Question: Solve for x.

Answer: 14

1. Let the three vertices of the triangle be A, B, and C. Given ∠CAB=30° and ∠ACB=90°. According to the definition of a right triangle, △ABC is a right triangle.

2. According to the triangle angle sum theorem, ∠ABC=180°-∠BAC-∠ACB=180°-30°-90°=60°.

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

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

直角三角形; 正弦; 余弦

Triangle Angle Sum Theorem

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

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

Definition of Right Triangle

In the figure of this problem, <note>in triangle ABC, angle ACB is a right angle (90 degrees)</note>, so this triangle is a right triangle. <note>Side 14√3 and side x are the legs, side y is the hypotenuse</note>.

Properties of 30°-60°-90° Triangle

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

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


Question: We know that if the sum of two acute angles equals one right angle, then these two angles are complementary angles, abbreviated as complementary. As shown in the figure, ∠A and ∠B are complementary, and we have: sinA=rac{opposite side of ∠A}{hypotenuse}=rac{a}{c}, 	ext{cos}B=rac{adjacent side of ∠B}{hypotenuse}=rac{a}{c}, therefore we know sinA=	ext{cos}B. Notice that in △ABC, ∠A+∠B=90°, that is, ∠B=90°-∠A, ∠A=90°-∠B, thus we have: sin(90°-A)=	ext{cos}A, 	ext{cos}(90°-A)=sinA. Try to complete the following multiple-choice question: If α is an acute angle, and 	ext{cos}α=rac{4}{5}, then the value of sin(90°-α) equals ()

Answer: \frac{4}{5}

1. Given ∠α is an acute angle, and cosα=##4/5##.

2. According to the property of complementary angles, let the complementary angle of ∠α be ∠B, then ∠B=90°-∠α.

3. ##According to the cosine function, cos(∠α)=b/c. According to the sine function, sin(∠B)=b/c, so cos(∠α)=sin(∠B). Given ∠B=90°-∠α, then cos(∠α)=sin(90°-∠α)##.

4. Substitute the given condition cosα=##4/5## into the above equation, we get sin(90°-∠α)=##4/5##.

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

直角三角形; 正弦; 余弦

Definition of Sine Function

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

<note>In the figure of this problem, in the right triangle ABC,</note> angle B is an acute angle, <note>side AC is the opposite side of angle B, and side AB is the hypotenuse</note>. According to the definition of the sine function, <note>the sine value of angle B is equal to the ratio of the opposite side AC to the hypotenuse AB, i.e., sin(B) = AC / AB = b / c.</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 <note>ABC</note>, side <note>AC</note> is the adjacent side of angle <note>∠α</note>, and side <note>AB</note> is the hypotenuse. Therefore, according to the definition of the cosine function, the cosine value of angle <note>∠α</note> is equal to the ratio of the adjacent side <note>AC</note> to the hypotenuse <note>AB</note>, that is, cos(∠α) = AC / AB = <note>b</note> / <note>c</note>.


Question: As shown in the figure, points A, B, and C are on the circle. In △ABC, ∠ABC=70°, ∠ACB=30°, D is the midpoint of arc BAC. Connect DB and DC, then the degree of ∠DBC is ()

Answer: 50°

1. In △ABC, it is known that ∠ABC=70°, ∠ACB=30°. According to the triangle angle sum theorem, we get ∠BAC=180°-∠ABC-∠ACB=180°-70°-30°=80°.

2. Because D is the midpoint of arc BAC, ##arc BD=arc CD. From the figure, it can be seen that ∠DCB is the inscribed angle of arc BD, and ∠DBC is the inscribed angle of arc CD. Given arc BD=arc CD, then ∠DCB=∠DBC##.

3. ##According to the definition of an isosceles triangle, given ∠DCB=∠DBC, then triangle BDC is an isosceles triangle##.

####

##4. According to the triangle angle sum theorem, the sum of the three interior angles of any triangle is 180°. In triangle BDC, ∠DBC+∠BDC+∠DCB=180°##, because according to the inscribed angle theorem corollary 1, we have: ∠BDC=∠BAC=80°. Then ##∠DBC=(1/2)×(180°-∠BDC)= (1/2)×(180°-80°)=50°##.

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

圆; 圆周角; 弧; 中点; 等腰三角形

Definition of Inscribed Angle

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

<note>In circle O, the vertex of angle BAC (point A) is on the circumference</note>, and the two sides of angle BAC intersect circle O at <note>point B and point C</note>. Therefore, <note>angle BAC is an inscribed angle</note>. <note>In circle O, the vertex of angle BDC (point D) is on the circumference</note>, and the two sides of angle BDC intersect circle O at <note>point B and point C</note>. Therefore, <note>angle BDC is an inscribed angle</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 three non-collinear points <note>A, B, C</note> and their connecting line segments <note>AC, AB, BC</note>. <note>Points A, B, C</note> are the three vertices of the triangle, and <note>line segments AC, AB, BC</note> are the three sides of the triangle. <note>Triangle BDC</note> is a geometric figure composed of three non-collinear points <note>D, B, C</note> and their connecting line segments <note>BC, BD, DC</note>. <note>Points D, B, C</note> are the three vertices of the triangle, and <note>line segments BC, BD, DC</note> are the three sides of the triangle.

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 BDC, <note>side BD and side CD are equal</note>, therefore <note>triangle BDC is an isosceles triangle</note>.

Triangle Angle Sum Theorem

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

Properties of Isosceles Triangle

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

In the <note>isosceles triangle DBC</note>, <note>side DB and side DC are equal</note>. Therefore, according to the properties of isosceles triangles, <note>the angles opposite the equal sides are equal</note>, that is, <note>angle DBC = angle DCB</note>.

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

<note>In circle O, the inscribed angles ∠BAC and ∠BDC corresponding to arc BC are equal</note>. According to Corollary 1 of the Inscribed Angle Theorem, <note>the inscribed angles ∠BAC and ∠BDC corresponding to the same arc BC are equal</note>, i.e., <note>∠BAC=∠BDC</note>.


Question: For a pair of similar figures, use the given area to find x.

Answer: 8

1. Given that the two figures are similar figures, and the area of the first figure ##ABCD (counterclockwise vertex order A is the top left vertex)## is 8 square inches, the area of the second figure ##EFGH (counterclockwise vertex order E is the top left vertex)## is 32 square inches.

2. ##From the figure, the area ratio of the first figure to the second figure## is 8:32, which is 1:4.

3. ##According to the theorem that the area ratio of similar polygons is equal to the square of the similarity ratio:## the square of the length ratio is 1:4, that is (ratio)^2 = 1/4.

4. To solve the corresponding length ratio, we need to take the square root, so the length ratio is 1:2.

5. From the given conditions, the corresponding side length of the first figure is 4 inches. Using the length ratio of 1:2, the corresponding side x of the second figure satisfies 4: x = 1:2.

6. By proportion calculation, x = 4 × 2 = 8.

7. Through the above reasoning, the final answer for the length x of the second figure is 8 inches.

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>In the figure of this problem, figures ABCD and EFGH are similar figures</note>. According to the definition of similar figures, <note>the ratios of their corresponding sides are equal, that is, the ratio of side AD to side EH is equal to the ratio of side AB to side EF</note>.

Area Ratio Theorem of Similar Polygons

If the ratio of the side lengths of two similar polygons is k, then the ratio of their areas is equal to k squared (k²).

<note>Polygons ABCD and EFGH are similar figures</note>, <note>the ratio of side AD to side EH is 4/x, which means the similarity ratio is 4/x</note>. Therefore, <note>the ratio of the area of figure EFGH to the area of figure ABCD is equal to the square of 4/x</note>. That is: Area ratio = (4/x)².


Question: As shown in the figure, points A, B, and C are three points on ⊙O. If ∠A = 40°, then the degree of ∠BOC is ()

Answer: 80°

1. Given that points A, B, and C are three points on circle O, ∠A = 40°.

2. ##According to the Inscribed Angle Theorem##, in a circle, the central angle is twice the inscribed angle that subtends the same arc. Therefore, ∠BOC = 2 × ∠A.

3. Specifically for this problem, ∠BOC = 2 × 40°.

5. Through the above reasoning, the final answer is 80°.

Definition of Circle

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

<note>In circle O, point O is the center</note>, <note>the radii are OB and OC</note>. All points in the figure that are equidistant from point O as OB and OC are on <note>circle O</note>.

Definition of Inscribed Angle

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

Definition of Central Angle

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

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

Inscribed Angle Theorem

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

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


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

Answer: 180

1. Given that the base length of the parallelogram is 15 inches, and the height is unknown. According to the properties of a parallelogram, the height can be calculated using the properties of a right triangle.

2. Draw an auxiliary line from one vertex perpendicular to the opposite side (as shown in the figure), dividing the parallelogram into a right triangle and a trapezoid. The hypotenuse of the right triangle is 13 inches, and one of the legs is 5 inches.

3. According to the Pythagorean theorem (also known as the Pythagoras theorem), in a right triangle, the sum of the squares of the two legs is equal to the square of the hypotenuse. The formula for the Pythagorean theorem is: ##a^2 + b^2 = c^2##, in this problem c = 13 inches, a = 5 inches, solve for b.

4. Substitute the known values into the Pythagorean theorem formula: ##5^2 + b^2 = 13^2##, calculate to get 25 + b^2 = 169, further calculate to get ##b^2## = 144, thus b = √144 = 12 inches.

5. Through the above calculations, the height is found to be 12 inches.

6. According to the area formula for a parallelogram: Area = base × height, the area of this parallelogram is 15 inches × 12 inches = 180 square inches.

7. The final answer is that the area of the parallelogram is 180 square inches.

平行四边形; 平行线; 直角三角形; 线段

Definition of Parallelogram

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

In the diagram of this problem, the quadrilateral is a <note>parallelogram</note>, with <note>two pairs of opposite sides parallel and equal</note>. The <note>base length is 15 inches</note>, and the <note>opposite side is also 15 inches</note>. The other pair of <note>opposite sides are 13 inches each</note>.

Definition of Right Triangle

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

<note>By drawing an auxiliary line from one vertex of the parallelogram to the opposite side, a right triangle is formed</note>. In this right triangle, <note>one leg is 5 inches long</note>, <note>the other leg is 12 inches long</note>, <note>the hypotenuse is 13 inches long</note>. Therefore, this triangle is a right 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.

In the figure of this problem, in the <note>right triangle</note>, one <note>right side is 5 inches long</note>, the <note>hypotenuse is 13 inches long</note>, and the other <note>right side is the height</note>. According to the Pythagorean Theorem, the <note>square of the hypotenuse is equal to the sum of the squares of the two right sides</note>, i.e., <note>13^2 = 5^2 + height^2</note>. Substituting into the calculation: <note>13^2 = 5^2 + height^2</note> -> <note>169 = 25 + height^2</note> -> <note>height^2 = 144</note> -> <note>height = √144 = 12 inches</note>.

Area Formula of a Parallelogram

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

<note>The length of the base of the parallelogram is 15 inches</note>, and the corresponding <note>height is the vertical distance from the base to the opposite side</note>, denoted as <note>12 inches</note>. Therefore, according to the area formula of the parallelogram, the area of the parallelogram is equal to the base length <note>15 inches</note> multiplied by the corresponding height <note>12 inches</note>, i.e., A = 15 inches × 12 inches.

Height of a Parallelogram

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

In the figure of this problem, in the parallelogram, <note>the top and bottom sides are parallel</note>. From the top vertex, <note>draw a line segment perpendicular to the bottom side</note>, <note>the length of this line segment is the height of the parallelogram</note>.


Question: Find the value of x. Round to the nearest tenth.

Answer: 14.1

1. From the figure, we know the triangle has two equal sides of length x, the hypotenuse is 20, and one angle is a right angle. According to the definition of an isosceles right triangle, the figure is an isosceles right triangle.

2. According to the Pythagorean theorem (Pythagoras' theorem), the sum of the squares of the two legs of an isosceles right triangle is equal to the square of the hypotenuse, we can write the equation: x^2 + x^2 = 20^2.

3. Solving the equation: 2x^2 = 400 => x^2 = 200.

4. Taking the positive value of x (since the side length cannot be negative), we get x = √(200).

5. Further calculation gives x ≈ 14.142, and finally rounding to the nearest tenth as required, x ≈ 14.1.

6. Through the above reasoning, the final answer is x ≈ 14.1.

直角三角形; 等腰三角形; 余弦

Pythagorean Theorem

In an <note>isosceles right triangle</note>, let the two legs be <note>x and x</note>, and the hypotenuse be <note>20</note>. According to the Pythagorean Theorem, <note>x^2 + x^2 = 20^2</note>.

Definition of Isosceles Right Triangle

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

<note>One of the angles is a right angle (90 degrees)</note>, <note>the two equal legs of the right angle are x</note>, and the hypotenuse is 20. <note>According to the definition of an isosceles right triangle</note>, this figure is an isosceles right triangle.


Question: As shown in the figure, AB is the diameter of ⊙O, the measure of ∠ADC is 35°, then the measure of ∠BOC is ()

Answer: 110°

1. Given that the degree of ∠ADC is 35°, according to the Inscribed Angle Theorem, we get ∠AOC = 2 × ∠ADC = 2 × 35° = 70°.

2. Since AB is the diameter, according to the definition of a straight angle, we get ∠AOC + ∠BOC = 180°.

3. From ∠AOC = 70° and ∠AOC + ∠BOC = 180°, we get ∠BOC = 180° - 70° = 110°.

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

Inscribed Angle Theorem

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

Definition of Inscribed Angle

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

In the diagram of this problem, <note>ray x rotates around endpoint x until it forms a straight line with the initial side</note>, forming <note>straight angle AOB</note>. According to the definition of a straight angle, the measure of a straight angle is 180 degrees, i.e., <note>angle AOB = 180 degrees</note>.

Definition of Central Angle

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


Question: In the figure below, angle ABC is intersected by parallel lines l and m. What is the measure of angle ABC? Please express your answer in degrees.

Answer: 71

1. Draw a line z through point B, and line z is parallel to line l. Given that l is parallel to m, according to the transitivity of parallel lines, we can deduce that z is parallel to m.

2. According to the parallel axiom 2 of parallel lines, the alternate interior angles are equal, thus ∠ABC = 38° + 33°.

3. Therefore, ∠ABC = 38° + 33° = 71°.

4. The final value of angle ABC is 71°.

平行线; 同位角; 内错角

Definition of Parallel Lines

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

<note>Draw a line z through point B</note>, and <note>line z is parallel to two parallel lines l and m</note>. <note>Line l and line z lie in the same plane and they do not intersect</note>, so according to the definition of parallel lines, <note>line l and line z are parallel lines</note>. <note>Line m and line z lie in the same plane and they do not intersect</note>, so according to the definition of parallel lines, <note>line m and line z are parallel lines</note>. <note>Line l and line m lie in the same plane and they do not intersect</note>, so according to the definition of parallel lines, <note>line l and line m 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 problem diagram, a <note>line z</note> is drawn through point B, and <note>line z is parallel to the two parallel lines l and m</note>. The two parallel lines <note>l and z are intersected by the third line AB</note>, forming the following geometric relationships: 1. Corresponding angles: none. 2. Alternate interior angles: <note>∠ABz and ∠BAl are equal</note>, i.e., <note>38°</note>. 3. Consecutive interior angles: <note>the transversal supplementary angles of ∠ABz and ∠BAl are supplementary</note>. The two parallel lines <note>z and m are intersected by the third line BC</note>, forming the following geometric relationships: 1. Corresponding angles: none. 2. Alternate interior angles: <note>∠zBC and ∠BCm are equal</note>, i.e., <note>33°</note>. 3. Consecutive interior angles: <note>the transversal supplementary angles of ∠zBC and ∠BCm are supplementary</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>In the figure of this problem, draw a line z through point B, and line z is parallel to two parallel lines l and m</note>. Two parallel lines <note>l and z are intersected by a line AB, where ∠ABz and ∠BAl are located between the two parallel lines and on opposite sides of the intersecting line AB</note>, therefore <note>∠ABz and ∠BAl are alternate interior angles</note>. Alternate interior angles are equal, that is <note>∠ABz and ∠BAl</note>. Two parallel lines <note>z and m are intersected by a line BC, where ∠zBC and ∠BCm are located between the two parallel lines and on opposite sides of the intersecting line BC</note>, therefore <note>∠zBC and ∠BCm are alternate interior angles</note>. Alternate interior angles are equal, that is <note>∠zBC and ∠BCm</note>.

Transitivity of Parallel Lines

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

In the figure of this problem, <note>line m and line z are each parallel to line l</note>. According to the transitivity of parallel lines, if <note>line m is parallel to line l</note>, and <note>line z is also parallel to line l</note>, then <note>line z and line m are parallel to each other</note>. Therefore, <note>line z is parallel to line m</note>.


Question: In \odot R, find T V. Round to two decimal places.

Answer: 18.44

1. Given that R is the center of circle R, points T, U, and V are on circle R, UR is the radius of circle R. According to the definition of a circle, UR = TR = VR; according to the definition of a chord, TV is a chord on circle R; TV is perpendicular to UR and intersects at point S, according to the perpendicular bisector theorem, TS = SV.

2. Given US = 5 and SR = 6, then TR = UR = VR = US + SR = 5 + 6 = 11.

3. Since TV is perpendicular to UR and intersects at point S, according to the definition of a right triangle, triangle TSR is a right triangle. According to the Pythagorean theorem, TS^2 + SR^2 = TR^2, that is, TS^2 + 6^2 = 11^2.

5. Given TS = SV, therefore calculate TV as: TV = 2 * TS = 2 * 9.22 = 18.44.

6. Round the value of TV to the nearest hundredth, the final calculated result is 18.44.

圆; 垂线; 直线; 直角三角形; 弦

Definition of Chord

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

<note>In circle R</note>, <note>point T and point V are any two points on the circle</note>, <note>the line segment TV connects these two points</note>, so <note>the line segment TV is a chord of circle R</note>.

Definition of Circle

In the circle in the diagram of this problem, <note>point R is the center of the circle</note>, the radius is <note>UR</note>, points <note>T, U, V are on circle R</note>, so <note>UR = TR = VR</note>.

Definition of Right Triangle

<note>Angle TSR is a right angle (90 degrees)</note>, therefore triangle TSR is a right triangle.

Pythagorean Theorem

<note>∠TSR is a right angle (90°)</note>, <note>sides TS and SR are the legs, and side TR is the hypotenuse</note>, so according to the Pythagorean Theorem, <note>TS^2 + SR^2 = TR^2</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>UR is perpendicular to chord TV</note>, <note>UR is the radius = 1/2 diameter</note>, then according to the Perpendicular Diameter Theorem, <note>UR bisects chord TV, that is, TS = SV.</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 R, point R is the center</note>, <note>points T, U, and V are any points on the circle</note>, <note>segments RT, RU, and RV are segments from the center to any point on the circle</note>, therefore <note>segments RT, RU, and RV are the radii of the circle</note>.


Question: Find z in the given parallelogram.

Answer: 4.5

1. Let the four vertices of the parallelogram be A, B, C, D, and the two diagonals AC and BD intersect at point O.

2. Given that the figure is a parallelogram, according to the properties of a parallelogram, we know that the diagonals bisect each other when they intersect.

3. From the previous step, we get AO = OC, and from the figure, we know AO = 3z - 4 and OC = z + 5.

4. Establish the equation by setting 3z - 4 = z + 5, and rearrange the equation: 3z - z = 5 + 4.

6. Further simplification gives z = 4.5.

7. Through the above reasoning, the final answer is z = 4.5.

Definition of Parallelogram

<note>Quadrilateral ABCD is a parallelogram</note>, <note>sides AB and CD are parallel and equal</note>, <note>sides AD and BC are parallel and equal</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, <note>the opposite angles ∠ABC and ∠ADC</note> are equal. <note>The diagonals AC and BD bisect each other</note>, that is, the intersection point divides <note>the diagonal AC into two equal segments AO and OC</note>, and divides <note>the diagonal BD into two equal segments BO and OD</note>.


Question: As shown in the figure, in △ABC, ∠BAC=90°, AD⊥BC at point D, AE bisects ∠DAC, ∠B=50°, find the degree measure of ∠DAE ()

Answer: 25°

1. In △ABC, it is known that ∠BAC=90°, ∠B=50°.

2. According to the triangle angle sum theorem, in △ABC, ∠C=180°-∠BAC-∠B=180°-90°-50°=40°.

3. Since AD⊥BC at point D, according to the definition of perpendicular lines, ∠ADC=90°.

####

##4##. In △ADC, it is known that ∠ADC=90°, ∠C=40°.

##5##. According to the triangle angle sum theorem, in △ADC, ∠DAC=180°-∠ADC-∠C=180°-90°-40°=50°.

7. Since AE bisects ∠DAC, according to the definition of angle bisectors, ∠DAE=1/2*∠DAC=1/2×50°=25°.

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

直角三角形; 垂线; 普通三角形; 三角形的外角

Definition of Angle Bisector

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

<note>The vertex of angle DAC is point A</note>, from <note>point A a line AE is drawn</note>, this line <note>divides angle DAC into two equal angles</note>, namely <note>∠DAE and ∠EAC are equal</note>. Therefore, <note>line AE is the angle bisector of angle DAC</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.

<note>The angle ∠ADC formed by the intersection of line AD and line BC is 90 degrees</note>, therefore according to the definition of perpendicular lines, <note>line AD and line BC are perpendicular to each other</note>.

Triangle Angle Sum Theorem

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

<note>In triangle ABC, angles ABC, BAC, and BCA are the three interior angles of triangle ABC</note>, according to the Triangle Angle Sum Theorem, <note>angle ABC + angle BAC + angle BCA = 180°</note>. <note>In triangle ADC, angles ADC, DAC, and DCA are the three interior angles of triangle ADC</note>, according to the Triangle Angle Sum Theorem, <note>angle ADC + angle DAC + angle DCA = 180°</note>.


Question: Quadrilateral W X Y Z is a rectangle. If m \angle 1 = 30, find the degree measure of \angle 3.

Answer: 60

1. Given that quadrilateral WXYZ is a rectangle, ##let diagonals WZ and YX intersect at O. According to the properties of diagonals in a rectangle, the diagonals are equal and bisect each other. Therefore, WZ = XY, WO = ZO = XO = YO.##

####

##2. Since quadrilateral WXYZ is a rectangle, according to the definition of a rectangle, ∠WXZ = 90°. Given that ∠1 = 30°, then ∠2 = 90° - 30° = 60°.##

##3. Given that ZO = XO, according to the definition of an isosceles triangle, triangle XOZ is an isosceles triangle; and according to the properties of an isosceles triangle, ∠2 = ∠3.##

##4. From the above, we know that ∠2 = ∠3 = 60°.##

####

##5. After the above calculations, ∠3 = 60°.##

矩形; 邻补角; 内错角

Definition of Rectangle

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

<note>Quadrilateral WXYZ is a rectangle</note>, and its <note>interior angles ∠WXZ, ∠XZY, ∠ZYW, ∠YWX are all right angles (90 degrees)</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 WXYZ, <note>side WX is parallel and equal to side YZ</note>, <note>side WY is parallel and equal to side XZ</note>. <note>Diagonals WZ and XY are equal and bisect each other</note>, that is, <note>the intersection point O of diagonals WZ and XY is the midpoint of both diagonals</note>. Therefore, <note>segment WO is equal to segment ZO</note>, <note>segment YO is equal to segment XO</note>.

Definition of Isosceles Triangle

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

<note>In triangle XOZ, side XO and side ZO are equal</note>, therefore triangle XOZ is an isosceles triangle.

Properties of Isosceles Triangle

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

<note>In the isosceles triangle XOZ, side XO and side ZO are equal</note>, therefore, according to the properties of isosceles triangles, <note>the angles opposite the equal sides are equal, i.e., ∠2 = ∠3</note>.


Question: As shown in the figure, a student sees a tree by the lake. He estimates the distance between himself and the tree to be 20m. The reflection of the top of the tree in the water is 5m away from him. The student's height is 1.7m. What is the height of the tree in meters?

Answer: 5.1

1. As shown in the figure, a student sees a tree by the lake. He measures the distance between himself and the tree to be 20 meters, and the distance between the top of the tree's reflection in the water and himself is 5 meters. His height is 1.7 meters.

2. We can denote the height of the tree as x meters.

3. According to the problem, the tree and the person form two similar right triangles. Using the properties of similar triangles, the corresponding sides of the two triangles are proportional, which gives us: [distance of the tree's reflection] / [distance between the person and the tree minus the reflection distance] = [height of the person] / [height of the tree].

4. Establishing the ratio relationship, the equation is: \( \frac{5}{20-5} = \frac{1.7}{x} \).

5. Calculating, we get: \( 5 \times x = 1.7 \times 15 \).

6. Further calculation gives x = \( \frac{1.7 \times 15}{5} \).

7. The final result is x = 5.1.

8. Therefore, the height of the tree is 5.1 meters.

普通三角形; 反射; 直角三角形

Definition of Similar Triangles

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

Original: <note>三角形ABC（A为树顶，B为树底，C为倒影点）</note>和<note>三角形DEF（D为同学的眼睛，E为同学的脚，F为倒影点）</note>是相似三角形。根据相似三角形的定义有：<note>∠BAC = ∠EDF</note>, <note>∠BCA = ∠DFE</note>, <note>∠ABC = ∠DEF</note>；<note>AB/DE = BC/EF = AC/DF</note>。即，##<note>5/(20-5)</note> = <note>1.7/x</note>##.######

    Translation: <note>Triangle ABC (A is the top of the tree, B is the base of the tree, C is the reflection point)</note> and <note>triangle DEF (D is the student's eyes, E is the student's feet, F is the reflection point)</note> are similar triangles. According to the definition of similar triangles: <note>∠BAC = ∠EDF</note>, <note>∠BCA = ∠DFE</note>, <note>∠ABC = ∠DEF</note>; <note>AB/DE = BC/EF = AC/DF</note>. That

Similarity Theorem for Triangles (AA)

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

<note>In the diagram of this problem</note>, in <note>the diagram of this problem</note>, in triangle ABC and triangle DEF, <note>if angle DEF is equal to angle BAC</note>, and <note>angle DEF is equal to angle ABC</note>, then <note>triangle ABC is similar to triangle DEF</note>.


Question: Quadrilateral W X Y Z is a rectangle. If m \angle 1 = 30, find the measure of \angle 2.

Answer: 60

1. Given that WXYZ is a rectangle. ##According to the definition of a rectangle, ∠WXZ is one of the interior angles of the rectangle, so ∠WXZ = 90°##

2. ##Since ∠WXZ = ∠1 + ∠2, ∠1 = 30°, ∠WXZ = 90°, we can deduce that ∠2 = 60°##

3. ##Through the above reasoning, the final answer is 60 degrees##.

矩形; 对顶角; 邻补角

Definition of Rectangle

<note>Quadrilateral WXYZ is a rectangle</note>, whose <note>interior angles ∠WXZ, ∠XZY, ∠ZYW, ∠YWX are all right angles (90 degrees)</note>, and <note>side WX is parallel and equal in length to side YZ</note>, <note>side WY is parallel and equal in length to side XZ</note>.


Question: Find y.

Answer: \frac { 5 \sqrt { 2 } } { 2 }

1. In △ABC, it is known that ∠CAB = 45°, ∠ACB = 90°. According to the definition of a right triangle, △ABC is a right triangle. According to the definition of the tangent function, tan∠CAB = CB/CA = tan45° = 1. From the figure, we know CB = 5, so CA = CB = 5. According to the definition of an isosceles triangle, the right △ABC is an isosceles right triangle.

2. According to the properties of an isosceles triangle, ∠CBA = ∠CAB = 45°.

3. Given that CD ⊥ AB, then ∠CDA and ∠CDB are equal to 90°. According to the definition of a right triangle, △ACD and △CDB are both right triangles. According to the definition of the tangent function, tan∠CAD = CD/AD, tan∠DBC = CD/BD. Since ∠CAD = ∠DBC = 45°, tan45° = 1, so y = CD = AD = BD = ?AB = ?x.

4. In the isosceles right △ABC, according to the Pythagorean theorem, CA? + CB? = AB?, that is, 5? + 5? = x?, calculating gives x = 5√2.

5. Therefore, y = ?x = 5√2 / 2.

6. Based on the above derivation, we obtain the final solution as y = 5√2 / 2.

直角三角形; 等腰三角形; 余弦; 正弦

Definition of Isosceles Triangle

In the figure of this problem, <note>right triangle ABC</note>, <note>sides CA and CB are equal</note>, so the right triangle ABC is an isosceles triangle.

Properties of Isosceles Triangle

In the figure of this problem, <note>isosceles triangle ABC</note>, <note>sides CA and CB are equal</note>. Therefore, according to the properties of isosceles triangles, the angles opposite the equal sides are equal, that is, <note>∠CBA=∠CAB=45°##</note>.

Pythagorean Theorem

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

<note>In the isosceles right triangle ABC, ∠ACB is a right angle (90 degrees)</note>, <note>side CA and side CB are the legs</note>, <note>side AB is the hypotenuse</note>, so according to the Pythagorean Theorem, <note>CA²+CB²=AB²</note>.

Definition of Right Triangle

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

<note>In the diagram of this problem, in triangle ABC, ∠ACB is a right angle (90 degrees)</note>, therefore triangle ABC is a right triangle. Side CA and side CB are the legs, and side AB is the hypotenuse. <note>In △ACD and △CDB, ∠CDA and ∠CDB are equal to 90°</note>, thus △ACD and △CDB are right triangles, sides AD, CD, BD are the legs, and CA, CB are the hypotenuses.

Definition of Tangent Function

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

<note>In the figure of this problem, in the isosceles right triangle ABC, ∠CAB = 45°</note>, <note>side CB is the opposite leg of ∠CAB</note>, <note>side CA is the adjacent leg of ∠CAB</note>, so the tangent value of ∠CAB is equal to the length of side CB divided by the length of side CA, that is, <note>tan∠CAB = CB / CA = tan 45° = 1</note>, so <note>CA = CB = 5</note>. In the right triangle △ACD, <note>∠CAD = 45°</note>, <note>side CD is the opposite leg of ∠CAD</note>, <note>side AD is the adjacent leg of ∠CAD</note>, so the tangent value of ∠CAD is equal to the length of side CD divided by the length of side AD, that is, <note>tan∠CAD = CD / AD = tan 45° = 1</note>, so <note>AD = CD = y</note>. In the right triangle △BCD, <note>∠DBC = 45°</note>, <note>side CD is the opposite leg of ∠DBC</note>, <note>side BD is the adjacent leg of ∠DBC</note>, so ∠DBC


Question: As shown in the figure, in the rhombus ABCD, M and N are on AB and CD respectively, and AM = CN. MN intersects AC at point O. Connect BO. If ∠DAC = 28°, then the degree of ∠OBC is ()

Answer: 62°

1. Given quadrilateral ABCD is a rhombus, according to the properties of a rhombus, AB is parallel and equal to CD, AD is parallel and equal to BC.

2. Given point M and point N are on sides AB and CD respectively, and AM is equal to CN. Connect MN, and extend MN to intersect AC at point O. Connect BO.

3. In △AMO and △CNO, we have ∠MAO=∠NCO (##alternate interior angles## are equal), ∠AMO=∠CNO (##alternate interior angles## are equal), AM=CN (given),

4. According to the ASA (angle-side-angle) congruence criterion, we can conclude △AMO≌△CNO.

5. Since corresponding sides are equal, AO=CO.

6. Also, according to the properties of rhombus ABCD, AB=BC, and the diagonals of the rhombus are perpendicular bisectors of each other.

7. Therefore, BO is perpendicular to AC and intersects at point O, giving ∠BOC=90°.

8. Given ∠DAC=28°, since BC is parallel to AD, ∠BCA=∠DAC=28° (##alternate interior angles## are equal).

9. In △BOC, since BO is perpendicular to AC, ∠BOC=90°, so ∠OBC=90°-∠BCA=90°-28°.

10. Through the above reasoning, we finally conclude that the measure of ∠OBC is 62°.

菱形; 内错角; 等腰三角形; 平行线

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, quadrilateral ABCD is a rhombus</note>, AB=BC=CD=DA; <note>diagonals AC and BD are perpendicular to each other and bisect each other</note>, that is, <note>diagonals AC and BD intersect at point O</note>, and <note>angles AOB, BOC, COD, DOA are all right angles (90 degrees)</note>, and <note>AO=OC and BO=OD</note>.

Definition of Alternate Interior Angles

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

In the figure of this problem, <note>AC intersects AD and BC</note>, where <note>∠DAC and ∠BCA are between the two parallel lines</note>, and <note>on opposite sides of the transversal AC</note>, thus <note>∠DAC and ∠BCA are alternate interior angles</note>. <note>Alternate interior angles are equal, i.e., ∠DAC=∠BCA</note>. Similarly, <note>MN intersects AB and CD</note>, where <note>∠MAO and ∠OCN are between the two parallel lines</note>, and <note>on opposite sides of the transversal MN</note>, thus <note>∠MAO and ∠OCN are alternate interior angles</note>. <note>Alternate interior angles are equal, i.e., ∠MAO=∠OCN</note>. Similarly, <note>MN intersects AB and CD</note>, where <note>∠AMO and ∠CNO are between the two parallel lines</note>, and <note>on opposite sides of the transversal MN</note>, thus <note>∠AMO and ∠CNO are alternate interior angles</note>. <note>Alternate interior angles are equal, i.e., ∠AMO=∠CNO</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 this problem, <note>AD is parallel and equal to BC, i.e., AD∥BC</note>. At the same time, AC intersects AD and BC, forming <note>alternate interior angles ∠DAC=∠BCA</note>, <note>AB is parallel and equal to CD, i.e., AB∥CD</note>. At the same time, MN intersects AB and CD, forming <note>alternate interior angles ∠AMO=∠CNO</note>, <note>∠MAO=∠OCN</note>.

Congruence Theorem of Triangles (ASA)

If two angles and the included side of one triangle are respectively equal to two angles and the included side of another triangle, then the two triangles are congruent.

In <note>triangles AOM and CON</note>, <note>∠AMO=∠CNO</note>, <note>∠MAO=∠OCN</note>, and <note>side AM is equal to side CN</note>. Since the two angles and the included side of these triangles are respectively equal, according to the Angle-Side-Angle (ASA) Congruence Theorem of Triangles, it can be concluded that <note>triangle AMO is congruent to triangle CNO</note>.


Question: As shown in the figure, ∠1=∠2, ∠3=30°, then ∠4 equals ()

Answer: 150°

1. Given ∠1=∠2. ##According to the definition of corresponding angles##, line a and line b are parallel.

2. ##Because line a and line b are parallel, according to Parallel Postulate 2, the same-side interior angles are supplementary, i.e., ∠3+∠4=180°##.

3. ##Given ∠3=30°, substituting into the above equation we get 30°+∠4=180°, then ∠4=180°-30°##.

5. ##Through the above reasoning, the final answer is ∠4 equals 150°##.

平行线; 内错角; 同位角; 三角形的外角; 邻补角

Definition of Corresponding Angles

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

<note>In the diagram of this problem, two parallel lines a and b are intersected by a transversal line d</note>, where <note>angle ∠1 and angle ∠2 are on the same side of the transversal d, and on the same side of the two intersected lines a and b</note>, therefore, <note>angle ∠1 and angle ∠2 are corresponding angles. Corresponding angles are equal, that is, angle ∠1 is equal to angle ∠2</note>.

Definition of Consecutive Interior Angles

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

<note>Two lines a and b are intersected by a third line c</note>, <note>angles 3 and 4 are on the same side of the intersecting line c</note>, and inside the intersected lines a and b, so <note>angles 3 and 4 are consecutive interior angles</note>. <note>Consecutive interior angles 3 and 4 are supplementary</note>, that is, <note>angle 3 + angle 4 = 180 degrees</note>.

Parallel Postulate 2 of Parallel Lines

In the figure of this problem, the lines <note>a and b are parallel</note>, and the lines <note>c and d are transversal lines</note>, forming the following geometric relationships: 1. Corresponding angles: <note>∠1 and ∠2 are equal</note>. 2. Consecutive interior angles: <note>∠3 and ∠4 are supplementary, that is, ∠3 + ∠4 = 180 degrees</note>. These relationships illustrate that when two parallel lines are intersected by a third line, <note>corresponding angles are equal</note>, <note>consecutive interior angles are supplementary</note>.

Definition of Parallel Lines

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

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


Question: As shown in the figure, given ∠α=130°, then ∠β=()

Answer: 40°

1. Given ∠α = 130°.

2. Let the small triangle be triangle ABC, given ∠ABC=90°, according to the exterior angle theorem of the triangle, ∠BCA=∠α-∠ABC=130°-90°=40°.

3. According to the definition of vertical angles, ∠BCA=∠β=40°.

4. Through the above reasoning, the final answer is 40°.

邻补角; 对顶角; 直角三角形

Definition of Triangle

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

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

Exterior Angle Theorem of Triangle

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

In the figure of this problem, in triangle ABC, <note>angle α is an exterior angle of the triangle</note>, <note>angle ABC and angle BCA are the two interior angles that are not adjacent to the exterior angle α</note>. According to the Exterior Angle Theorem of Triangle, <note>the exterior angle α is equal to the sum of the two non-adjacent interior angles ABC and BCA</note>, that is, <note>angle α = angle ABC + angle BCA</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 intersect at point <note>C</note>, forming four angles. According to the definition of vertical angles, <note>angle BCA and angle β are vertical angles</note>. Since vertical angles are equal, <note>angle BCA = angle β</note>.


Question: As shown in the figure, quadrilateral ABCD and A′B′C′D′ are homothetic figures with point O as the homothetic center. If OA′:A′A=2:1 and the area of quadrilateral A′B′C′D′ is 12cm^{2}, then the area of quadrilateral ABCD is ()

Answer: 27cm^{2}

1. Given quadrilateral ABCD and A′B′C′D′ are homothetic figures with point O as the homothetic center, which is a special type of similar figures, and OA′:A′A=2:1.

2. Since OA=OA'+AA', therefore, OA′:OA=2:3.

3. According to the theorem that the area ratio of similar polygons is equal to the square of the similarity ratio, the area ratio of quadrilateral A′B′C′D′ to quadrilateral ABCD is (2:3)²=4:9.

4. Given the area of quadrilateral A′B′C′D′ is 12 cm², therefore the area of quadrilateral ABCD is 12 cm² × (9/4) = 27 cm².

5. Through the above reasoning, the final answer is 27 cm².

位似; 普通四边形; 点

Area Ratio Theorem of Similar Polygons

If the ratio of the side lengths of two similar polygons is k, then the ratio of their areas is equal to k squared (k²).

<note>Quadrilateral ABCD and quadrilateral A′B′C′D′ are similar figures with point O as the center of similarity</note>, the ratio of <note>side OA' to side OA is 2/3</note>, so <note>the ratio of the area of quadrilateral A′B′C′D′ to the area of quadrilateral ABCD is (2/3)², which is 4/9</note>.


Question: In Rt△ABC, ∠C=90°, AB=2, BC=1, then the value of sinB is ()

Answer: \frac{√{3}}{2}

1. Given that in Rt△ABC, ∠C = 90°, and AB = 2, BC = 1.

2. According to the Pythagorean theorem, in right triangle ABC, the square of the hypotenuse is equal to the sum of the squares of the two legs. That is, AB² = AC² + BC².

3. Substituting the given conditions, we get 2² = AC² + 1², which means 4 = AC² + 1.

4. Solving this equation, we get AC² = 3, so AC = √3.

5. According to the definition of the sine function, sin∠B = opposite side/hypotenuse, where the opposite side is AC and the hypotenuse is AB.

7. Through the above reasoning, the final answer is √3/2.

直角三角形; 正弦; 线段

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, in the right triangle ABC, <note>∠B is an acute angle</note>, <note>side AC is the opposite side of ∠B</note>, <note>side AB is the hypotenuse</note>. According to the definition of the sine function, <note>the sine value of ∠B is equal to the ratio of the opposite side AC to the hypotenuse AB</note>, that is, <note>sin∠B = AC / AB</note>.

Pythagorean Theorem

<note>In the figure of this problem, in the right-angled triangle ABC, angle C is a right angle (90 degrees)</note>, sides AC and BC are the legs, and side AB is the hypotenuse. Therefore, according to the Pythagorean Theorem, <note>AB² = AC² + BC²</note>. Given the conditions of the problem, AB=2 and BC=1, substituting the data yields <note>2² = AC² + 1²</note>, that is, 4 = <note>AC²</note> + 1, thus obtaining <note>AC²</note> = 3, resulting in AC = √3.


Question: Find x.

Answer: 6 \sqrt { 6 }

1. Let the triangle be triangle ABC. In the figure, a right angle mark is drawn at the intersection of lines AB and AC, indicating that these two lines are perpendicular to each other. Therefore, angle BAC is a right angle in the triangle, with BC as the hypotenuse. In the figure, a line segment passing through point A connects to BC, with the intersection point labeled as point D. A right angle mark is also drawn at the intersection of lines AD and BC, indicating that these two lines are perpendicular to each other, i.e., AD⊥BC, so ∠ADB and ∠ADC are both right angles.

2. According to the definition of a right triangle, since ∠ADB=∠ADC=90°, triangles ADB and ADC are both right triangles.

3. According to the property of the altitude on the hypotenuse in a right triangle, since AD⊥BC, AD is the altitude on the hypotenuse BC of right triangle ABC. Therefore, the two triangles formed by the altitude on the hypotenuse are similar to the original triangle, i.e., triangle ADB is similar to triangle ABC, triangle ADC is similar to triangle ABC, so triangle ADB is similar to triangle ADC.

4. According to the definition of similar triangles, the corresponding sides of two similar triangles are proportional, so CD/AD=AD/BD. Given CD=x, AD=36, BD=6x, substituting these values, we get: x/36=36/6x, cross-multiplying yields: 6x^2=36^2.

直角三角形; 正弦; 余弦

Definition of Triangle

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

Definition of Right Triangle

In the figure of this problem, ##In the figure of this problem, <note>in triangle ABC, angle BAC is a right angle (90 degrees)</note>, therefore <note>triangle ABC is a right triangle</note>. The sides <note>AB and AC are the legs</note>, <note>side BC is the hypotenuse</note>. <note>In triangle ABD, angle ADB is a right angle (90 degrees)</note>, therefore <note>triangle ABD is a right triangle</note>. <note>Sides AD and BD are the legs</note>, <note>side AB is the hypotenuse</note>. <note>In triangle ADC, angle ADC is a right angle (90 degrees)</note>, therefore <note>triangle ADC is a right triangle</note>. <note>Sides AD and CD are the legs</note>, <note>side AC is the hypotenuse</note>##.######

In a right triangle, the two triangles formed by the altitude drawn to the hypotenuse are similar to the original triangle.

<note>In the diagram of this problem, in the right triangle ABC, angle BAC is a right angle (90 degrees)</note>, from vertex A, draw <note>altitude AD</note> to the hypotenuse BC. According to the <note>Property of the Altitude on the Hypotenuse in a Right Triangle</note>, <note>altitude AD divides the right triangle ABC into two new right triangles ADB and ADC</note>. <note>Triangle ADB is similar to triangle ABC</note>, <note>triangle ADC is also similar to triangle ABC</note>. Therefore, <note>triangle ADB ? triangle ABC</note>, <note>triangle ADC ? triangle ABC</note>, and <note>triangle ADB ? triangle ADC</note>.

Definition of Similar Triangles

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

In the figure of this problem, <note>triangle ADB and triangle ADC are similar triangles</note>. According to the <note>definition of similar triangles</note>, we have: <note>angle ADB = angle ADC</note>, <note>angle BAD = angle DAC</note>, <note>angle ABD = angle CAD</note>; <note>CD/AD = AD/BD = AC/AB</note>.


Question: Quadrilateral W X Y Z is inscribed in \odot V. Find m \angle X.

Answer: 120

1. Given that quadrilateral WXYZ has its four vertices inscribed in circle V, then WXYZ is a cyclic quadrilateral of circle V. ∠ YZW = 60°.

2. According to the (Inscribed Angle Theorem Corollary 3) cyclic quadrilateral opposite angles supplementary theorem, the sum of the interior angles of opposite angles of a cyclic quadrilateral equals 180 degrees. Therefore, we can deduce ∠WXY + ∠WZY = 180° and ∠XWZ + ∠XYZ = 180°.

3. From ∠WXY + ∠WZY = 180°, we get: ∠WXY = 180° - ∠WZY = 180° - 60° = 120°.

4. Through the above reasoning, the final answer is ∠WXY = 120°.

圆内接四边形; 圆周角; 圆

Cyclic Quadrilateral

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

In the figure of this problem, <note>the four vertices W, X, Y, Z of quadrilateral WXYZ are all on circle V</note>. Therefore, <note>quadrilateral WXYZ is a cyclic quadrilateral</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, quadrilateral WXYZ is inscribed in circle V. Therefore, <note>∠WXY</note> + <note>∠WZY</note> = 180° and <note>∠XWZ</note> + <note>∠XYZ</note> = 180°. According to the theorem, <note>∠XWZ</note> + <note>∠XYZ</note> = 180° gives <note>∠XYZ</note> = 180° - <note>∠XWZ</note> = 180° - 95° = 85°, and <note>∠WXY</note> + <note>∠WZY</note> = 180° gives <note>∠WXY</note> = 180° - <note>∠WZY</note> = 180° - 60° = 120°.


Question: As shown in the figure, AB is the diameter of ⊙O, ∠D=33°, then the degree of ∠AOC is ()

Answer: 114°

1. Given that AB is the diameter of ⊙O, according to the Inscribed Angle Theorem, ∠BDC=33°.

2. The Inscribed Angle Theorem tells us that the central angle subtended by the same arc is twice the inscribed angle, so ∠BOC=2∠BDC=66°.

3. AB is the diameter, therefore ∠AOB=180°.

4. ##∠AOC##+∠BOC=180°, thus we can deduce ∠AOC=180°-∠BOC.

5. Substituting the value of ∠BOC, we get ∠AOC=180°-66°=114°.

6. Through the above reasoning, the final answer is 114°.

圆; 圆心角; 圆周角; 直角三角形; 线段

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

Definition of Inscribed Angle

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

In <note>circle O</note>, <note>the vertex D of angle BDC is on the circumference</note>, and the two sides of angle BDC intersect <note>circle O at points B and C</note> respectively. Therefore, <note>angle BDC is an inscribed 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>Line segment AB is the diameter</note>, connecting <note>the center O of the circle and points A and B on the circumference</note>, with a length of 2 times the radius, i.e., <note>AB = 2r</note>.

Inscribed Angle Theorem

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

<note>In circle O, points B, D, and C are on the circle</note>, the central angle corresponding to arc BC and arc BDC is <note>∠BOC</note>, and the inscribed angle is <note>∠BDC</note>. According to the Inscribed Angle Theorem, <note>∠BDC is equal to half of the central angle ∠BOC corresponding to arc BC</note>, i.e., <note>∠BDC = 1/2 ∠BOC</note>.

Definition of Linear Pair of Angles

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

In the figure of this problem, <note>angle BOC and angle AOC have a common side OC</note>, <note>their other sides OA and OB are extensions in opposite directions</note>, so <note>angle BOC and angle AOC are adjacent supplementary angles</note>.


Question: As shown in the figure, lines AB and CD intersect at point O, ray OM bisects ∠AOC, and ON⊥OM. If ∠AOC=70°, then the measure of ∠CON is ()

Answer: 55°

1. Given ON is perpendicular to OM, ##according to the definition of perpendicular lines##, it follows that ∠MON=90°.

2. Given OM bisects ∠AOC, and ∠AOC=70°, according to the definition of angle bisector, it follows that ##∠MOC=∠AOM=?∠AOC=?*70°##=35°.

3. ##From the figure, ∠MON=∠MOC+∠CON##, it follows that ∠CON=∠MON-∠MOC.

4. Substituting the values, ∠CON=90°-35°=55°.

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

对顶角; 射线; 垂线; 直角三角形

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>ON and OM</note> intersect to form an angle <note>∠MON is 90 degrees</note>, so according to the definition of perpendicular lines, <note>the lines ON and OM are perpendicular to each other</note>.

Definition of Angle Bisector

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

<note>The vertex of angle AOC is point O</note>, from <note>point O, a ray OM is drawn</note>, this ray divides <note>angle AOC into two equal angles</note>, namely <note>∠AOM and ∠MOC are equal</note>. Therefore, <note>ray OM is the angle bisector of angle AOC</note>.


Question: Find x.

Answer: 35

1. According to the figure in the problem, the figure is a circle with four angles, namely 150°, 85°, 90°, and x°, and the four angles intersect at the center of the circle. ##According to the definition of central angles, the four angles are central angles##.

2. ##According to the properties of central angles##, the sum of the central angles equals the full angle. According to the definition of the full angle, the full angle is 360°, so the sum of the central angles equals the full angle, which is 360°. Therefore, these angles can be expressed as 360° in the equation.

3. ####Thus, 150° + 85° + 90° + x° = 360°.

4. By calculation, 150° + 85° + 90° = 325°.

5. ##Solve the equation 325° + x° = 360°##.

6. Calculation gives ##x° = 360° - 325° = 35°##.

7. Therefore, through the above reasoning, the final answer is ##x = 35##.

Properties of Central Angles

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

<note>In the figure of this problem, according to the properties of central angles</note>, the degree measure of a central angle is equal to the degree measure of the arc it intercepts. Since the entire circumference corresponds to 360°, <note>the sum of all central angles is equal to the full angle, which is 360°</note>.

Definition of Full Angle

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

<note>A ray rotates around its endpoint to form a full angle</note>. According to <note>the definition of a full angle</note>, <note>a full angle measures 360 degrees</note>.

Definition of Central Angle

Original text: <note>Center O</note>, the four central angles are <note>150°</note>, <note>85°</note>, <note>90°</note>, and <note>x°</note>. The vertices of these angles are at <note>Center O</note>, and each angle's sides are the circle's <note>radii</note>.


Question: Find the value of t in the parallelogram.

Answer: 7

1. Given that quadrilateral WXZY is a parallelogram. According to the properties of a parallelogram, the diagonals bisect each other, let the intersection point be point O, thus we have OW = OZ and OY = OX.

2. According to the markings in the figure, it is known that OW = 8 and OZ = 2t - 6. Since the diagonals bisect each other, we have OW = OZ, that is 8 = 2t - 6.

3. To solve the equation 8 = 2t - 6, first move 6 to the left side, obtaining 8 + 6 = 2t.

4. Adding both sides gives 14 = 2t.

5. To solve for t, divide both sides of the equation by 2, obtaining t = 7.

6. Through the above reasoning steps, the final answer is that the value of t is 7.

Definition of Parallelogram

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

<note>Quadrilateral WXZY</note> is a parallelogram, which means <note>WX is parallel and equal to YZ</note>, and <note>WY is parallel and equal to XZ</note>.

Properties of Parallelogram Theorem

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

In the <note>parallelogram WXZY</note>, <note>angles YWX and YZX are equal</note>, <note>angles WXZ and WYZ are equal</note>; <note>sides WX and YZ are equal</note>, <note>sides WY and XZ are equal</note>; <note>diagonals WZ and XY bisect each other</note>, that is, <note>the intersection point divides diagonal WZ into two equal segments OW and OZ</note>, <note>and divides diagonal XY into two equal segments OX and OY</note>.


Question: Solve for x.

Answer: 15

1. According to the definition of a right triangle, the triangle in the figure has one right angle, so the triangle is a right triangle, with one leg of the right angle being 9, the other leg of the right angle being x-3, and the hypotenuse being x. Applying the Pythagorean theorem, we get x^2 = (x-3)^2 + 9^2.

2. Expand and simplify the equation x^2 = (x-3)^2 + 81, first calculating (x-3)^2 to get x^2 - 6x + 9.

4. Eliminate the x^2 terms from the equation to get 0 = -6x + 90.

5. Solve the equation 0 = -6x + 90 to get 6x = 90.

7. Therefore, through the above reasoning, the final answer is 15.

Definition of Right Triangle

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

<note>A triangle contains a 90-degree angle, represented by a red right angle symbol</note>. The <note>two legs of the right triangle are 9 and x-3, and the hypotenuse is x</note>.

Pythagorean Theorem

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

In the figure of this problem, in the right triangle, <note>one leg is 9</note>, <note>the other leg is x-3</note>, <note>the hypotenuse is x</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>##x^2 = (x-3)^2 + 9^2##</note>.


Question: Find m \angle M P Q.

Answer: 101

1. Given the conditions, in △MNP, ##∠PNM## = 45°, ##∠PMN## = 56°.

2. According to the triangle angle sum theorem, which states: the sum of the interior angles of a triangle is 180° (i.e., the sum of the degrees of the three interior angles of any triangle is 180°). In this problem, we can use this theorem to calculate the unknown angle ##∠MPN##.

3. According to the triangle angle sum theorem, we can find ##∠MPN = 180° - ∠PNM - ∠PMN## = 180° - 45° - 56° = 79°.

4. Notice that points Q, P, N are collinear, thus the line QPN forms a straight angle, which is 180°.

5. Therefore, we can obtain ##∠MPQ + ∠MPN## = 180°.

6. Substituting the specific values, we get ##∠MPQ## + 79° = 180°.

7. Thus, we can find ##∠MPQ## = 180° - 79° = 101°.

8. Through the above reasoning, the final answer is ##∠MPQ## = 101°.

普通三角形; 三角形的外角

Triangle Angle Sum Theorem

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

<note>In triangle MNP, angle MPN, angle MNP, and angle PMN are the three interior angles of triangle MNP</note>, according to the Triangle Angle Sum Theorem, <note>angle MPN + angle MNP + angle PMN = 180°</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 PQ rotates around endpoint P until it forms a straight line with the initial side</note>, forming <note>straight angle QPN</note>. According to the definition of a straight angle, the measure of a straight angle is 180 degrees, that is, <note>angle QPN = 180 degrees</note>.


Question: As shown in the figure: AB∥DE, ∠B=30°, ∠C=110°, the measure of ∠D is ()

Answer: 100°

1. Draw auxiliary line CF through point C, making CF∥AB.

2. Since AB∥DE, according to the transitivity of parallel lines, it is known that AB∥DE∥CF.

3. Since ∠B = 30°, ∠BCF and ∠B are alternate interior angles. According to Parallel Postulate 2, it follows that ∠BCF = 30°.

4. Given ∠C = 110°, it can be deduced that ∠DCF = ∠C - ∠BCF = 110° - 30° = 80°.

5. Given that ∠D and ∠DCF are consecutive interior angles, according to Parallel Postulate 2, ∠D = 180° - ∠DCF = 100°.

6. Through the above reasoning, it is finally concluded that the measure of ∠D is 100°.

平行线; 同位角; 内错角; 三角形的外角

Definition of Parallel Lines

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

In the figure of this problem, <note>line AB and line DE lie in the same plane</note>, and <note>they do not intersect</note>, so according to the definition of parallel lines, <note>line AB and line DE are parallel lines</note>. <note>CF, as an auxiliary line, is also parallel to AB and DE</note>.

Transitivity of Parallel Lines

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

<note>Line DE and line CF are respectively parallel to line AB</note>. According to the transitivity of parallel lines, if <note>line DE is parallel to line AB</note>, and <note>line CF is also parallel to line AB</note>, then <note>line DE and line CF are parallel to each other</note>. Therefore, <note>line DE is parallel to line CF</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 AB and CF</note> are intersected by <note>a line BC</note>, where <note>angle ABC and angle BCF are located between the two parallel lines and on opposite sides of the intersecting line BC</note>, thus <note>angle ABC and angle BCF are alternate interior angles</note>. Alternate interior angles are equal, that is, <note>angle ABC is equal to angle BCF</note>.

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

<note>In the figure of this problem, two lines CF and DE are intersected by a third line CD</note>, <note>the two angles FCD and CDE are on the same side of the intersecting line CD and within the intersected lines CF and DE</note>, so <note>angles FCD and CDE are consecutive interior angles</note>. <note>Consecutive interior angles FCD and CDE are supplementary, that is, angle FCD + angle CDE = 180 degrees</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>In the figure of this problem, two parallel lines AB and CF are intersected by a third line BC</note>, forming the following geometric relationships: <note>alternate interior angles: angle ABC and angle BCF are equal</note>. Two parallel lines CF and DE are intersected by a third line CD, forming the following geometric relationships: <note>same-side interior angles: angle FCD and angle CDE are supplementary, that is, angle FCD + angle CDE = 180 degrees</note>. These relationships illustrate that when two parallel lines are intersected by a third line, <note>alternate interior angles are equal, and same-side interior angles are supplementary</note>.


Question: Find m ∠ B.

Answer: 120

1. Given the four interior angles of quadrilateral ABCD in the figure, according to the theorem of the sum of interior angles of a quadrilateral, the sum of the four interior angles in any quadrilateral is 360°, so ∠A + ∠B + ∠C + ∠D = 360°, i.e., x° + 2x° + 2x° + x° = 360°.

2. Simplifying gives: 6x = 360, so x = 60.

3. Substitute the obtained value of x into ∠B = 2x°, and finally calculate: ∠B = 120°.

4. Through the above reasoning, the final answer is ∠B = 120°.

梯形; 邻补角; 同旁内角

Definition of Quadrilateral

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

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

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

In the figure of this problem, the quadrilateral <note>ABCD</note> has four interior angles, <note>angle A, angle B, angle C, and angle D</note>. According to the <note>Sum of Interior Angles of a Quadrilateral Theorem</note>, the sum of these four interior angles is 360°, that is, <note>∠A + ∠B + ∠C + ∠D = 360°</note>.


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

Answer: 23.4

1. Given that the side length of the regular hexagon is 3 meters. According to the definition of a regular hexagon, the regular hexagon can be divided into 6 congruent equilateral triangles.

2. Each equilateral triangle has a side length of 3 meters. By the triangle angle sum theorem, each interior angle of an equilateral triangle is 60 degrees.

3. As shown in the figure, with vertex B as the center point, connect the adjacent vertices A and C to form the equilateral triangle ABC. Here, AD is perpendicular to BC, and ##AD## = DC. ##In triangle BCD, by the triangle angle sum theorem, we know ∠BDC = 180° - 90° - 60° = 30°. According to the properties of a 30°-60°-90° triangle##, half of BC, which is ##DC##, is equal to 1.5 meters.

4. Use the Pythagorean theorem to calculate the length of BD####, where the expression of the Pythagorean theorem is c^2 = a^2 + b^2. Let the vertical height from B to D be h, then (3)^2 = (1.5)^2 + h^2. Solving this, we get ####h = 1.5√3.

5. Next, calculate the area of each equilateral triangle. The area A of an equilateral triangle can be calculated using the formula A = 1/2 * base * height, where the base is 3 meters and the height is 1.5√3 meters. Therefore, the area of the equilateral triangle is 1/2 * 3 * 1.5√3 ≈ 3.9 square meters.

6. The entire regular hexagon is composed of six equilateral triangles, so the area is 6 times the area of one equilateral triangle. Therefore, the area of the regular hexagon is 6 * 3.9 ≈ 23.4 square meters.

7. Thus, the final area of the regular hexagon is approximately 23.4 square meters.

正多边形; 直角三角形; 正弦

Regular Hexagon

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

In the context of a <note>Regular Hexagon</note>, each interior angle is equal and each side length is equal. Specifically, each interior angle of a regular hexagon is <note>120 degrees</note>, and each side length is equal, i.e., <note>AB=BC=CD=DE=EF=FA=3 meters</note>. A regular hexagon can be divided into <note>6 congruent equilateral triangles</note>.

Definition of Equilateral Triangle

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

<note>Triangle ABC is an equilateral triangle</note>.<note>Side AB, side BC, and side CA are equal in length, each being 3 meters</note>, and<note>angle BAC, angle ABC, and angle BCA are equal in degrees, each being 60°</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>In the diagram of this problem, the line segment BD perpendicular to the opposite side AC from vertex B is the altitude from vertex B</note>.<note>The line segment BD forms a right angle (90 degrees) with side AC</note>, which indicates that <note>the line segment BD is the perpendicular distance from vertex B to the opposite side AC</note>.

Triangle Angle Sum Theorem

<note>In triangle BCD</note>, angles ∠BDC, ∠BCD, and ∠DBC are the three interior angles of <note>triangle BCD</note>. According to the Triangle Angle Sum Theorem, <note>angle ∠BDC + angle ∠BCD + angle ∠DBC</note> = 180°.

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.

<note>Line BD</note> passes through <note>the midpoint D of segment AC</note> and <note>line BD is perpendicular to segment AC</note>. Therefore, <note>line BD is the perpendicular bisector of segment AC</note>.

Pythagorean Theorem

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

Area Formula of a Triangle

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

<note>Side AC is the base</note>, <note>Line segment BD is the height</note>. According to the area formula of a triangle, the area of triangle ABC is equal to <note>the base AC multiplied by the height BD and then divided by 2</note>, that is, Area = (AC * BD) / 2 = (3 * 1.5√3) / 2 ≈ 3.9 square meters.


Question: In the figure above, four circles have the same center, and their radii are 1, 2, 3, and 4 respectively. What is the ratio of the area of the smaller shaded ring to the area of the larger shaded ring?

Answer: 03:07

1. Given four concentric circles with radii of 1, 2, 3, and 4 respectively. The area of the small shaded ring is the area of the circle with radius 2 minus the area of the circle with radius 1.

2. The area of the large shaded ring is the area of the circle with radius 4 minus the area of the circle with radius 3.

3. According to the formula for the area of a circle A = πr^2, for the small shaded ring, its area is π(2^2) - π(1^2) = π(4 - 1) = 3π.

4. For the large shaded ring, its area is π(4^2) - π(3^2) = π(16 - 9) = 7π.

5. Calculate the ratio of the area of the small shaded ring to the area of the large shaded ring, which is 3π/7π.

6. After simplification, the ratio is 3/7.

7. Therefore, the final conclusion is that the area ratio of the small shaded ring to the large shaded ring is 3:7.

Area Formula of a Circle

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

<note>Four circles are concentric</note>, <note>the radii of the circles are 1, 2, 3, and 4 respectively</note>. According to the area formula of a circle, <note>the area A of a circle is equal to the circumference π multiplied by the square of the radius r</note>, i.e., A = πr^2. The specific calculations are as follows: <note>the area of the circle with radius 1 is A_1=π(1^2)=π</note>, <note>the area of the circle with radius 2 is A_2=π(2^2)=4π</note>, <note>the area of the circle with radius 3 is A_3=π(3^2)=9π</note>, <note>the area of the circle with radius 4 is A_4=π(4^2)=16π</note>.


Question: If C W = W F and E D = 30, find D F?

Answer: 15

1. Extend both sides of WF to intersect the minor arc ED at point X and the major arc ED at point Y. According to the definition of the diameter, XY is the diameter of circle W.

2. Given that XY ⊥ ED, according to the perpendicular bisector theorem, XY bisects ED, thus EF = FD = ED/2.

3. Since ED = 30, therefore FD = 30/2 = 15.

圆; 弦; 垂线; 垂直平分线; 圆心角

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 W, diameter XY is perpendicular to chord ED</note>, then according to the Perpendicular Diameter Theorem, <note>diameter XY bisects chord ED, that is, EF=FD</note>, and <note>diameter XY bisects the arcs subtended by chord ED, that is, arc EXD = arc EYD</note>.

Definition of Chord

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

<note>Point E and point D are any two points on the circle</note>, <note>line segment ED connects these two points</note>, so <note>line segment ED is the chord of the circle</note>, and it is known that <note>ED = 30</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>XY is the diameter</note>, connecting <note>the center O</note> and <note>the points X and Y on the circumference</note>, with a length of <note>2 times the radius</note>, that is, <note>XY = 2WX</note>.


Question: The line segment is tangent to the circle. Find the value of x.

Answer: 3

1. Given that line segment WX and line segment XY are tangent to circle Z at points W and Y respectively, connect WZ, ZY, and ZX.

2. According to the definition and properties of a tangent to a circle, tangent WX and tangent XY are perpendicular to the radii WZ and YZ at the points of tangency W and Y, respectively, i.e., ∠ZWX=90°, ∠ZYX=90°.

3. According to the definition of a right triangle, △WZX and △ZYX are right triangles.

4. Since WZ and ZY are radii of circle Z, WZ=ZY. According to the criteria for congruence of right triangles (hypotenuse-leg), ∠ZWX=∠ZYX=90°, ZX=ZX, WZ=ZY, right triangles WZX and ZYX are congruent.

5. According to the definition of congruent triangles, right triangles WZX and ZYX are congruent, so WX=XY, i.e., 2x+9=3x+6.

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

Definition of Tangent to a Circle

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

<note>Circle Z and line WX have exactly one common point W, which is called the point of tangency</note>. Therefore, line WX is the tangent to circle Z. <note>Circle Z and line XY have exactly one common point Y, which is called the point of tangency</note>. Therefore, line XY is the tangent to circle Z.

Property of the Tangent Line to a Circle

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

<note>In the diagram of this problem, in circle Z, point W is the point of tangency between line WX and the circle, and segment WZ is the radius of the circle.</note> According to the property of the tangent line to a circle, <note>the tangent WX is perpendicular to the radius WZ at the point of tangency W, i.e., ∠ZWX = 90 degrees.</note> <note>In circle Z, point Y is the point of tangency between line XY and the circle, and segment ZY is the radius of the circle.</note> According to the property of the tangent line to a circle, <note>the tangent XY is perpendicular to the radius ZY at the point of tangency Y, i.e., ∠ZYX = 90 degrees.</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 Z, point Z is the center</note>, point W is any point on the circle, <note>line segment WZ is the line segment from the center to any point on the circle</note>, therefore <note>line segment WZ is the radius of the circle</note>. <note>In circle Z, point Z is the center</note>, point Y is any point on the circle, <note>line segment YZ is the line segment from the center to any point on the circle</note>, therefore <note>line segment YZ is the radius of the circle</note>.

Definition of Right Triangle

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

<note>Angle ZWX is a right angle (90 degrees)</note>, thus triangle ZWX is a right triangle. <note>Side ZW and side WX are the legs</note>, <note>side ZX is the hypotenuse</note>. <note>Angle ZYX is a right angle (90 degrees)</note>, thus triangle ZYX is a right triangle. <note>Side ZY and side YX are the legs</note>, <note>side ZX is the hypotenuse</note>.

Right Triangle Congruence (Hypotenuse, One Leg)

Two right triangles are congruent if their hypotenuses and one corresponding leg are equal.

In the figure of this problem, the right triangles <note>ZWX and ZYX</note>, assuming <note>angles ZWX and ZYX are both right angles (90 degrees)</note>. If <note>side ZX equals side ZX (hypotenuse)</note>, and <note>side ZW equals side ZY (one leg)</note>, then according to the right triangle congruence (hypotenuse, one leg), these two right triangles <note>ZWX and ZYX are congruent</note>.

Definition of Congruent Triangles

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

<note>Triangles ZWX and ZYX are congruent triangles</note>, the <note>corresponding sides and corresponding angles of triangle ZWX are equal to those of triangle ZYX</note>, namely:
<note>side WZ = side ZY</note>,
<note>side WX = side YX</note>,
<note>side ZX = side ZX</note>,
simultaneously, <note>the corresponding angles are also equal</note>:
<note>angle ZWX = angle ZYX</note>,
<note>angle WZX = angle YZX</note>,
<note>angle WXZ = angle YXZ</note>.


Question: If X N = 6, X M = 2, and X Y = 10, find N Z.

Answer: 24

1. Given conditions are XN = 6, XM = 2, XY = 10, ##from the figure, it can be seen that the sides MN and YZ are marked as parallel, i.e., MN∥YZ##.

2. ##According to the theorem of proportional segments in parallel lines, since in triangle YXZ there is a line MN parallel to one side YZ of triangle YXZ, and it intersects the other two sides XY and XZ, the segments intercepted by line MN are proportional to the corresponding segments of the original triangle YXZ, i.e., XM/XY = XN/XZ##.

3. ##Given XN = 6, XM = 2, and XY = 10, substituting the known values into XM/XY = XN/XZ, we get: 2/10 = 6/XZ, cross-multiplying gives: XZ = 6*10/2 = 30##.

####

##4. Since XZ = XN + NZ##, we can derive NZ = XZ - XN = 30 - 6 = 24.

##5##. Through the above reasoning, the final answer is NZ = 24.

直角三角形; 普通三角形

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 YXZ is composed of three non-collinear points Y, X, Z</note> and their <note>connecting line segments XY, XZ, YZ</note>. <note>Points Y, X, Z are the three vertices of the triangle</note>, <note>line segments XY, XZ, YZ 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 MN and line YZ are located in the same plane, and they do not intersect</note>, therefore according to the definition of parallel lines, <note>line MN and line YZ are parallel lines</note>.

Proportional Segments Theorem

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

<note>Line MN is parallel to side YZ</note>, and it intersects the other two sides XY and XZ at <note>points M and N</note>. Then, according to the <note>Proportional Segments Theorem</note>, we have: <note>segment XM/segment XY = segment XN/segment XZ</note>, that is, the intercepted segments are proportional to the corresponding segments of the original triangle.


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

Answer: 54

1. Given that this is a right triangle, where one leg is 12 ft, the other leg is 9 ft, and the hypotenuse is 15 ft.

2. According to the area formula of a right triangle, the area is equal to half the product of the two legs. The area formula for a right triangle is: Area = 1/2 * base * height.

3. In this problem, the base is 12 ft and the height is 9 ft, so Area = 1/2 * 12 ft * 9 ft.

4. Calculating, we get: Area = 1/2 * 108 square ft = 54 square ft.

5. Therefore, the area of the right triangle is 54 square ft.

直角三角形; 普通三角形

Definition of Right Triangle

<note>The angle is a right angle (90 degrees)</note>, therefore the triangle is a right triangle. <note>The sides of 12 feet and 9 feet are the legs, and the side of 15 feet is the hypotenuse</note>.

Area of Right Triangle

The area of a right triangle is equal to half the product of the two legs that form the right angle, i.e., Area = 1/2 * base * height.

In the figure of this problem, in the right triangle, <note>the right angle is located at the intersection of the base and the height</note>, where <note>the base is 12 feet</note> and <note>the height is 9 feet</note>, so the area of the right triangle is equal to half the product of these two perpendicular sides, that is, <note>Area = 1/2 * 12 feet * 9 feet = 54 square feet</note>.


Question: The area of trapezoid G H J K is 188.35 square feet. If H J is 16.5 feet, find G K.

Answer: 26.8

1. Given that the area of trapezoid G H J K is 188.35 square feet, the length of H J is 16.5 feet, ##let the vertical height from H to G K be HL, H L = 8.7 feet##.

2. According to the formula for the area of a trapezoid, trapezoid area = 1/2 * (upper base + lower base) * height, i.e., 188.35 = 1/2 * (HJ + GK) * height.

3. In this formula, the length of upper base H J is 16.5 feet, the height HL is 8.7 feet, we can substitute these values into the formula: 188.35 = 1/2 * (16.5 + G K) * 8.7.

4. First, calculate 1/2 * 8.7 = 4.35, so the formula can be simplified to 188.35 = 4.35 * (16.5 + G K).

5. Calculate 4.35 * 16.5 = 71.775, thus the formula is further simplified to 188.35 = 71.775 + 4.35 * G K.

6. Subtract 71.775 from 188.35 to find the value of 4.35 * G K: 188.35 - 71.775 = 116.575.

7. Finally, use division to find the value of G K: ##G K = 116.575 / 4.35 ≈ 26.8##.

8. Therefore, through the detailed reasoning steps above, the answer obtained is G K = 26.8 feet.

Definition of Trapezoid

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

In the figure of this problem, the quadrilateral <note>G H J K</note> has side <note>G K</note> and side <note>H J</note> parallel, while side <note>G H</note> and side <note>J K</note> are not parallel. Therefore, according to the definition of a trapezoid, the quadrilateral <note>G H J K</note> is a trapezoid because it has exactly one pair of parallel sides.

Definition of Altitude

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

In the diagram of this problem, the line segment HL perpendicular to the opposite side GK (or its extension) from vertex H is the altitude of vertex H. The line segment HL forms a right angle (90 degrees) with the side GK (or its extension), which indicates that the line segment HL is the perpendicular distance from vertex H to the opposite side GK (or its extension), with a height of 8.7 feet.

Trapezoid Area Formula

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

In the figure of this problem, trapezoid <note>G H J K</note> has side <note>HJ</note> and side <note>GK</note> as the two parallel sides, and side <note>HL</note> as the height between them, so the area of the trapezoid is <note>(HJ + GK) * HL / 2</note>.


Question: As shown in the figure, △ABC≌△ADE, if ∠B=70°, ∠C=30°, ∠DAC=35°, then the degree of ∠EAC is ()

Answer: 45°

1. Given ∠B equals 70 degrees and ∠C equals 30 degrees, according to the triangle angle sum theorem, we get ∠BAC = 180 degrees - ∠B - ∠C = 180 degrees - 70 degrees - 30 degrees, which means ∠BAC = 80 degrees.

2. From the given condition △ABC is congruent to △ADE, according to the definition of congruent triangles, we get ∠EAD equals ∠BAC, therefore ∠EAD equals 80 degrees.

3. Since ∠EAC equals ∠EAD minus ∠DAC, i.e., ∠EAC = 80 degrees minus 35 degrees, so ∠EAC equals 45 degrees.

4. Through the above reasoning, the final answer is 45 degrees.

普通三角形; 三角形的外角; 对顶角

Triangle Angle Sum Theorem

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

<note>In triangle ABC, angle B, angle C, and angle BAC are the three interior angles of triangle ABC</note>, according to the Triangle Angle Sum Theorem, <note>angle B + angle C + angle BAC = 180°</note>, that is <note>∠BAC = 180° - ∠B - ∠C = 180° - 70° - 30° = 80°</note>.

Definition of Congruent Triangles

In the figure of this problem, <note>triangle ABC and triangle ADE are congruent triangles</note>, <note>the corresponding sides and corresponding angles of triangle ABC are equal to those of triangle ADE</note>, that is:
<note>side AB = side AD</note>
<note>side BC = side DE</note>
<note>side CA = side AE</note>
At the same time, the corresponding angles are also equal:
<note>angle BAC = angle DAE</note>
<note>angle ABC = angle ADE</note>
<note>angle BCA = angle DEA</note>.


Question: The area of trapezoid G H J K is 75 square meters. Find the height.

Answer: 5

1. Given the area of trapezoid G H J K is 75 square meters, according to the trapezoid area formula: 'Trapezoid area = (upper base + lower base) * height / 2', draw height KD perpendicular to GH, then let the height of the trapezoid be h meters.

2. The upper base GH of trapezoid G H J K is 17 meters long, and the lower base K J is 13 meters long. Substitute these known values into the trapezoid area formula to get: 75 = (17 + 13) * h / 2.

3. Calculate the part inside the parentheses in the formula: 17 + 13 = 30. Replace the parentheses part in the formula with 30 to get: 75 = 30 * h / 2.

4. Cross multiply to get: 75 * 2 = 30 * h.

5. Calculate the left side 75 * 2 = 150, then: 150 = 30 * h.

6. Divide both sides of the formula by 30 to get: h = 150 / 30.

7. Calculate 150 / 30 = 5 to get: h = 5.

8. Through the above reasoning, it is concluded that the height of trapezoid G H J K is 5 meters.

Definition of Trapezoid

In the figure of this problem, in quadrilateral G H J K, <note>side G H and side K J are parallel</note>, while <note>side G K and side H J are not parallel</note>. Therefore, according to the definition of a trapezoid, quadrilateral G H J K is a trapezoid because <note>it has exactly one pair of parallel sides</note>.

Trapezoid Area Formula

In <note>trapezoid G H J K</note>, <note>side G H and side K J are two parallel sides</note>, <note>side G H is 17 meters long</note>, <note>side K J is 13 meters long</note>, assuming the height between them is h meters, the area of the trapezoid is <note>(G H + K J) * h / 2</note>.

Definition of Altitude

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


Question: In \\odot M, F L = 24, H J = 48, and m \\widehat H P = 65. Find N J.

Answer: 24

1. Given FL = 24, HJ = 48, and ∠HNP = 65°. Find NJ.

2. According to the definition of radius: segment MK and segment MP are both radii of circle M.

3. Given MK is perpendicular to FG at point L, MP is perpendicular to HJ at point N, according to the perpendicular bisector theorem: MK perpendicularly bisects FG, MP perpendicularly bisects HJ. This means N is the midpoint of HJ, NJ = 1/2 * HJ.

4. Since HJ = 48, calculating gives NJ = 48 / 2.

5. Through the above reasoning, the final answer is 24.

圆; 直角三角形; 圆周角; 中点; 弦

Definition of Chord

<note>In the figure of this problem</note>, in circle M, point <note>F and point G are any two points on the circle</note>, <note>line segment FG connects these two points</note>, so <note>line segment FG is a chord of circle M</note>. Similarly, point H and point J are any two points on the circle, <note>line segment HJ connects these two points</note>, so <note>line segment HJ is a chord of circle M</note>.

Definition of Radius

<note>In circle M, point M is the center of the circle, and point K is any point on the circle</note>, <note>the line segment MK is a line segment from the center to any point on the circle</note>, thus <note>the line segment MK is the radius of the circle</note>. Similarly, <note>point M is the center of the circle, and point P is any point on the circle</note>, <note>the line segment MP is a line segment from the center to any point on the circle</note>, thus <note>the line segment MP is the radius of the circle</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>The diameter on radius KM and the diameter on radius MP are respectively perpendicular to chords FG and HJ</note>, then according to the Perpendicular Diameter Theorem, <note>the diameter on radius KM and the diameter on radius MP respectively</note> bisect <note>chord FG</note> and HJ, that is, <note>FG = 2 * FL</note> and <note>HJ = 2 * NJ</note>, and <note>the diameter on radius KM and the diameter on radius MP respectively</note> bisect the arcs corresponding to <note>chord FG</note> and HJ, that is, <note>arc FK = arc KG</note> and <note>arc HP = arc JP</note>.

Midpoint of a Line Segment

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

Original text: <note>The midpoint of line segment HJ is point N</note>. According to the definition of the midpoint of a line segment, <note>point N divides line segment HJ into two equal parts</note>, that is, <note>the lengths of line segments HN and NJ are equal</note>. That is, <note>HN = NJ</note>.