Question: Find x.

Answer: \frac { 17 } { 2 } \sqrt { 3 }

1. ##Let the three vertices of the triangle be ABC, then the hypotenuse AB=17, ∠B is 60° and ∠C is 90°. According to the definition of a right triangle, this triangle is a right triangle.##

2. ##According to the triangle angle sum theorem, ∠A+∠B+∠C=180°, given ∠B=60° and ∠C=90°, then ∠A=180°-∠B-∠C=180°-60°-90°=30°.##

3. ##According to the definition of the sine function, for the right triangle ABC, sin(∠B)=opposite side/hypotenuse##, therefore sin(60°)=x/17.##

4. ##According to the properties of a 30°-60°-90° triangle, the sides y:x:hypotenuse=1:√3:2, and according to the definition of the sine function,## sin(60°)=√3/2. Substituting this into sin(60°)=x/17, we get √3/2 = x/17.##

5. Solving the equation √3/2 = x/17, we get x = (17√3)/2.##

6. After calculation, we get x ≈ 14.722.##

7. Through the above reasoning, the final answer is (17√3)/2.##

直角三角形; 正弦; 余弦; 正切

Definition of Right Triangle

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

Definition of Sine Function

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

In the figure of this problem, <note>in the right triangle, ∠B = 60° is an acute angle</note>, <note>side x is the opposite side of ∠B</note>, <note>side 17 is the hypotenuse</note>. According to the definition of the sine function, the sine value of ∠B is equal to the ratio of the opposite side x to the hypotenuse 17, that is, <note>sin(60°) = x / 17</note>.

Triangle Angle Sum Theorem

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

<note>In triangle ABC, ∠A, ∠B, and ∠C are the three interior angles of triangle ABC</note>, according to the Triangle Angle Sum Theorem, <note>∠A + ∠B + ∠C = 180°</note>. Among them, <note>∠C = 90°</note>, <note>∠B = 60°</note>, so <note>∠A = 30°</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 <note>30°-60°-90° triangle ABC</note>, <note>∠A is 30°</note>, <note>∠B is 60°</note>, and <note>∠C is 90°</note>. <note>Side 17 is the hypotenuse</note>, <note>side y is opposite the 30° angle</note>, and <note>side x is opposite the 60° angle</note>. According to the properties of a 30°-60°-90° triangle, <note>side y is half of side 17</note>, and <note>side x is √3 times side y</note>. That is: <note>y = 17/2</note>, <note>x = y * √3</note>.


Question: Find x.

Answer: 1.4

1. Given conditions: ##∠LMP##=60°, ∠MPL=60°, ####According to the triangle angle sum theorem, we know the third angle of the triangle##: ∠LMP=180°-∠LMP-∠MPL=180°-60°-60°=60°##.

2. From the above conclusion, we know that ##each angle in △MLP is 60°, so △MLP is an equilateral triangle (according to the definition of an equilateral triangle)##.

3. Since △MLP is an equilateral triangle, all three sides are equal, thus, ML=MP=LP.

4. Since MP=4, therefore ##ML=PL=4##.

####

##5##. Because LP=5x-3 and LP=4, we can set up the equation: 5x - 3 = 4.

##6##. Solve the equation 5x - 3 = 4, we get 5x = 7, so x = 7/5##=1.4##.

##7##. Through the above reasoning, the final answer is ##x=1.4##.

等边三角形; 直角三角形; 内错角

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 figure of this problem, triangle MLP is an equilateral triangle</note>. <note>The lengths of side ML, side MP, and side LP are equal</note>, and <note>the measures of angle MLP, angle MPL, and angle LMP are equal, each being 60°</note>.

Triangle Angle Sum Theorem

<note>In the diagram for this problem, in triangle MLP, angle MLP, angle MPL, and angle LMP are the three interior angles of triangle MLP</note>, according to the Triangle Angle Sum Theorem, <note>angle MLP + angle MPL + angle LMP = 180°</note>.


Question: As shown in the figure, in ⊙A, it is known that chord BC=8, DE=6, and ∠BAC+∠EAD=180°, then the radius of ⊙A is ()

Answer: 5

1. Given: chord BC=8, DE=6, ∠BAC+∠EAD=180°. Draw diameter CF, connect BF.

2. According to the definition of the inscribed angle, ∠CBF is an inscribed angle; according to (corollary 2 of the inscribed angle theorem) the inscribed angle subtended by the diameter is a right angle, we get ∠CBF is a right angle. According to the definition of a right triangle, triangle CBF is a right triangle.

3. Given condition ∠BAC+∠EAD=180°, according to the definition of a straight angle, ∠CAF is a straight angle, ∠CAF=180°. That is, ∠BAC+∠BAF=180°.

4. Since ∠BAC+∠BAF=180°, ∠BAC+∠EAD=180°, therefore ∠DAE=∠BAF.

5. According to the definition of the central angle, ∠BAF and ∠DAE are central angles; and because ∠DAE=∠BAF, according to the central angle theorem, we get DE=BF, so BF=6.

6. According to the Pythagorean theorem, in the right triangle CBF, we can derive CF^2=BC^2+BF^2, that is CF=√(BC^2 + BF^2)=√(8^2 + 6^2)=√(64 + 36)=√100=10.

7. The radius of circle A is half of the diameter, therefore the radius of ⊙A is CF/2=10/2=5.

圆; 弦; 圆周角; 圆内接四边形

Definition of Chord

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

In the diagram of this problem, in circle A, <note>point C and point B are any two points on the circle</note>, <note>line segment CB connects these two points</note>, so <note>line segment CB is a chord of circle A</note>; in circle A, <note>point D and point E are any two points on the circle</note>, <note>line segment DE connects these two points</note>, so <note>line segment DE is a chord of circle A</note>; in circle A, <note>point B and point F are any two points on the circle</note>, <note>line segment BF connects these two points</note>, so <note>line segment BF is a chord of circle A</note>.

Definition of Inscribed Angle

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

In <note>circle A</note>, the vertex of ∠CBF is <note>on the circumference</note>, and the two sides of ∠CBF intersect <note>circle A at points F and C</note>. Therefore, <note>∠CBF is an inscribed angle</note>.

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

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

<note>In circle A, the angle subtended by the diameter CF at the circumference, angle CBF, is a right angle (90 degrees)</note>.

Definition of Right Triangle

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

In the figure of this problem, in triangle CBF, <note>angle CBF is a right angle (90 degrees)</note>, so triangle CBF is a right triangle. <note>Side CB and side BF are the legs, and side CF is the hypotenuse</note>.

Definition of Straight Angle

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

<note>Ray AC rotates around endpoint A to form a straight line with the initial side</note>, forming <note>straight angle CAF</note>. According to the <note>definition of a straight angle</note>, the measure of a straight angle is <note>180 degrees</note>, that is, <note>angle CAF = 180 degrees</note>.

Pythagorean Theorem

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

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

Definition of Central Angle

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

In the diagram of this problem, <note>in circle A, point B and point F are two points on the circle, and the center is point A</note>. The angle <note>∠BAF formed by the lines AB and AF is called the central angle</note>; <note>in circle A, point D and point E are two points on the circle, and the center is point A</note>. The angle <note>∠DAE formed by the lines AD and AE is called the central angle</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>CF is the diameter</note>, connecting <note>the center A</note> and <note>the points C and F on the circumference</note>, with a length of <note>2 times the radius</note>, that is, <note>AC = AF</note>.


Question: Find m \angle 3.

Answer: 66

1. ##Let the triangle containing ∠3 be △ABC, it is known that ∠ABC=72°, ∠CAB=42°, ∠3=∠ACB, find the measure of ∠3.##

2. ##According to the triangle angle sum theorem, we know that ∠ABC+∠CAB+∠3=180°.##

3. Given the measures of ##∠ABC and ∠CAB##, their sum is 42° + 72° = 114°.

5. Through the above reasoning, the final answer is 66°.

普通三角形; 三角形的外角

Triangle Angle Sum Theorem

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

Definition of Triangle

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

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


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

Answer: 91.9

1. Given that one side of the parallelogram is 13 yd.

2. To calculate the area of the parallelogram, the length of the base and the corresponding height are needed. The base (13 yd) is given, but the height needs to be calculated.

3. In this parallelogram, use trigonometric functions to calculate the height. According to the problem, a line can be drawn from the vertex perpendicular to the base, forming a right triangle.

4. The hypotenuse of this right triangle is the side of the parallelogram (i.e., 10 yd), and the 45° angle is one of the acute angles of the right triangle.

5. According to the sine function, we have sin(45°) = height / 10 yd.

6. Since sin(45°) = √2 / 2, from the equation, we get height = 10 yd * (√2 / 2) = 10√2/2 yd = 5√2 yd.

7. Knowing the height, the area of the parallelogram can be calculated using the area formula: Area = base × height.

8. Therefore, the area of the parallelogram = 13 yd × 5√2 yd = 65√2 yd^2.

9. Calculate the approximate value of 65√2, rounding the result to the first decimal place, then 65√2 ≈ 91.92 yd^2.

10. After the above reasoning, the final answer is: the area of the parallelogram is approximately 91.9 yd^2.

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 (≅).

The quadrilateral is a parallelogram, with side <note>13 yd parallel and equal to the opposite side 13 yd</note>, and side <note>10 yd parallel and equal to the opposite side 10 yd</note>.

Definition of Right Triangle

In the diagram of this problem, an auxiliary line forms a <note>right triangle</note> by dropping perpendicularly from the vertex to the base. One of the interior angles of this <note>right triangle</note> is <note>45°</note>, and the other interior angle is <note>90°</note>, thus this triangle is a <note>right triangle</note>.

Definition of Sine Function

In the figure of this problem, in the right triangle, <note>the 45° angle is an acute angle</note>, <note>the dashed line is the opposite side of the 45° angle</note>, and the side of the parallelogram. According to the definition of the sine function, <note>the sine value of the 45° angle is equal to the ratio of the opposite side to the hypotenuse</note>, that is, <note>sin(45°) = height / 10 yd</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, <note>the length of the base of the parallelogram is 13 yd</note>, <note>the corresponding height is the perpendicular distance from the base to the opposite side, denoted as 5√2 yd</note>. Therefore, according to the area formula of a parallelogram, the area of the parallelogram is equal to <note>the base length of 13 yd</note> multiplied by <note>the corresponding height of 5√2 yd</note>, that is, <note>A = 13 yd × 5√2 yd = 65√2 yd##^2##</note>, calculating <note>the approximate value of 65√2</note> gives <note>65√2 ≈ 91.92 yd##^2##</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, <note>the dashed line perpendicular to the opposite side from the vertex of the parallelogram is the altitude of that vertex</note>. <note>The dashed line forms a right angle (90 degrees) with the base</note>, which indicates that the dashed line is the vertical distance from the vertex to the opposite side.


Question:
Point D is the center of the circle. What is m
angle A B C?

Answer: 90

1. According to the description in the figure, we know that D is the center of the circle, ##CA passes through the center D, intersecting the circle at points C and A, which fits the definition of a diameter, so CA is the diameter of circle D.##

2. ##According to the definition of an inscribed angle, ∠ABC is the inscribed angle of the diameter CA.##

3. ##Based on (Corollary 2 of the Inscribed Angle Theorem) the inscribed angle subtended by a diameter is a right angle, so ∠ABC = 90°.##

Center of a Circle

The center of a circle is the fixed point equidistant from all points on the circumference of the circle.

<note>Point D is the center of the circle</note>, indicating that all distances from <note>point D to any point are equal</note> in the diagram.

Definition of Diameter

<note>Line segment CA is the diameter</note>, connecting <note>center D</note> and <note>points C and A on the circumference</note>, with a length of 2 times the radius, i.e., <note>CA = 2 * radius</note>.

Definition of Inscribed Angle

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

In the diagram of this problem, since <note>CA is the diameter</note>, the inscribed angle <note>∠ABC is a right angle</note>, i.e., <note>∠ABC = 90 degrees</note>.


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

Answer: 1.8

1. Given that the radius ##XW## = 2 cm, and the central angle ##∠WXY## = 52°.

2. According to the formula for the area of a sector, the area of the sector X = (θ/360) * π * r², where r is the radius and θ is the central angle in degrees.

3. Substituting the values, the area of the sector X = (52/360) * π * 2² = (52/360) * π * 4.

####

##4. Further simplifying, we get X ≈ 0.1444 * π * 4##.

##5. Continuing to simplify, X ≈ 0.5776π##.

##6.## Calculating, we get ##X## ≈ 0.5776 * 3.1416 ≈ 1.81416.

##7.## Rounding the result to one decimal place, we get 1.8.

##8.## Therefore, the area of the sector is approximately 1.8 square centimeters.

Definition of Sector

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

<note>In the figure of this problem</note>, the sector is the figure formed by the center point <note>O</note>, the two radii <note>XW and XY</note>, and the arc <note>WY</note>. The lengths of <note>XW and XY</note> are both <note>2 cm</note>, and it includes the central angle <note>∠WXY</note>, which measures <note>52°</note>.

Definition of Central Angle

In the diagram of this problem, <note>in circle X</note>, <note>point W and point Y are two points on the circle</note>, <note>the center of the circle is point X</note>. <note>The angle ∠WXY formed by the lines XW and XY is called the central angle</note>.

Formula for the Area of a Sector

The area \( X \) of a sector can be calculated using the formula \( X = \frac{\theta}{360} \times \pi \times r^2 \), where \( \theta \) is the measure of the central angle in degrees, and \( r \) is the radius length.

In the <note>sector WXY</note>, the degree measure of the <note>central angle WXY</note> is <note>52 degrees</note>, and the length of the <note>radius XW</note> is <note>2</note>. According to the <note>formula for the area of a sector</note>, the area A of the sector can be calculated using the formula X = (θ/360) * π * r², where θ is the degree measure of the central angle and r is the length of the radius. Therefore, the <note>area of sector WXY is X = (θ/360) * π * r²</note>.

Definition of Radius

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

<note>In circle X, point X is the center</note>, <note>point W is any point on the circle</note>, <note>line segment XW is the line segment from the center to any point on the circle</note>, therefore <note>line segment XW is the radius of the circle</note>.


Question: Express the value of \cos B as a decimal, accurate to two decimal places.

Answer: 0.38

1. From the figure, it can be seen that ∠ACB=90°, according to the definition of a right triangle, triangle ABC is a right triangle, AC=24, BC=10.

2. According to the cosine function, cosine is the ratio of the adjacent side (BC) to the hypotenuse (AB), i.e., cos B = BC / AB.

3. Substitute the known values, cos B = 10 / 26.

4. Calculate the ratio and round to two decimal places: cos B = 10 / 26 ≈ 0.3846153846, rounded to two decimal places is 0.38.

5. The final approximate value of cos B in decimal form is 0.38.

Definition of Triangle

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

Definition of Right Triangle

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

Cosine Function

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

In the figure of this problem, in the right triangle ABC, <note>side BC is the adjacent side of angle ∠B</note>, <note>side AB is the hypotenuse</note>. Therefore, according to the definition of the cosine function, <note>the cosine of angle ∠B is equal to the ratio of the adjacent side BC to the hypotenuse AB</note>, that is, <note>cos(∠B) = BC / AB</note>.


Question: Find x.

Answer: 8 \sqrt { 3 }

1. The geometric figure described in this problem is a right triangle, where one of the legs has a length of 8 and the hypotenuse has a length of 16. We need to find the length of the other leg x.

2. According to the Pythagorean theorem, that is: in a right triangle, the sum of the squares of the legs is equal to the square of the hypotenuse. It is expressed as: a^2 + b^2 = c^2.

3. In this problem, it is known that c = 16 (hypotenuse), one leg a = 8, and the other leg b = x.

4. Substitute the known values into the Pythagorean theorem formula, that is: 8^2 + x^2 = 16^2.

5. Calculate the square values of each term, that is: 64 + x^2 = 256.

6. Move 64 to the other side of the equation, obtaining: x^2 = 256 - 64.

7. Simplify the equation, obtaining: x^2 = 192.

8. Take the square root of both sides of the equation, obtaining: x = √192.

9. Further simplify to obtain: x = 8√3.

10. Through the above reasoning, the final answer is x = 8√3.

Definition of Triangle

<note>In the figure of this problem, let the quadrilateral be ABCD, and triangle BCD is a geometric figure composed of three non-collinear points B, C, D and their connecting line segments BC, CD, BD.</note> Points B, C, D are the three vertices of the triangle, <note>and line segments BC, CD, BD are the three sides of the triangle respectively.</note>

Definition of Right Triangle

<note>In the figure of this problem, ## let the quadrilateral be ABCD, and in triangle BCD, one angle is marked with a right angle sign</note>, <note>this angle is a right angle (90 degrees)</note>, therefore <note>this triangle is a right triangle</note>. <note>Side BC and side CD are the legs</note>, <note>side BD is the hypotenuse</note>.

Pythagorean Theorem

In the context of a <note>right triangle</note>, given that one <note>leg length is 8</note>, the <note>hypotenuse length is 16</note>, and the other <note>leg length is x</note>. Therefore, according to the <note>Pythagorean Theorem</note>, <note>##16^2 = 8^2 + x^2##</note>.


Question: In \odot S, m \widehat P Q R = 98. Find m \widehat P Q.

Answer: 49

1. Given that radius QS is perpendicular to PR, according to the perpendicular bisector theorem, radius SQ bisects chord PR and arc PQR, so arc PQ = arc RQ.

2. Since m \widehat P Q R = 98, therefore m \widehat P Q = 1/2 * m \widehat P Q R = 49.

3. Through the above reasoning, we can conclude that the value of m \widehat P Q is 49.

圆; 圆周角; 圆心角; 弦

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 S, point S is the center</note>, <note>the radius is QS</note>. All points in the diagram that are at a distance equal to QS from point S are on <note>circle S</note>.

Definition of Radius

In circle S, point S is the center of the circle, point Q is any point on the circle, the line segment QS is the line segment from the center to any point on the circle, therefore the line segment QS is the radius of the circle. <note>圆S中，点S是圆心，点Q是圆上的任意一点</note>，<note>线段QS是从圆心到圆上任意一点的线段</note>，因此<note>线段QS是圆的半径</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.

<note>In circle S, radius QS is perpendicular to chord PR</note>, then according to the Perpendicular Diameter Theorem, <note>radius QS bisects chord PR</note>, that is, <note>PT=RT</note>, and <note>radius QS bisects the two arcs subtended by chord PR</note>, that is, <note>arc PQ=arc RQ</note>.


Question: In Rt△ACB, ∠C=90°, BC=5, AC=12, find sinA=()

Answer: \frac{5}{13}

1. Given the right triangle △ACB, ∠C=90°, BC=5, AC=12.

2. According to the Pythagorean theorem, ##a? + b? = c?##, where a and b are the legs, and c is the hypotenuse.

3. In this problem, AC and BC are the legs, and AB is the hypotenuse.

4. Therefore, we can calculate AB = ##√(AC? + BC?) = √(12? + 5?)## = √(144 + 25) = √169 = 13.

5. Calculate sinA, sinA = opposite side / hypotenuse.

6. In △ACB, the opposite side to angle A is BC, and the hypotenuse is AB, so sinA = BC / AB = 5 / 13.

Through the above reasoning, the final answer is sinA = 5 / 13.

直角三角形; 正弦; 线段

Definition of Right Triangle

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

Definition of Sine Function

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

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

Pythagorean Theorem

In the figure of this problem, in the right triangle ABC, <note>angle ACB is a right angle (90 degrees)</note>, sides <note>AC and BC are the legs</note>, side <note>AB is the hypotenuse</note>, so according to the Pythagorean Theorem, <note>AB² = AC² + BC²</note>.


Question: In \odot X, A B = 30, C D = 30, and m \widehat C Z = 40. Find N D.

Answer: 15

1. Given that chords AB and CD are equal in length, AB = CD = 30. ####Calculate the length of ND.

2. ##In the figure, XZ is the radius of circle X, and CD is a chord on the circle. It is known that XZ⊥CD. According to the perpendicular bisector theorem, if a diameter is perpendicular to a chord, then the diameter bisects the chord and the arcs subtended by the chord, so CN=DN##.

3. ##Since CN=DN, ND=CD/2. Given CD=30, thus ND=30/2=15##.

####

##4##. ##Therefore##, the length of ND should be ND = 15.

圆; 弦; 圆心角; 圆周角

Definition of Chord

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

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

Definition of Radius

<note>In circle X, point X is the center of the circle, and point Z is any point on the circle</note>, <note>the line segment XZ is the line segment from the center of the circle to any point on the circle</note>, therefore <note>the line segment XZ is the radius of the circle</note>.

Definition of Circle

In the figure of this problem, in <note>circle X</note>, <note>point X is the center of the circle</note>. All points in the figure that are at a distance equal to the radius from <note>point X</note> are on <note>circle X</note>.

Perpendicular Diameter Theorem

<note>Radius XZ is perpendicular to chord CD</note>, then according to the Perpendicular Diameter Theorem, <note>radius XZ bisects chord CD, that is CN=DN</note>, and <note>radius XZ bisects the two arcs subtended by chord CD, that is arc CZ=arc DZ</note>.


Question: These two polygons are similar. Find U T.

Answer: 22.5

1. The given condition is that pentagon STUVR is similar to pentagon ABCDE. ##According to the definition of similar figures, if two figures are similar, then the ratios of their corresponding sides are equal##.

2. According to the problem statement, ST corresponds to BC, ##TU corresponds to CD, VU corresponds to DE, ST = 18 m, TU = (y + 2) m, AE = 3 m, BC = 4 m, CD = 5 m##.

3. Using the definition of similar polygons and equal ratios, let the similarity ratio be k, then ##ST/BC = TU/CD = k##.

4. Since ST = 18 m and BC = 4 m are given, the ratio is k = 18/4 = 4.5. ##TU/CD = (y + 2)/5##.

线段; 普通多边形; 位似

Definition of Similar Figures

Two geometric figures are similar if and only if their corresponding sides are proportional, and their corresponding angles are equal.

<note>Pentagon STUVR is similar to pentagon ABCDE</note>, and <note>ST = 18 m</note>, <note>TU = (y + 2) m</note>, <note>AE = 3 m</note>, <note>BC = 4 m</note>, <note>CD = 5 m</note>, <note>RV = x m</note>. According to the definition of similar polygons, <note>their corresponding sides are proportional and corresponding angles are equal</note>. Let the similarity ratio be k, then <note>ST/BC = 18/4 = k=4.5</note>, <note>TU/CD = (y + 2)/5 = k</note>.


Question: As shown in the figure, a math study interest group measured the height of tree AB. They measured the length of the tree's shadow BC to be 9m. At the same time, they also measured Xiaoliang's shadow length to be 1.5m, knowing that Xiaoliang's height is 1.8m. Then the height of tree AB is ()

Answer: 10.8m

1. Given that the length of the shadow BC of the tree AB in the sunlight is 9 meters, Xiaoliang's shadow length in the sunlight is 1.5 meters, and Xiaoliang's height is 1.8 meters.

2. According to the definition of similar triangles, the height of an object and its shadow length are in direct proportion at the same time.

3. Establish the proportional relationship between Xiaoliang and the tree's height and shadow length, expressed as 1.8/1.5 = height of tree AB/9.

4. Solve for the height of tree AB based on the proportional relationship, height of tree AB = 1.8/1.5 * 9.

5. Calculate to find that the height of tree AB is 10.8 meters.

6. Through the above reasoning, the final answer is 10.8 meters.

Definition of Similar Triangles

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

<note>In the diagram of this problem, triangle ABC and triangle DEF are similar triangles</note>. According to the definition of similar triangles: <note>∠BAC = ∠EDF</note>, <note>∠ABC = ∠DEF</note>, <note>∠BCA = ∠EFD</note>; <note>AB/DE = BC/EF = AC/DF</note>.

Similarity of Right Triangles

Two right triangles are similar if and only if one of their acute angles are equal.

In the problem diagram, right triangle ABC and right triangle DEF have right angles (90 degrees) at angle ABC and angle DEF, and angle BAC is equal to angle EDF. Therefore, according to the similarity theorem of right triangles, these two right triangles are similar. That is to say, the corresponding sides of triangle ABC and triangle DEF are proportional.


Question: Find y in the figure.

Answer: 6

1. In the figure, it is known that the lines are parallel, ##two parallel lines are intersected by a third line##.

2. ##According to the parallel axiom 2 of parallel lines and the definition of same-side interior angles, same-side interior angles are supplementary, the angles (8y + 2)° and (25y - 20)° in the figure are same-side interior angles. Thus: (8y + 2)° + (25y - 20)° = 180°##.

4. ##Solve the equation 33y - 18 = 180. First, rearrange the equation to 33y = 198, obtaining y = 6##.

5. Therefore, through the above reasoning, the final answer is ##y = 6##.

对顶角; 内错角; 同位角

Definition of Parallel Lines

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

In the diagram of this problem, <note>the two lines with red arrows</note> are located <note>in the same plane</note>, and they <note>do not intersect</note>. Therefore, according to <note>the definition of parallel lines</note>, <note>these two lines 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 are intersected by a third line</note>, forming the following geometric relationships:
1. Corresponding angles: None.
2. Alternate interior angles: <note>10x° and (25y - 20)° are equal</note>.
3. Consecutive interior angles: <note>(8y + 2)° and (25y - 20)° are supplementary, i.e., (8y + 2)° + (25y - 20)° = 180 degrees</note>.

These relationships illustrate that when <note>two parallel lines are intersected by a third line</note>, corresponding angles are equal, alternate interior angles are equal, and consecutive interior angles are supplementary.

Definition of Consecutive Interior Angles

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

<note>Two parallel lines are intersected by a third line</note>, where the angles measuring <note>(8y + 2)° and (25y - 20)° are on the same side of the intersecting line and within the intersected lines</note>, so <note>the angles are consecutive interior angles</note>. Consecutive interior angles are supplementary, that is, <note>(8y + 2)° + (25y - 20)° = 180 degrees</note>.


Question: As shown in the figure, quadrilateral ABCD is inscribed in ⊙O. If one of its exterior angles ∠DCE=64°, then ∠BOD=()

Answer: 128°

1. Given that quadrilateral ABCD is inscribed in circle O, ##according to (Corollary 3 of the Inscribed Angle Theorem) the opposite angles of a cyclic quadrilateral are supplementary##, we get ∠BAD + ∠BCD = 180°.

2. ##By the definition of a straight angle, we have ∠BCD + ∠DCE = 180° , and since## ∠DCE = 64°, we get ∠BCD = 116°.

3. Since ∠BAD + ∠BCD = 180°, we have ∠BAD = 180° - 116° = 64°.

4. ##According to the Inscribed Angle Theorem##, we have ∠BOD = 2 × ∠BAD = 2 × 64° = 128°.

5. Through the above reasoning, the final answer is 128°.

圆; 圆内接四边形; 三角形的外角; 对顶角; 圆心角

Definition of Inscribed Angle

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

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

Inscribed Angle Theorem

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

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

Definition of Central Angle

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

<note>In the diagram of this problem, in circle O, points B and D are two points on the circle, and the center of the circle is point O. The angle ∠BOD formed by the lines OB and OD is called the central angle.</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 <note>BC</note> rotates around the <note>endpoint C</note> to form a straight line with the initial side, creating a <note>straight angle BCE</note>. According to the definition of a straight angle, the measure of a straight angle is 180 degrees, i.e., <note>angle BCE = 180 degrees</note>.

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

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

In the figure of this problem, the vertices of the cyclic quadrilateral <note>ABCD</note> lie on the circle. According to the Diagonal Supplementary Theorem for Cyclic Quadrilateral, <note>the sum of each pair of opposite angles of quadrilateral ABCD is equal to 180°</note>. Specifically, <note>∠BAD + ∠BCD = 180°</note>.


Question: Given that the perimeter of triangle ABC = 25, find x. Assume that the line segments that appear to be tangent to the circle are indeed tangent.

Answer: 3

1. In the given geometry problem, it is known that the perimeter of ΔABC is 25. ##According to the tangent-segment theorem##, BC = AC = 3x, AB = 7.

2. According to the definition of the perimeter of a triangle, the perimeter is the sum of the three sides, therefore, we have AB + BC + CA = 25.

3. Thus, we get the equation: 7 + 3x + 3x = 25.

4. Simplifying the equation gives 7 + 6x = 25.

5. Moving 7 to the right side of the equation gives 6x = 18.

6. Dividing both sides by 6 gives x = 3.

7. Through the above reasoning, the final answer is: x = 3.

普通三角形; 切线; 线段; 圆

Tangent-Segment Theorem

If two tangent segments are drawn to a circle from an external point, then the lengths of these tangent segments are equal. Moreover, the line segment connecting the center of the circle to the external point bisects the angle formed by the two tangent segments.

In the figure of this problem, from an external point B, two tangents <note>BA and</note> <note>BC are drawn to the circle</note>, and their tangent lengths are equal, that is, <note>BA=BC</note>.

Definition of Tangent to a Circle

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

<note>The circle and the line AB have only one common point A, which is called the point of tangency</note>. Therefore, <note>the line BA is the tangent to the circle</note>. Similarly, in the diagram of this problem, <note>the circle and the line BC have only one common point C, which is called the point of tangency</note>. Therefore, <note>the line BC is the tangent to the circle</note>.

Formula for the Perimeter of a Triangle

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

The three sides of <note>triangle ABC</note> are <note>AB, AC, BC</note> respectively. According to the formula for the perimeter of a triangle, the perimeter of the triangle is equal to the sum of the lengths of its three sides, i.e., the perimeter <note>P=AB+AC+BC</note>.


Question: As shown in the figure, line AD∥BC. If ∠1=42°, ∠BAC=78°, then the degree of ∠2 is ()

Answer: 60°

1. According to the definition of alternate interior angles, ∠1 and ∠DAC are alternate interior angles, ∠2 and ∠ABC are alternate interior angles; given AD∥BC, ∠1= 42°, then according to the parallel postulate 2 of parallel lines, ∠1= ∠DAC= 42°, ∠2=∠ABC.

2. Given ∠BAC=78°, according to the triangle angle sum theorem, in triangle ABC, we have ∠ABC+∠1+∠BAC=180°, that is, ∠ABC = 180° - ∠1 - ∠BAC.

3. Substituting the values, we get ∠ABC = 180° - 42° - 78° = 60°.

4. According to the result obtained in step 1, we get ∠2=∠ABC = 60°.

5. Through the above reasoning, we finally get ∠2= 60°.

平行线; 内错角; 三角形的外角

Definition of Alternate Interior Angles

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

<note>Two parallel lines AD and BC</note> are intersected by <note>a line AC</note>, where <note>∠1 and ∠DAC are located between the two parallel lines and on opposite sides of the intersecting line AC</note>, thus <note>∠1 and ∠DAC are alternate interior angles</note>; <note>Two parallel lines AD and BC</note> are intersected by <note>a line AB</note>, where <note>∠2 and ∠ABC are located between the two parallel lines and on opposite sides of the intersecting line AB</note>, thus <note>∠2 and ∠ABC are alternate interior angles</note>.

Triangle Angle Sum Theorem

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

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

Parallel Postulate 2 of Parallel Lines

In this problem diagram, <note>line AD is parallel to BC, intersected by line AC</note>. According to this theorem, <note>∠1 and ∠DAC are alternate interior angles</note>, therefore <note>∠1 = ∠DAC</note>; <note>line AD is parallel to BC, intersected by line AB</note>. According to this theorem, <note>∠2 and ∠ABC are alternate interior angles</note>, therefore <note>∠2 = ∠ABC</note>.


Question: The diagonals of rhombus F G H J intersect at K. If G H = x + 9 and J H = 5 x - 2, find x.

Answer: 2.75

1. The given conditions are: diagonals ##F H and G J## intersect at point K, G H = x + 9 and J H = 5 x - 2. According to ##the definition of a rhombus, a rhombus is a quadrilateral with all sides equal##, therefore all sides are equal.

2. Since G H and J H are both sides of the rhombus, ####it can be concluded that G H = J H.

3. Substituting the given conditions, we get x + 9 = 5 x - 2.

4. By rearranging and calculating, ####move x to the ##same side## of the equation, we get 9 + 2 = 5 x - x. Simplifying, we get 4x = 11.

5. Finally, solving for x, we get x = 11 / 4.

6. Through the above reasoning, the final answer is x = 11/4##=2.75##.

Definition of Rhombus

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

<note>In the figure of this problem, all sides FG, GH, HJ, and JF of quadrilateral FGHJ are equal</note>, so quadrilateral FGHJ is a rhombus. Additionally, <note>the diagonals FH and GJ of quadrilateral FGHJ are perpendicular bisectors of each other</note>, that is, <note>the diagonals FH and GJ intersect at point K, and angle ∠FKG is a right angle (90 degrees)</note>, and <note>FK = KH and GK = KJ</note>.


Question: Four congruent triangles are cut from the four corners of a rectangle to form an octagon. What is the area of this octagon?

Answer: 528

1. Given that the length of the rectangle is 30 cm, the width is 20 cm, and the length of the legs of each isosceles right triangle cut off at each corner is 6 cm.

2. Four isosceles right triangles are cut off from the rectangle. The area of each triangle is (1/2) * (6 cm) * (6 cm) = 18 square cm.

3. Since four identical triangles are cut off, the total area of the cut-off triangles is 4 * 18 square cm = 72 square cm.

4. The total area of the rectangle is 30 cm * 20 cm = 600 square cm.

5. The area of the octagon is the original rectangle area minus the total area of the four cut-off triangles, which is 600 square cm - 72 square cm = 528 square cm.

6. Through the above reasoning, the final answer is 528 square cm.

矩形; 等腰三角形; 普通多边形

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>A quadrilateral is a rectangle</note>, its interior angles are all <note>right angles (90 degrees)</note>, and <note>opposite sides are parallel and equal in length</note>, with one pair of <note>opposite sides measuring 30 centimeters</note>, and the other pair of <note>opposite sides measuring 20 centimeters</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.

The original text: <note>Four isosceles right triangles</note> have <note>legs of 6 cm each</note>, where <note>each triangle has two equal legs</note>, and <note>each triangle has a right angle of 90 degrees</note>.

Area Formula of a Triangle

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

<note>Each isosceles right triangle has two perpendicular sides of 6 cm</note>, so the base is 6 cm and the height is also 6 cm. According to the <note>area formula of a triangle</note>, the area of the triangle is equal to the base multiplied by the height divided by 2, which is area = <note>(6 cm * 6 cm) / 2</note> = 18 square cm.

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 30 cm</note>, <note>the width is 20 cm</note>, so <note>the area of the rectangle = 30 cm * 20 cm = 600 square cm</note>.


Question: As shown in the figure, Xiaoqiang made a pinhole imaging device by himself, in which the length of the paper tube is 15cm. He prepared a candle with a length of 20cm and wants to obtain an image with a height of 4cm. The distance between the candle and the paper tube should be ()

Answer: 75cm

1. Let the position of the small hole be point O, and the vertical distance from the small hole to CD be OF. Given AB=20cm, OF=15cm, CD=4cm. According to the problem statement, the candle AB is placed vertically, and the image CD is at the same height as the candle AB.

2. Construct auxiliary line EF, making it perpendicular to both AB and CD, then EF is perpendicular to AB and CD.

3. Since CD is the image of AB formed through hole O, we have OA=OB and OC=OD. Therefore, according to the definition of an isosceles triangle, triangles OAB and OCD are isosceles triangles. Also, according to the definition of vertical angles, ∠COD and ∠AOB are the angles between lines AD and BC, so ∠COD=∠AOB, thus ∠OAB=∠OBA=∠OCD=∠ODC. According to the similarity theorem (AA), we get △OAB ∽△ODC.

4. According to the definition of similar triangles, the corresponding sides of the two triangles are proportional, so CD/AB = OF/OE.

5. Based on the given conditions: 4/20 = 15/OE.

6. Solving the equation, we get OE = 75 cm.

7. Through the above reasoning, the distance between the candle and the paper tube is 75 cm.

Definition of Perpendicular Lines

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

<note>The angle ∠EAB formed by the intersection of line EF and line AB is 90 degrees</note>, therefore according to the definition of perpendicular lines, line EF and line AB are perpendicular to each other; similarly, <note>The angle ∠ECD formed by the intersection of line EF and line CD is 90 degrees</note>, therefore according to the definition of perpendicular lines, line EF and line CD are perpendicular to each other.

Definition of Triangle

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

In the figure of this problem, <note>triangle ABO is a geometric figure formed by three non-collinear points A, B, O</note> and <note>its connecting line segments AO, AB, BO</note>. <note>Points A, B, O are the three vertices of the triangle</note>, <note>and the line segments AO, AB, BO are the three sides of the triangle</note>. <note>Triangle CDO is a geometric figure formed by three non-collinear points D, C, O</note> and <note>its connecting line segments CO, CD, DO</note>. <note>Points C, D, O are the three vertices of the triangle</note>, <note>and the line segments CO, DC, DO are the three sides of the triangle</note>.

Definition of Isosceles Triangle

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

<note>In the given diagram, in triangle ABO, side AO and side BO are equal, thus triangle ABO is an isosceles triangle.</note><note>In triangle CDO, side CO and side DO are equal, thus triangle CDO is an isosceles triangle.</note>

Definition of Similar Triangles

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

<note>OF and OE are the heights in triangles ABO and CDO</note>, and the two triangles are <note>similar triangles</note>. According to the definition of similar triangles: <note>angle AOB = angle COD</note>, <note>angle OAB = angle ODC</note>, <note>angle OBA = angle OCD</note>; <note>CD/AB = OF/OE</note>.

Similarity Theorem for Triangles (AA)

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

<note>Angle AOB is equal to angle COD</note>, and <note>angle OAB is equal to angle ODC</note>, so <note>triangle ABO is similar to triangle CDO</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 figure of this problem, in the isosceles triangle OAB, sides OA and OB are equal</note>. Therefore, according to the properties of the isosceles triangle, <note>the angles opposite the equal sides are equal, i.e., angle OAB = angle OBA</note>. Similarly, <note>in the isosceles triangle OCD, sides OC and OD are equal</note>. Therefore, according to the properties of the isosceles triangle, <note>the angles opposite the equal sides are equal, i.e., angle OCD = angle ODC</note>.

Triangle Angle Sum Theorem

In the figure of this problem, <note>in triangle OAB</note>, <note>angle OAB</note>, <note>angle OBA</note>, and <note>angle AOB</note> are <note>the three interior angles of triangle OAB</note>. According to the Triangle Angle Sum Theorem, <note>angle OAB + angle OBA + angle AOB = 180°</note>. Similarly, <note>in triangle OCD</note>, <note>angle OCD</note>, <note>angle ODC</note>, and <note>angle COD</note> are <note>the three interior angles of triangle OCD</note>. According to the Triangle Angle Sum Theorem, <note>angle OCD + angle ODC + angle COD = 180°</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).

<note>The line segment OE perpendicular to the opposite side AB from vertex O is the altitude of vertex O</note>. <note>The line segment OE forms a right angle (90 degrees) with side AB</note>, which indicates that <note>the line segment OE is the perpendicular distance from vertex O to the opposite side AB</note>. Similarly, <note>the line segment OF perpendicular to the opposite side CD from vertex O is the altitude of vertex O</note>. <note>The line segment OF forms a right angle (90 degrees) with side CD</note>, which indicates that <note>the line segment OF is the perpendicular distance from vertex O to the opposite side CD</note>.


Question: Find QS.

Answer: 25

1. From the figure, each of the three sides is marked with a single short line (small horizontal line) indicating that the lengths of these three sides are equal. That is, side RQ = side RS.

2. From the figure, it is known that side RQ = 5x and side RS = 6x - 5, so 5x = 6x - 5. Solving the equation 5x = 6x - 5, subtracting 5x from both sides gives 0 = x - 5, then adding 5 gives x = 5.

3. Substituting x = 5 into QS = 3x + 10 to calculate QS, we get QS = 3(5) + 10 = 15 + 10 = 25.

4. Therefore, through the above reasoning, the final answer is 25.

等边三角形; 等腰三角形

Definition of Isosceles Triangle

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

Properties of Isosceles Triangle

In the figure of this problem, in the isosceles triangle <note>QRS</note>, side <note>RQ</note> and side <note>RS</note> are equal. Therefore, according to the properties of isosceles triangles, the angles opposite the equal sides are equal, that is, <note>∠QRS = ∠QSR</note>.


Question: Quadrilateral E F G H is a rectangle. If F K = 32 feet, find E G.

Answer: 64

1. Given rectangle EFGH, K is the intersection point of diagonals EF and HG.

2. According to the properties of the diagonals of a rectangle, the diagonals are equal and the intersection point of the diagonals bisects the diagonals. Therefore FK = EK.

3. Given FK = 32 feet, so EK is also equal to 32 feet.

4. Because the intersection point of the diagonals bisects the diagonals, EG = 2 * EK.

5. Based on the conclusion of the previous step, EG = 2 * 32 feet = 64 feet.

Definition of Rectangle

<note>Quadrilateral EFGH is a rectangle</note>, with its interior angles <note>∠HEF</note>, <note>∠EFG</note>, <note>∠FGH</note>, and <note>∠GHE all being right angles (90 degrees)</note>, and <note>side EF is parallel and equal in length to side HG</note>, and <note>side EH is parallel and equal in length to side FG</note>.

Property of Diagonals in a Rectangle

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

In the figure of this problem, in rectangle EFGH, <note>side EF is parallel and equal to side HG</note>, <note>side EH is parallel and equal to side FG</note>. <note>Diagonals EG and FH are equal and bisect each other</note>, that is, <note>the intersection point K of diagonals EG and FH is the midpoint of both diagonals</note>. Therefore, <note>segment EK is equal to segment GK</note>, <note>segment FK is equal to segment HK</note>.


Question: Given a = 14, b = 48, and c = 50, find \\cos A.

Answer: 0.96

1. The given conditions are: a = 14, b = 48, c = 50, and △ABC is a right triangle.

####

##2. According to the cosine function##, for the right triangle △ABC with ∠A, the sides AB and AC are the hypotenuse and the adjacent side, then cos A = adjacent side / hypotenuse = b/c.

##3##. Replacing the known values, we can calculate cos A = b/c = 48/50=0.96.

####

##5##. Finally, we get cos A = ##0.96##.

Definition of Right Triangle

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

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

Cosine Function

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

In the right triangle <note>ABC</note>, side <note>AC</note> is the <note>adjacent side to angle CAB</note>, and side <note>AB is the hypotenuse</note>. Therefore, according to the definition of the cosine function, the <note>cosine of angle CAB</note> is equal to the ratio of the adjacent side <note>AC</note> to the hypotenuse <note>AB</note>, that is, <note>cos(CAB) = AC/AB = 45/50 = 0.96</note>.


Question: Find x.

Answer: 14

1. Given conditions: XT and ZT are secants of the circle, XW=12, WT=7, TY=6.

2. According to the Secant Theorem (also known as the Power of a Point Theorem), TW × XW = TY × YZ.

3. Substitute the known values into the Secant Theorem: 7 × 12 = 6 × x.

4. Calculate to get: x = 14.

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

Secant Line

A straight line that intersects a circle at two distinct points is called a secant line of the circle.

<note>Line XT</note> has two intersection points with <note>Circle O</note>, which are <note>Point W</note> and <note>Point Y</note>. According to the definition of a secant line, <note>Line XT</note> intersects <note>Circle O</note> at two distinct points, so <note>Line XT</note> is a <note>secant line of Circle O</note>.

Secant Line Theorem

In a circle, if two secant lines intersect at a point outside the circle, then the product of the lengths of the external segments and the internal segments of each secant line is equal.

<note>Circle O</note> has two <note>secant lines XT and ZT</note>, which intersect at <note>a point T outside the circle</note>. The external part of secant line XT is <note>WT (7)</note>, and the internal part is <note>XW (12)</note>; the external part of secant line ZT is <note>TY (6)</note>, and the internal part is <note>ZY (x)</note>. According to the <note>Secant Line Theorem</note>, the product of the external part and the internal part of secant line XT is equal to the product of the external part and the internal part of secant line TZ, that is, <note>7 * 12 = 6 * x</note>.


Question: If pentagon ABCDE is similar to pentagon PQRST, find SR.

Answer: \frac { 48 } { 11 }

1. The given condition is that pentagon ABCDE is similar to pentagon PQRST. According to the definition of similar figures, if two figures are similar, then corresponding angles are equal and corresponding sides are proportional.

2. Based on the similarity of pentagon ABCDE and PQRST, let the similarity ratio be k. Then AE corresponds to PT, i.e., AE / PT = k. According to the image data, the length of AE is 11, and the length of PT is 6.

3. According to the similarity ratio k = AE / PT = 11 / 6.

4. To find the length of side SR, the corresponding side is CD. According to the definition of similar figures, CD / SR = k. According to the image data, the length of CD is 8.

5. Substituting k = 11 / 6 and CD = 8, we can find the length of SR as SR = CD / k = 8 / (11 / 6).

6. After the above calculations, the length of SR is 48 / 11.

五边形; 普通多边形; 位似

Definition of Similar Figures

Two geometric figures are similar if and only if their corresponding sides are proportional, and their corresponding angles are equal.

In the figure of this problem, pentagon <note>A B C D E</note> and pentagon <note>P Q R S T</note> are similar. According to the definition of similar figures, the ratios of their corresponding sides are equal, i.e., <note>AB/PQ = BC/QR = CD/RS = DE/ST = EA/TP</note>. At the same time, their corresponding angles are also equal, i.e., <note>∠A = ∠P</note>, <note>∠B = ∠Q</note>, <note>∠C = ∠R</note>, <note>∠D = ∠S</note>, <note>∠E = ∠T</note>.


Question: As shown in the figure, a∥b, point B is on line b, and AB⊥BC, ∠1=36°, then ∠2=()

Answer: 54°

1. Let AB intersect line a at point A, and BC intersect line a at point C. Given that line AB is perpendicular to line BC, according to the definition of perpendicular lines, ∠ABC = 90°. According to the definition of a right triangle, △ABC is a right triangle.

2. Because a ∥ b, according to the definition of alternate interior angles, ∠BCA and ∠1 are alternate interior angles. According to Parallel Postulate 2, ∠BCA = ∠1 = 36°. In Rt△ABC, according to the property of complementary acute angles in a right triangle, ∠BCA + ∠CAB = 90°, that is, 36° + ∠CAB = 90°.

3. Therefore, ∠CAB = 90° - 36° = 54°.

4. According to the definition of vertical angles, ∠2 = ∠CAB = 54°.

5. Through the above reasoning, the final answer is 54°.

平行线; 同位角; 直角三角形

Definition of Right Triangle

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

<note>In the right triangle ABC, angle ABC is a right angle (90 degrees)</note>, <note>angle CAB and angle BCA are the two acute angles other than the right angle</note>, according to the complementary property of acute angles in a right triangle, <note>the sum of angle CAB and angle BCA is 90 degrees</note>, that is, <note>angle CAB + angle BCA = 90°</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 a and b are intersected by a third line BC</note>, forming a geometric relationship, <note>the alternate interior angles ∠BCA and ∠1 are equal, i.e., ∠BCA = ∠1=36°</note>.

Definition of Vertical Angles

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

In the figure of this problem, two intersecting lines <note>a and AB intersect at point A</note>, forming angles: <note>∠2, ∠CAB</note>. According to the definition of vertical angles, <note>∠2 and ∠CAB are vertical angles</note>, and since vertical angles are equal, <note>∠2 = ∠CAB##。##</note>

Definition of Alternate Interior Angles

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

<note>Two parallel lines a and b</note> are intersected by a line <note>BC</note>, where <note>∠1 and ∠BCA are located between the two parallel lines and on opposite sides of the transversal BC</note>, therefore <note>∠BCA and ∠1 are alternate interior angles</note>.

Definition of Perpendicular Lines

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

The lines <note>AB</note> and <note>BC</note> intersect to form an angle <note>∠ABC is 90 degrees</note>, so according to the <note>Definition of Perpendicular Lines</note>, <note>line AB</note> and <note>line BC</note> are perpendicular to each other.


Question: The diameters of circles \odot A, \odot B, and \odot C are 10, 30, and 10 units respectively. If A Z \cong C W and C W = 2, find Z X.

Answer: 3

1. According to the problem, the diameters of circle A, circle B, and circle C are 10, 30, and 10 units respectively. As shown in the figure, the diameter of circle B intersects at point X on circle A and point Y on circle C, the diameter of circle A intersects at point Z on circle B, and the diameter of circle C intersects at point W on circle B. From the figure, it can be seen that points A, Z, X, B, Y, W, and C are collinear.

2. Given that CW = 2 and AZ ≌ CW, it can be deduced that AZ = 2.

3. Since AX is the radius of circle A, and given that the diameter of circle A is 10, AX = 1/2 * 10 = 5. Also, since AZ + ZX = AX and AZ = 2, ZX is the length of the radius of circle A minus the length of segment AZ, i.e., ZX = 5 - 2 = 3.

4. According to the above reasoning process, the length of ZX is 3.

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 AX is a part of a straight line, including endpoint A and endpoint X and all points between them.</note><note>Line segment AX has two endpoints, which are A and X respectively, and every point on line segment AX is located between endpoint A and endpoint X.</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>Point A is the center of the circle</note>, <note>Point X is any point on the circle</note>, <note>Line segment AX is the line segment from the center to any point on the circle</note>, therefore <note>Line segment AX is the radius of the 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 circle A is 10</note>, which is twice the length of the radius, i.e., <note>10 = 2*radius</note>.


Question: Find x.

Answer: 12.75

1. Given that points D, E, and C belong to the same circle, and the line CE is the diameter, according to the corollary of the inscribed angle theorem, the inscribed angle subtended by the diameter is a right angle, thus ∠CDE is 90°.

2. Given that ∠DEC, ∠CDE, and ∠ECD are the three interior angles of △CDE, applying the triangle angle sum theorem, we get ∠CDE + ∠ECD + ∠DEC = 180°.

3. ∠DEC is given as 90°, based on the previous step, we get ∠DEC + ∠ECD = 180° - ∠CDE = 180° - 90° = 90°.

4. The problem states that ∠ECD = (5x - 12)° and ∠DEC = 3x°.

5. Based on the result of step 3, using the substitution method, we get 3x + (5x - 12) = 90°.

6. Simplify the equation 3x + 5x - 12 = 90, resulting in 8x - 12 = 90.

7. Move -12 to the other side, obtaining 8x = 90 + 12.

8. After performing the addition, we get 8x = 102.

9. Divide both sides of the equation by 8, solving for x = 102 ÷ 8.

圆; 圆内接四边形; 圆周角

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.

The vertex <note>D</note> of the angle <note>CDE</note> is on the circumference, and the two sides of the angle <note>CDE</note> intersect the circle at points E and <note>C</note> respectively. Therefore, the angle <note>CDE</note> is an inscribed angle.

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.

The angle subtended by the diameter <note>CE</note> is a right angle (90 degrees) <note>∠CDE</note>.

Triangle Angle Sum Theorem

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

<note>Angle DEC</note>, <note>Angle CDE</note>, and <note>Angle ECD</note> are the three interior angles of <note>Triangle CDE</note>. According to the Triangle Angle Sum Theorem, <note>Angle DEC + Angle CDE + Angle ECD = 180°</note>.


Question: Use the Pythagorean theorem to find the length of the hypotenuse of the right triangle.

Answer: 13

1. Let the three vertices of the triangle be A, B, C. From the given conditions, ∠ACB is a right angle. According to the definition of a right triangle, △ABC is a right triangle. BC is 5 feet, AC is 12 feet, and the length of the hypotenuse AB is required.

2. According to the Pythagorean theorem, the formula is: In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. That is, AB^2 = AC^2 + BC^2.

3. Substitute the given side lengths into the Pythagorean theorem formula: AB^2 = 12^2 + 5^2.

4. Calculate to get: AB^2 = 144 + 25.

5. Further calculation: AB^2 = 169.

6. Find the length of AB: AB = √169.

7. The calculation result is: AB = 13.

8. Finally, the length of the hypotenuse AB is 13 feet.

Definition of Right Triangle

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

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

Pythagorean Theorem

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

<note>In the figure of this problem, the legs of the right triangle are 5 feet and 12 feet respectively</note>, <note>the hypotenuse is AB</note>. According to the Pythagorean Theorem, <note>the square of the hypotenuse is equal to the sum of the squares of the two legs</note>, that is, <note>AB^2 = 5^2 + 12^2 = 25 + 144 = 169</note>. Therefore, the hypotenuse <note>AB's length is √169 = 13 feet</note>.


Question: As shown in the figure, given that AB and AD are chords of ⊙O, ∠BOD=50°, then the degree of ∠BAD is ()

Answer: 25°

1. Given AB and AD are chords of circle O, ∠BOD=50°. O is the center of the circle, then according to the definition of central angle, ∠DOB is a central angle; according to the definition of inscribed angle, ∠BAD is an inscribed angle.

2. On circle O, arc BD corresponds to the central angle ∠BOD and the inscribed angle ∠BAD. According to the inscribed angle theorem, ∠BAD=1/2 * ∠DOB

3. Substitute the given data into the formula, i.e., ∠BAD = 1/2 * 50°.

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

圆; 弦; 圆心角; 圆周角

Definition of Arc

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

<note>In the diagram of this problem, there are two points B and D on circle O</note>, arc BD is a segment of the curve connecting these two points. According to the definition of arc, <note>arc BD is a segment of the curve between two points B and D on the circle</note>.

Definition of Inscribed Angle

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

Definition of Central Angle

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

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

Inscribed Angle Theorem

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

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


Question: Using parallelogram J K L M, find m \angle J M L.

Answer: 109

1. Given that shape J K L M is a parallelogram, it can be concluded that ∠J K L and ∠J M L are opposite angles. According to the properties of a parallelogram, opposite angles are equal, i.e., ∠J K L = ∠J M L.

2. Given that ∠J K L is equal to 109°, therefore, based on the property of equal opposite angles, it can be concluded that ∠J M L is also equal to 109°.

3. Based on the above reasoning, the measure of ∠J M L is 109°.

Definition of Parallelogram

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

In the figure of this problem, the quadrilateral <note>J K L M</note> is a parallelogram, with side <note>JK parallel and equal to side LM</note>, and side <note>JM parallel and equal to side KL</note>.

Properties of Parallelogram Theorem

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

In the figure of this problem, in the parallelogram J K L M, <note>the opposite angles ∠J K L and ∠J M L are equal</note>, <note>the opposite angles ∠K J M and ∠K L M are equal</note>; the sides <note>J K and L M are equal</note>, <note>J M and K L are equal</note>; <note>the diagonals J L and K M bisect each other</note>, <note>that is, the intersection point divides the diagonal J L into two equal segments, and the diagonal K M into two equal segments</note>.


Question: As shown in the figure, it is known that points A, B, and C are on ⊙O, point C is on the minor arc AB, and ∠AOB=130°, then the degree of ∠ACB is ()

Answer: 115°

1. As shown in the figure, take a point D on the major arc AB, and connect AD, BD. ##According to the definition of the inscribed angle, ∠ADB is an inscribed angle; according to the definition of the central angle, ∠AOB is a central angle.##

2. Given ∠AOB=130°, according to the inscribed angle theorem, ##we can get ∠ADB=1/2∠AOB=1/2 * 130°=65°##.

3. ##Quadrilateral ADBC is a cyclic quadrilateral of ⊙O, so according to (corollary 3 of the inscribed angle theorem) the cyclic quadrilateral opposite angle supplementary theorem, we can get: ∠ADB+∠ACB=180°.##

4. ##Given ∠ADB=65°, so ∠ACB=180°-65°=115°.##

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

圆; 圆心角; 圆周角; 弧; 弦

Definition of Arc

<note>Arc ACB</note> is a segment of a curve connecting these two points. According to the <note>Definition of Arc</note>, <note>Arc ACB</note> is a segment of a curve between two points <note>A and B</note> on a circle.

Definition of Inscribed Angle

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

Definition of Central Angle

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

<note>All four vertices A, B, C, and D of quadrilateral ADBC lie on the same circle (⊙O)</note>. This circle is called the circumcircle of quadrilateral ADBC. Therefore, quadrilateral ADBC is a cyclic quadrilateral. According to the properties of cyclic quadrilaterals, <note>the sum of opposite angles is equal to 180 degrees, ##i.e., ∠ADB + ∠ACB = 180 degrees##</note>.

Inscribed Angle Theorem

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

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

In the figure of this problem, the quadrilateral ADBC is inscribed in ⊙O. According to the Diagonal Supplementary Theorem for Cyclic Quadrilateral, <note>the sum of each pair of opposite angles of quadrilateral ADBC is equal to 180°</note>. Specifically, <note>∠ADB + ∠ACB = 180°</note>.


Question: As shown in the figure, ⊙O is the circumcircle of △ABC. Connect OA and OB, ∠AOB=50°, then the degree of ∠C is ()

Answer: 25°

1. Given that circle O is the circumcircle of triangle ABC, ∠AOB=50°.

2. According to the definition of inscribed angle and central angle, ∠ACB is an inscribed angle and ∠AOB is a central angle. According to the inscribed angle theorem, in circle O, the central angle is equal to twice the inscribed angle subtended by the same arc.

3. Therefore, ∠ACB is half of ∠AOB, i.e., ∠ACB=1/2 × 50° = 25°.

圆; 圆心角; 圆周角; 普通三角形; 圆内接四边形

Circumcircle of a Triangle

The circumcircle of a triangle is a circle that passes through all three vertices of the triangle.

The three vertices of <note>triangle ABC</note> are <note>point A, point B, and point C</note>. A circle can be drawn through these three vertices, which is called the <note>circumcircle of triangle ABC</note>. The <note>center O</note> is the <note>center of the circumcircle of triangle ABC</note>, and <note>points A, B, and C are all on this circle</note>. Therefore, <note>the radius of circle O is equal to OA, OB, and OC</note>, that is, <note>OA=OB=OC</note>.

Definition of Central Angle

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

Inscribed Angle Theorem

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

Definition of Inscribed Angle

In the diagram of this problem, in circle O, the vertex of angle ACB (point C) is on the circumference, and the two sides of angle AOB intersect circle O at point A and point B respectively. Therefore, angle ACB is an inscribed angle.


Question: Find the value of x, accurate to two decimal places.

Answer: 43.86

1. Let the vertex of the 70° angle be A, the vertex of the 90° angle be B, and the vertex of the last angle be C.

2. In the right triangle ABC, given the hypotenuse is x and the length of the adjacent leg is 15, we can use the cosine theorem for calculation. The cosine theorem states: In a right triangle, cos(∠CAB) = adjacent leg / hypotenuse, i.e., cos(70°) = AB / AC.

3. By substituting the known values, we get cos(70°) = 15 / x.

4. According to the trigonometric table or calculator, the value of cos(70°) is approximately 0.3420.

5. Substituting the value of cos(70°) into the equation, we get 0.3420 = 15 / x.

6. By cross-multiplying, we get x = 15 / 0.3420.

7. Dividing 15 by 0.3420, we get x ≈ 43.86.

8. After the above calculations and verification, the final answer is x ≈ 43.86.

直角三角形; 正弦; 余弦; 正切

Definition of Right Triangle

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

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

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.

In the <note>right triangle ABC</note>, side <note>AB is the adjacent side to angle BAC</note>, and side <note>AC is the hypotenuse</note>. Therefore, according to the definition of the cosine function, <note>the cosine of angle BAC is equal to the ratio of the adjacent side AB to the hypotenuse AC</note>, that is, <note>cos(∠BAC) = AB / AC</note>.


Question: Find the value of x.

Answer: 163

1. Given conditions: There is a central angle x° within a circle, and the other two angles are 65° and 132°.

2. Let the center of the circle be point O, and the three points on the circle connected to the center be points A, B, and C (from the leftmost side in a clockwise direction), so ∠AOB=65°, ∠AOC=132°, ∠BOC=x°. According to the definition of a full angle, it satisfies ∠AOB+∠AOC+∠BOC= 65° + 132° + x° = 360°.

3. Solve the equation: 65° + 132° + x° = 360°, which becomes 197° + x° = 360°.

4. Solve for x°, obtaining x° = 360° - 197°.

5. After calculation, we get x° = 163°.

6. Through the above reasoning steps, confirm that the value of x° is 163°.

Definition of Full Angle

An angle formed when a ray revolves around its endpoint for one complete rotation is called a full angle.

In the figure of this problem, <note>ray OA around the center of the circle</note> rotates a full circle, forming a <note>full angle</note>. According to the <note>definition of full angle</note>, <note>the degree of a full angle is 360 degrees</note>, that is, <note>full angle 65° + 132° + x° = 360°</note>.


Question: As shown in the figure, given OA=OB=OC and ∠ACB=30°, then the measure of ∠AOB is ()

Answer: 60°

1. Given OA=OB=OC, ##draw ⊙O, with O as the center and OA as the radius to draw a circle. According to the definition of a circle##, it is concluded that A, B, and C are on the circle with O as the center and OA as the radius.

2. ##According to the definition of the inscribed angle, ∠ACB is an inscribed angle and equals 30°. According to the definition of the central angle, ∠AOB is a central angle##.

3. ####According to the inscribed angle theorem,## ∠ACB=?∠AOB##.

4. ##It is calculated that## ∠AOB=2∠ACB.

5. ##Because## ∠ACB=30°, it is concluded that ∠AOB=2×30°=60°.

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

等腰三角形; 圆心角; 圆周角

Definition of Circle

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

<note>Construct ⊙O, with O as the center and OA as the radius to draw a circle</note>, all points in the figure that are at a distance equal to OA from point O are on circle O. <note>Because OA=OB=OC, points A, B, and C are on the circle with O as the center and OA as the radius</note>.

Definition of Inscribed Angle

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

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

Definition of Central Angle

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

Inscribed Angle Theorem

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


Question: Find BC.

Answer: 6

1. From the parallel markings in the figure, we know AE∥CD. According to Parallel Postulate 2 and the definition of alternate interior angles, we have: ∠EAB=∠DCB, ∠AEB=∠CDB.

2. According to the AA criterion for similar triangles, the corresponding angles in triangles AEB and CDB are equal, i.e., ∠EAB=∠DCB, ∠AEB=∠CDB. Therefore, triangle AEB is similar to triangle CDB.

3. According to the definition of similar triangles, the corresponding sides are proportional, so BD/BE=BC/BA. From the figure, we know BD=3, BE=5, BC=2x-8, BA=x+3. Substituting the known values: (2x-8)/(x+3)=3/5.

4. Cross-multiplying gives: 3*(x+3)=5*(2x-8). Simplifying the equation: 3x+9=10x-40.

5. Finally, simplifying to: 7x=49, solving gives: x=7.

6. Substituting the obtained value of x: 2x-8, so BC=2*7-8=6.

等边三角形; 线段; 对顶角; 三角形的外角

Definition of Triangle

In the figure of this problem, <note>triangle ABE</note> is a geometric figure formed by <note>three non-collinear points A, B, E</note> and their connecting <note>line segments AE, AB, BE</note>. <note>Points A, B, E are the three vertices of the triangle</note>, and <note>line segments AE, AB, BE are the three sides of the triangle</note>. <note>Triangle CDB</note> is a geometric figure formed by <note>three non-collinear points C, D, B</note> and their connecting <note>line segments CD, DB, CB</note>. <note>Points C, D, B are the three vertices of the triangle</note>, and <note>line segments CD, DB, CB 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.

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

In the figure of this problem, <note>two parallel lines AE and CD are intersected by a third line AC</note>, forming the following geometric relationships:
1. Corresponding angles: None.
2. Alternate interior angles: <note>angle EAB and angle DCB are equal</note>.
3. Same-side interior angles: None.
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.
<note>Two parallel lines AE and CD are also intersected by a fourth line DE</note>, forming the following geometric relationships:
1. Corresponding angles: None.
2. Alternate interior angles: <note>angle AEB and angle CDB are equal</note>.
3. Same-side interior angles: None.
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.

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 diagram of this problem, <note>two parallel lines AE and CD are intersected by a line AC</note>, where <note>angle EAB and angle DCB are between the two parallel lines and on opposite sides of the intersecting line AC</note>, therefore <note>angle EAB and angle DCB are alternate interior angles</note>. <note>Alternate interior angles are equal, i.e., angle EAB is equal to angle DCB</note>. <note>Two parallel lines AE and CD are also intersected by a line DE</note>, where <note>angle AEB and angle CDB are between the two parallel lines and on opposite sides of the intersecting line DE</note>, therefore <note>angle AEB and angle CDB are alternate interior angles</note>. <note>Alternate interior angles are equal, i.e., angle AEB is equal to angle CDB</note>.

Similarity Theorem for Triangles (AA)

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

The original text: <note>Angle AEB is equal to angle CDB</note>, and <note>angle EAB is equal to angle DCB</note>, therefore <note>triangle AEB is similar to triangle CDB</note>.

Definition of Similar Triangles

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

<note>△ABE</note> and <note>△BDC are similar triangles</note>. According to the definition of similar triangles: <note>∠EAB = ∠DCB</note>, <note>∠AEB = ∠CBD</note>, <note>∠ABE = ∠CBD</note>; <note>AB/CB = EB/DB = AE/DC</note>.


Question: Given \odot A, \odot B, and \odot C have diameters of 10, 30, and 10 units respectively. If A Z \cong C W and C W = 2, find Y W.

Answer: 3

1. From the problem, we know that the diameters of circle A, circle B, and circle C are 10, 30, and 10 units respectively. As shown in the figure, the diameter of circle B intersects the circle at points Z and W, and the diameter of circle C intersects circle B's diameter at point Y. From the figure, we can see that points Y, W, and C are collinear.

3. Since CY is the radius of circle C, and the diameter of circle C is known to be 10, therefore CY = 5. Also, since CW + YW = CY and CW = 2, YW is the length of circle C's radius minus the length of segment CW, i.e., YW = 5 - 2 = 3.

4. According to the above reasoning process, the length of YW is 3.

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>\odot A's diameter is 10</note>, so <note>A's radius AX = 5</note>; <note>\odot B's diameter is 30</note>, so <note>B's radius BX = 15</note>; <note>\odot C's diameter is 10</note>, so <note>C's radius CY = 5</note>.

Definition of Radius

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

In the figure of this problem, in circle A, <note>point A is the center of the circle</note>, <note>point X is any point on the circle</note>, <note>line segment AX is the line segment from the center to any point on the circle</note>, therefore <note>line segment AX is the radius of circle A</note>. In circle B, <note>point B is the center of the circle</note>, <note>point X is any point on the circle</note>, <note>line segment BX is the line segment from the center to any point on the circle</note>, therefore <note>line segment BX is the radius of circle B</note>. In circle C, <note>point C is the center of the circle</note>, <note>point Y is any point on the circle</note>, <note>line segment CY is the line segment from the center to any point on the circle</note>, therefore <note>line segment CY is the radius of circle C</note>.

Definition of Diameter

<note>\odot A's diameter is 10</note>, therefore A's radius <note>AX = 5</note>; <note>\odot B's diameter is 30</note>, therefore B's radius <note>BX = 15</note>; <note>\odot C's diameter is 10</note>, therefore C's radius <note>CY = 5</note>.


Question: Find BC.

Answer: 9

1. In △ABC, it is known that AC = 9, BC = 9, ∠A = 60°.

2. According to the definition of isosceles triangle, it is known that △ABC is an isosceles triangle.

3. According to the theorem of equilateral triangle determination (60-degree angle in isosceles triangle), it is known that △ABC is an equilateral triangle.

4. According to the definition of equilateral triangle, it is known that BC = AB = AC = 9.

5. Finally, it is concluded that BC = 9.

等边三角形; 等腰三角形; 普通三角形

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>△ABC is an equilateral triangle</note>, therefore <note>AB = BC = AC = 9</note>. And <note>∠BAC = ∠ABC = ∠BCA = 60°</note>.

Definition of Isosceles Triangle

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

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

In the figure of this problem, it is known that △ABC is an isosceles triangle, <note>side AB is equal to side AC</note>, and <note>there is an interior angle of 60°, i.e., ∠A=60°</note>. According to the Equilateral Triangle Identification Theorem, <note>if an isosceles triangle has an interior angle of 60°, then the lengths of its three sides are equal, and all three interior angles are 60°</note>, therefore it can be determined that <note>△ABC is an equilateral triangle</note>.


Question: Circles G, J, and K all intersect at L. If G H = 10, find F H.

Answer: 20

1. Given conditions: circles G, J, K intersect at point L. It is known that G H = 10.

2. ##Since FH is the diameter of circle G, therefore F H = 2 * GH = 20.##

####

##3##. Answer the question: F H equals 20.

Definition of Radius

<note>Point G is the center of the circle</note>, <note>Point H is any point on the circle</note>, <note>Line segment GH is the line segment from the center of the circle to any point on the circle</note>, therefore <note>Line segment GH is the radius of circle G</note>.

Definition of Diameter

<note>FH is the diameter</note>, connecting <note>the center G</note> and <note>points F and H on the circumference</note>, with a length of <note>2 times the radius</note>, i.e., <note>FH = 2*GH=20</note>.


Question: As shown in the figure, in △ABC, AB=AC, ∠A=36°, the perpendicular bisector of AB, DE, intersects AC at D and intersects AB at E. Then the measure of ∠BDC is ()

Answer: 72°

1. Given DE is the perpendicular bisector of AB, according to the definition of the perpendicular bisector, DE is the perpendicular bisector of AB.

2. According to the property of the perpendicular bisector, point D is on the perpendicular bisector DE, thus DA = DB.

3. ∠BDC is an exterior angle of triangle ABD, according to the exterior angle theorem, ∠BDC = ∠DAB + ∠DBA.

4. In triangle ABD, DA = DB, according to the definition of an isosceles triangle, triangle ABD is an isosceles triangle. Furthermore, according to the property of an isosceles triangle, ∠DAB = ∠DBA.

5. Therefore, ∠BDC = 2∠DAB. Given ∠A = 36°, thus ∠BDC = 2×36° = 72°.

6. Through the above reasoning, the final answer is 72°.

等腰三角形; 垂直平分线; 对顶角; 垂线; 三角形的外角

Definition of Isosceles Triangle

In the diagram of this problem, in triangle ABC, <note>side AB and side AC are equal</note>, therefore triangle ABC is an isosceles triangle. In triangle ABD, <note>side DA and side DB are equal</note>, therefore triangle ABD is an isosceles triangle.

Definition of Perpendicular Bisector

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

In the figure of this problem, <note>line DE passes through the midpoint E of segment AB</note>, and <note>line DE is perpendicular to segment AB</note>. Therefore, line DE is <note>the perpendicular bisector of segment AB</note>.

Properties of Isosceles Triangle

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

In the figure of this problem, <note>in the isosceles triangle ABC, sides AB and AC are equal</note>. Therefore, according to the properties of isosceles triangles, <note>the angles opposite the equal sides are equal, that is, angle ABC = angle ACB</note>. <note>In the isosceles triangle ABD, sides DB and DA are equal</note>. Therefore, according to the properties of isosceles triangles, <note>the angles opposite the equal sides are equal, that is, angle DAB = angle DBA</note>.

Properties of the Perpendicular Bisector

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

In the figure of this problem, the perpendicular bisector of segment AB is <note>line DE</note>, and <note>points D and E are on line DE</note>. According to the properties of the perpendicular bisector, <note>the distance from point D to both endpoints A and B of segment AB is equal, i.e., AD = BD</note>; <note>the distance from point E to both endpoints A and B of segment AB is equal, i.e., EA = EB</note>.

Exterior Angle Theorem of Triangle

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

In the figure of this problem, in triangle ABD, <note>angle BDC is an exterior angle of the triangle</note>, <note>angle DBA and angle DAB are the two interior angles not adjacent to the exterior angle BDC</note>. According to the Exterior Angle Theorem of Triangle, <note>the exterior angle BDC is equal to the sum of the two non-adjacent interior angles DBA and DAB</note>, that is, <note>angle BDC = angle DBA + angle DAB</note>.

Definition of Exterior Angle of a Polygon

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

In the diagram of this problem, an interior angle of polygon ABD is <note>∠BDA</note>, and the angle formed by extending the adjacent sides <note>BD</note> and <note>AD</note> of this interior angle is called the exterior angle of the interior angle <note>∠BDA</note>.


Question: Find x.

Answer: 58

1. The given condition is: ##the triangle has a right angle, so according to the definition of a right triangle, the triangle is a right triangle##, with one leg of the right angle being 16 and the other leg of the right angle being 10. The size of angle x is required.

2. According to the ##definition of the tangent function##: In a right triangle, the tangent function tan(angle) is equal to the ratio of the length of the opposite side to the length of the adjacent side. For angle x, the opposite side is 16 and the adjacent side is 10.

3. Therefore, tan(x) = 16/10 = 1.6.

4. ##According to the table, tan58°≈1.6003, so x=58##.

直角三角形; 正弦; 余弦; 正切; 三角形的外角

Definition of Right Triangle

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

In the figure of this problem, <note>one of the interior angles of the triangle is 90 degrees</note>, so the triangle is a right triangle. <note>The sides with lengths 16 and 10 are the legs of the right angle</note>.

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>Angle x is an acute angle</note>, <note>the side with a length of 16 is the opposite side to angle x</note>, <note>the side with a length of 10 is the adjacent side to angle x</note>, so <note>the tangent value of angle x is equal to the length of the side with a length of 16 divided by the length of the side with a length of 10</note>, i.e., ##tan(x) = 16 / 10 = 1.6##.