PN</note>, and is perpendicular to the base <note>LN</note>, forming a <note>right angle ∠MPL (90 degrees)</note>. Therefore, the segment <note>MP</note> is both the angle bisector of the vertex angle, the median of the base, and the altitude.

Triangle Angle Sum Theorem

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

In <note>triangle LMP</note>, <note>angle LMP, angle L, and angle LPM are the three interior angles of triangle LMP</note>. According to the Triangle Angle Sum Theorem, <note>angle LMP + angle L + angle MPL = 180°</note>.


Question: If m \angle 5 = 7 x - 5 and m \angle 4 = 2 x + 23, find x.

Answer: 18

1. Given that line l is parallel to line m and is cut by line p, and ∠5 = 7x - 5 and ∠4 = 2x + 23.

2. According to the parallel postulate 2 and the definition of consecutive interior angles, ∠5 and ∠4 are consecutive interior angles, their sum should be 180°.

3. Therefore, we can set up an equation: (7x - 5) + (2x + 23) = 180.

4. Simplifying this equation, we get 9x + 18 = 180.

5. By subtracting 18, we establish the equation 9x = 162.

6. By dividing by 9, we calculate x = 18.

7. Through the above reasoning, we finally obtain the answer x = 18.

Definition of Parallel Lines

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

Parallel Postulate 2 of Parallel Lines

<note>In the figure of this problem, two parallel lines l and m are intersected by a third line p</note>, forming the following geometric relationship: <note>Consecutive interior angles: angle 4 and angle 5 are supplementary, that is, angle 4 + angle 5 = 180 degrees</note>. These relationships illustrate that when two parallel lines are intersected by a third line, the consecutive interior angles are supplementary.

Definition of Consecutive Interior Angles

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

In the diagram of this problem, <note>line l and line m are intersected by line p</note>, <note>angle 4 and angle 5 are on the same side of the intersecting line p and within the intersected lines l and m</note>, so angles 4 and 5 are consecutive interior angles. <note>Consecutive interior angles 4 and 5 are supplementary, that is, angle 4 + angle 5 = 180 degrees</note>.


Question: Find the area of the shaded region formed by the circle and the regular polygon. Round the result to the nearest tenth.

Answer: 76.4

1. Given the figure is a regular pentagon inscribed in a circle, with the circle's radius being 10 cm.

2. Calculate the area of the circle using the circle area formula ##A=πr?##, where r is the radius.

3. The radius r of the circle is 10 cm, so the area of the circle A = π ×## (10 cm) ? = 100π cm?=314.159 cm?##.

4. From the fact that a regular pentagon is a type of regular polygon, it can be divided into 5 congruent isosceles triangles.

5. In each of these isosceles triangles, ##the central angle of each triangle with the circle's center as the vertex is## 360°/5 = 72°.

6. Use the ##triangle area formula (using the sine function)## to calculate the area of a single isosceles triangle, using the formula: A = 1/2 × a × b × sin(C), where a and b are the two radii, also equal to the radius 10 cm, and angle C is the central angle 72°.

7. The area of the triangle = 1/2 × 10 cm × 10 cm × sin(72°) = 50 × ##sin(72°) cm?##.

8. We calculate sin(72°), which is approximately 0.9511, so the area of the triangle ≈ ##47.55 cm?##.

9. The pentagon is composed of 5 isosceles triangles, so the area of the pentagon: 5 × ##47.55 cm? = 237.75 cm?##.

10. To find the shaded area, which is the area of the entire circle minus the area of the inner pentagon, calculate: ##314.159 cm? - 237.75 cm?=76.4cm?##.

####

##11##. After the above reasoning, the final answer is approximately ##76.4cm?##.

圆; 正多边形; 扇形; 弦; 圆心角

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 original text: <note>The center of the circle is the black dot</note>, <note>A vertex of the regular pentagon is any point on the circle</note>, <note>The red segment is the line segment from the center of the circle to any point on the circle</note>, therefore <note>The red segment is the radius of the circle</note>, with a length of 10 cm.

Definition of Regular Polygon

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

In the diagram of this problem, <note>a regular pentagon is inscribed in a circle</note>, <note>all sides are of equal length and all interior angles are equal</note>. Therefore, <note>a regular pentagon is a regular polygon</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.

The original text: <note>The vertex angle of the isosceles triangle is 72°</note>, and the two legs are <note>the radius of the circle, 10 cm</note>. According to the <note>Definition of Sine Function</note>, <note>sin(72°) = opposite side / hypotenuse</note>.

Area Formula of a Circle

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

<note>In the diagram of this problem, the radius of the circle is 10 cm</note>, according to the area formula of a circle, <note>the area A of the circle is equal to the circumference π multiplied by the square of the radius 10 cm</note>, that is, <note>A = π(10 cm)^2</note>.

Triangle Area Formula (Using Sine Function)

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

<note>In the given figure</note>, the regular pentagon is divided into <note>5 congruent isosceles triangles</note>, each with a vertex angle of <note>72°</note> and two equal sides of <note>the circle's radius of 10 cm</note>. Using the triangle area formula to calculate the area of a single isosceles triangle, the specific formula is <note>A = 1/2 × a × b × sin(C)</note>, where <note>a and b are both the circle's radius (10 cm)</note>, and <note>angle C is 72°</note>, thus the area = 1/2 × 10 cm × 10 cm × sin(72°).


Question: In rhombus ABCD, AB = 2x + 3 and BC = 5x. Find m \angle AEB.

Answer: 90

1. Given that in the rhombus ABCD, the diagonals AC and BD intersect at point E.

####

2. According to the properties of the diagonals of a rhombus, angles AEB and CEB are both right angles (90 degrees).

3. From this, it can be concluded that angle AEB = 90 degrees.

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

Definition of Rhombus

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

In the figure of this problem, in <note>rhombus ABCD</note>, all sides <note>AB, BC, CD, and DA are equal</note>, so quadrilateral ABCD is a rhombus. Additionally, the <note>diagonals AC and BD of quadrilateral ABCD are perpendicular bisectors of each other</note>, that is, the <note>diagonals AC and BD intersect at point E</note>, and the <note>angle AEB is a right angle (90 degrees)</note>, and <note>AE = EC</note> and <note>BE = ED</note>.

Properties of the Diagonals of a Rhombus

In a rhombus, the diagonals bisect each other and are perpendicular to each other.

In the figure of this problem, in <note>rhombus ABCD</note>, <note>diagonals AC and BD bisect each other and are perpendicular to each other</note>. Specifically, <note>point E is the intersection of diagonals AC and BD, and AE=EC and BE=DE</note>. Additionally, <note>angles AEB and CEB are right angles (90 degrees)</note>, so <note>diagonals AC and BD are perpendicular to each other</note>.


Question: Find x in the figure.

Answer: 14

1. Let ∠A = (7x + 9)°, ∠B = (7y - 4)°, ∠C = (11x - 1)°, ∠D = (2y + 5)°. From the figure, we know there are three parallel lines, denoted as P//Q//Z (from top to bottom), and two other rays OM and ON (from top to bottom).

2. According to the definition of consecutive interior angles, ∠A and ∠B are consecutive interior angles, ∠C and ∠D are consecutive interior angles. Based on the parallel lines axiom 2, we have ∠A + ∠B = 180°, ∠C + ∠D = 180°.

5. Simplifying 7x + 7y = 175, we get: x + y = 25.

6. Calculating the second set of equations: (11x - 1) + (2y + 5) = 180.

9. Simplifying the equation, we get: 11x + 50 - 2x = 176 => 9x + 50 = 176.

10. Solving for x, we get: 9x = 126 => x = 14.

11. Through the above reasoning, the final answer is 14.

<note>Two parallel lines P and Q are intersected by a third line OM</note>, <note>angles ∠A and ∠B are on the same side of the intersecting line OM and within the intersected lines P and Q</note>, so <note>∠A and ∠B are consecutive interior angles</note>. <note>Consecutive interior angles ∠A and ∠B are supplementary, that is, ∠A + ∠B = 180 degrees</note>. Similarly, <note>two parallel lines Q and Z are intersected by ON</note>, <note>angles ∠C and ∠D are on the same side of the intersecting line ON and within the intersected lines Q and Z</note>, so <note>∠C and ∠D are consecutive interior angles</note>. <note>Consecutive interior angles ∠C and ∠D are supplementary, that is, ∠C + ∠D = 180 degrees</note>.

Parallel Postulate 2 of Parallel Lines

In the diagram of this problem, <note>two parallel lines P and Q are intersected by a third line OM</note>, <note>forming same-side interior angles: ∠A and ∠B are supplementary</note>, that is, <note>∠A + ∠B = 180°</note>. <note>Two parallel lines Q and Z are intersected by a third line ON</note>, <note>forming same-side interior angles: ∠C and ∠D are supplementary</note>, that is, <note>∠C + ∠D = 180°</note>. These relationships illustrate that <note>when two parallel lines are intersected by a third line, the same-side interior angles are supplementary</note>.


Question: The area of the shaded region A is known. Find x. A = 66 cm^2.

Answer: 13

1. Given condition: The shaded part is a semicircle, area A = 66 ##cm^2##.

2. According to the area formula of a circle, the area formula of a circle is A = πr^2, where A represents the area, π is the pi, and r is the radius of the circle. The area of the semicircle is half of the circle, so the area formula of the semicircle is (1/2)π##r^2##, where r is the radius of the circle.

3. According to the given condition, (1/2)π##r^2## = 66.

4. Simplify the equation, we get ##r^2## = (66 * 2) / π.

5. Solve for r, we get r = √((132) / π).

6. From the figure, we know that x is part of the diameter of the circle, i.e., x = 2r.

7. Therefore, x = 2 * √((132) / π).

8. Through the above reasoning, the final answer is 2 * √((132) / 3.14159). ##Approximately equal to 13##.

Definition of Circle

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

<note>In circle K, point K is the center</note>, <note>the radius is x/2</note>. All <note>points that are at a distance of x/2 from point K are on circle K</note>.

Definition of Diameter

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

In the figure of this problem, <note>JL is the diameter</note>, connecting <note>the center K and points J and L on the circumference</note>, with a length equal to 2 times the radius, i.e., <note>JL = 2r</note>.

Definition of Semicircle

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

<note>The shaded part is a semicircle of the circle</note>, formed by <note>diameter KL and arc JL</note>.

Area Formula of a Circle

In <note>circle K</note>, the radius of the circle is <note>JK</note>. According to the area formula of a circle, the area A of the circle is equal to the mathematical constant <note>π</note> multiplied by the square of the radius <note>JK</note>, which is A = <note>πJK^2</note>.


Question: Using parallelogram A B C D, find m \angle D A B.

Answer: 101

1. According to the information in the figure, ##in the parallelogram ABCD, the two diagonals divide a triangle ABD##.

2. ##It is known that in triangle ABD, ∠ADB=59°, ∠ABD=20°##.

3. ##According to the triangle angle sum theorem, ∠ABD+∠ADB+∠DAB=20°+59°+∠DAB=180°, then ∠DAB=180°-20°-59°=101°##.

4. After calculation, we get ##∠DAB = 101°##.

5. Therefore, the calculated degree of ##∠DAB## is ##101°##.

平行四边形; 邻补角; 平行线; 普通四边形

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

Definition of Triangle

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

The geometric figure <note>triangle ABD</note> is composed of <note>three non-collinear points A, B, and D</note> and their connecting <note>line segments AB, AD, and BD</note>. <note>Points A, B, and D</note> are the three vertices of the triangle, and <note>line segments AB, AD, and BD</note> are the three sides of the triangle.

Triangle Angle Sum Theorem

<note>In the diagram of this problem, ##triangle ABD## has three interior angles ##∠ABD, ∠ADB, and ∠DAB##</note>, according to the Triangle Angle Sum Theorem, <note>∠DAB + ∠ABD + ∠ADB = 180°</note>.