Question: If two sides of a triangle are 12 and 7 respectively, which of the following cannot be the perimeter of the triangle?

Answer: 38

1. According to the triangle inequality theorem, the sum of any two sides of a triangle is greater than the third side, and the difference of any two sides is less than the third side.

2. Let the third side be x, according to the triangle inequality theorem, we get 12 + 7 > x > 12 - 7, i.e., 19 > x > 5.

3. According to the formula for the perimeter of a triangle, the perimeter of a triangle is equal to the sum of the lengths of its three sides, i.e., perimeter P = a + b + c. So 5 + 12 + 7 < P < 19 + 12 + 7, i.e., 24 < x < 38.

4. From the above, we know that the option that does not satisfy the triangle side formula is 38.

Definition of Triangle

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

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

Theorem of Triangle Inequality

In any triangle, the sum of the lengths of any two sides is greater than the length of the third side, and the absolute difference of the lengths of any two sides is less than the length of the third side.

In the figure of this problem, let the triangle be triangle ABC, <note>side AC, side AB, and side BC</note> form the triangle. According to the theorem of triangle inequality, the sum of any two sides is greater than the third side, that is, <note>side AB + side AC > side BC</note>, <note>side AC + side BC > side AB</note>, <note>side AB + side BC > side AC</note>. The difference of any two sides is less than the third side, that is, <note>side AC - side AB < side BC</note>, <note>side BC - side AB < side AC</note>, <note>side BC - side AC < side AB</note>.

Formula for the Perimeter of a Triangle

<note>Let the triangle be triangle ABC</note>, given <note>the three sides of triangle ABC are AB, AC, and BC</note>, according to <note>the formula for the perimeter of a triangle</note>, <note>the perimeter of the triangle is equal to the sum of the lengths of its three sides</note>, i.e., <note>perimeter P = AB + AC + BC</note>.


Question: U J = 9, V J = 3, and Z T = 18. Find S J.

Answer: 6

As shown in the figure, ST = TU, SY = YZ, ZV = VU, so J is the intersection point of the medians of triangle SZU, which is the centroid.

Since JV = 3, according to the centroid theorem, SJ = 6.

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

等腰三角形; 线段; 中点

Centroid Theorem

The centroid of a triangle is the point where its three medians intersect, and it divides each median into two segments, with the longer segment being twice the length of the shorter segment and connecting to the vertex.

In <note>triangle SZU</note>, point J is the <note>centroid of the triangle</note>. The triangle's <note>three medians are the line segments SV, ZT, and UY</note>, which <note>intersect at point J</note>. According to the Centroid Theorem, point J divides each <note>median into two segments in a 2:1 ratio, with the longer segment connecting to the vertex</note>. For example, in the line segment SV, <note>point J divides it into two segments</note>, with the <note>longer segment being SJ (connecting to the vertex) and the shorter segment being JV</note>.

Definition of Median of a Triangle

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

In the diagram of this problem, triangle SZU, <note>vertex S is a vertex of the triangle</note>, <note>side ZU is the side opposite vertex S</note>. <note>Point V is the midpoint of side ZU</note>, <note>line segment SV is the segment from vertex S to midpoint V of side ZU</note>, therefore <note>SV is a median of triangle SZU</note>. Similarly, <note>line segments ZT and UY are also medians of triangle SZU</note>.

Midpoint of a Line Segment

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

<note>The midpoint of line segment SU is point T</note>. According to the definition of the midpoint of a line segment, <note>point T divides line segment SU into two equal parts</note>, that is, <note>line segment ST = TU</note>. <note>The midpoint of line segment SZ is point Y</note>. According to the definition of the midpoint of a line segment, <note>point Y divides line segment SZ into two equal parts</note>, that is, <note>line segment SY = YZ</note>. <note>The midpoint of line segment ZU is point V</note>. According to the definition of the midpoint of a line segment, <note>point V divides line segment ZU into two equal parts</note>, that is, <note>line segment ZV = VU</note>.


Question: Find CD, if AC = x - 3, BE = 20, AB = 16, and CD = x + 5.

Answer: 40

1. Given that in ##△ABC and △AED##, AC = x - 3, CD = x + 5, AB = 16, BE = 20.

2. Observing the figure, ##BC is parallel to ED, according to the parallel axiom 2 of parallel lines, ∠BCA = ∠EDA, ∠ABC = ∠AED##.

3. ##Since ∠BCA = ∠EDA, ∠ABC = ∠AED, by the similarity theorem (AA), △ABC and △AED are similar triangles##.

4. According to the proportionality of corresponding sides of similar triangles, we have AB/##AE## = AC/##AD##.

5. Substituting the known values into the proportion, we get ##16/36 = (x - 3)/(2x + 2)##.

6. Simplifying the proportion, ##16 * (2x + 2) = 36 * (x - 3)##.

7. Expanding the equation, we get ##32x + 32 = 36x - 108##.

8. Rearranging the equation, ##32x - 36x = -108 - 32##.

9. Simplifying, we get -4x = -140.

10. Solving the equation, we find x = 35.

11. Substituting to calculate CD, CD = x + 5 = 35 + 5.

12. Through the above reasoning, the final answer is CD = 40.

Definition of Similar Triangles

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

<note>In this problem, triangle ABC and triangle AED are similar</note>. According to the definition of similar triangles, we have: <note>AB/AE = AC/AD</note>. Using the given side lengths, we derive a proportion equation <note>16/36 = (x - 3)/(2x + 2)</note>, thus solving for <note>x = 35</note>, and further calculating CD. <note>In the figure of this problem, two parallel lines BC and ED are intercepted by the third lines AD and AE</note>, forming the following geometric relationships: <note>corresponding angles: angle ACB and angle ADE are equal</note>, <note>angle ABC and angle AED are equal</note>.

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>, <note>in triangles ABC and AED</note>, if <note>angle ACB is equal to angle ADE</note>, and <note>angle ABC is equal to angle AED</note>, then <note>triangle ABC is similar to triangle AED</note>.

Definition of Corresponding Angles

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

<note>Two parallel lines BC and ED</note> are intersected by two lines AE and AD, where <note>angle ACB and angle ADE are on the same side of the transversal AD, on the same side of the intersected lines BC and ED</note>, thus <note>angle ACB and angle ADE are corresponding angles</note>. Also, <note>angle ABC and angle AED are on the same side of the transversal AE, on the same side of the intersected lines BC and ED</note>, thus <note>angle ABC and angle AED are corresponding angles</note>. <note>Corresponding angles are equal, that is, angle ACB is equal to angle ADE, angle ABC is equal to angle AED</note>.


Question: As shown in the figure, △ABC is inscribed in ⊙O, ∠C=20°, then the degree of ∠OAB is ()

Answer: 70°

1. Given △ABC inscribed in ⊙O, and ∠C=20°.

2. According to the definition of the inscribed angle, ∠C is an inscribed angle. According to the definition of the central angle, ∠AOB is a central angle. According to the inscribed angle theorem, the inscribed angle is equal to half of the central angle that subtends the same arc. Therefore, we get ∠C=½∠AOB, and by calculation, ∠AOB=2∠C=40°.

3. According to the definition of the radius, OA=OB. According to the definition of an isosceles triangle, △OAB is an isosceles triangle. According to the properties of an isosceles triangle, ∴∠OAB=∠OBA.

4. In △OAB, according to the triangle angle sum theorem, we get ∠OAB+∠OBA+∠AOB=180°.

5. Since ∠OAB=∠OBA and ∠AOB=40°, we have 2∠OAB+40°=180°.

6. After simple calculation, we get 2∠OAB=140°, ∴∠OAB=70°.

7. Therefore, we deduce that the measure of ∠OAB is 70°.

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

Definition of Central Angle

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

Definition of Isosceles Triangle

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

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

Inscribed Angle Theorem

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

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

Triangle Angle Sum Theorem

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

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

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 OAB</note>, <note>sides OA and OB are equal</note>. Therefore, according to the properties of an isosceles triangle, the angles opposite the equal sides are equal, i.e., <note>∠OAB = ∠OBA</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 the diagram of this problem, in circle O, point O is the center of the circle, points A and B are any points on the circle, segments OA and OB are segments from the center to any point on the circle,</note> therefore <note>segments OA and OB are the radius of the circle</note>##.


Question: As shown in the figure, if ∠ABC=30°, then the degree of ∠AOC is ()

Answer: 60°

1. Observe and understand the figure, it is known that ∠ABC = 30°.

2. In the circle, ∠AOC is the central angle, while ∠ABC is the inscribed angle corresponding to arc AC.

3. According to the inscribed angle theorem, in a circle, the inscribed angle is equal to half of the central angle corresponding to the same arc, that is, ∠AOC = 2∠ABC.

4. Therefore, it can be deduced that ∠AOC = 2 × 30°.

Definition of Inscribed Angle

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

Definition of Central Angle

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

Inscribed Angle Theorem

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 AC</note> is ∠AOC, and the inscribed angle is ∠ABC. According to the Inscribed Angle Theorem, ∠AOC is equal to twice the inscribed angle <note>∠ABC</note> corresponding to <note>arc AC</note>, that is, <note>∠AOC = 2 × ∠ABC</note>.


Question: m \angle 3 = x - 12 and m \angle 6 = 72. Find x.

Answer: 84

1. According to the figure, ∠2 and ∠6 are corresponding angles formed by two parallel lines being intersected by a transversal.

2. According to the parallel postulate 2, if two parallel lines are intersected by a transversal, then each pair of corresponding angles are equal, thus ∠2 = ∠6.

3. According to the definition of vertical angles, we know ∠2 = ∠3 = x - 12, and ∠6 = 72.

4. Since ∠2 = ∠3 = ∠6, therefore x - 12 = 72.

5. Solve the equation x - 12 = 72, we get x = 72 + 12.

6. Calculate x = 72 + 12 = 84.

7. Through the above reasoning, the final answer is 84.

同位角; 平行线; 对顶角

Definition of Parallel Lines

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

In the diagram of this problem, <note>the vertical line on the left and the vertical line on the right</note> are in the same plane, <note>and they do not intersect</note>. Therefore, according to the <note>definition of parallel lines</note>, the vertical line on the left and the vertical line on the right are parallel lines.

Definition of Corresponding Angles

<note>In the figure of this problem, the two parallel lines are the left line and the right line intersected by a transversal</note>, where <note>angle 2 and angle 6 are on the same side of the transversal</note>, <note>on the same side of the intersected lines</note>, therefore <note>angle 2 and angle 6 are corresponding angles</note>. <note>Corresponding angles are equal, that is, angle 2 equals angle 6</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 context of <note>the vertical parallel lines on the left and the vertical parallel lines on the right intersected by a transversal forming angles</note>, <note>∠2 and ∠6 are corresponding angles</note>, therefore <note>∠2 = ∠6</note>.

Definition of Vertical Angles

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

<note>Two lines intersect</note>, forming four angles: <note>Angle 1, Angle 2, Angle 3, and Angle 4</note>. According to the definition of vertical angles, <note>Angle 1 and Angle 4 are vertical angles</note>, <note>Angle 2 and Angle 3 are vertical angles</note>. Since vertical angles are equal in measure, <note>Angle 1 = Angle 4</note>, <note>Angle 2 = Angle 3</note>.


Question: A square is tangent to a line at point P in the figure above. What is the value of x?

Answer: 30

1. Given that a square is tangent to the line l at point P, let the four points of the square be A, P, C, D, and ∠APC = 90°.

####

2. According to the definition of a straight angle, let the straight angle be ∠EPF, it can be known that x° + 2x° = 180° - ∠APC = 180° - 90° = 90°.

3. Derive the equation x° + 2x° = 90°, that is, 3x° = 90°.

4. Solve the equation 3x° = 90°, and get x = 30.

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

正方形; 直线; 切线; 垂线; 对称

Definition of Square

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

<note>In the figure of this problem, the four sides of the square are equal, namely AP, PC, CD, and DA</note>, and <note>the four internal angles are all right angles (90 degrees)</note>, so this is a square.

Definition of Straight Angle

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

The ray <note>l</note> rotates around the endpoint <note>P</note> to form a straight line with the initial side, forming a <note>straight angle EPF</note>. According to the definition of a straight angle, the measure of a straight angle is 180 degrees, that is, <note>angle EPF = 180 degrees</note>.


Question: Find x.

Answer: 9.5

1. Given that the segments in the triangle are of equal length, according to the markings in the figure, the two segments are equal (i.e., $3x+11=5x-8$).

2. Rearrange and combine like terms in the above equation to get $3x+11=5x-8$ => $11+8=5x-3x$ => $2x=19$.

3. Divide the result by 2 to get $x=rac{19}{2}=9.5$.

4. Through the above reasoning, the final answer is $x=9.5$.

Definition of Line Segment

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

In the diagram of this problem, <note>line segment AD</note> is a part of a straight line, including <note>endpoint A</note> and <note>endpoint D</note> and all points between them. Line segment AD has two endpoints, which are <note>A</note> and <note>D</note>, and <note>every point on line segment AD is located between endpoint A and endpoint D</note>. <note>Line segment BD</note> is a part of a straight line, including <note>endpoint B</note> and <note>endpoint D</note> and all points between them. Line segment BD has two endpoints, which are <note>B</note> and <note>D</note>, and <note>every point on line segment BD is located between endpoint B and endpoint D</note>.


Question: Find x. Assume that all segments that appear to be tangent are tangent.

Answer: 4

1. According to the figure and the problem, the segment outside the circle is tangent to the circle. Let the tangent be segment AB, and the point of tangency between the tangent and the circle be point D.

2. Connect the center of the circle and point D to form triangle CDA. According to the property of the tangent to a circle, the tangent is perpendicular to the radius passing through the point of tangency, i.e., CD⊥AB, so ∠CDA=90°. According to the definition of a right triangle, triangle CDA is a right triangle.

3. According to the Pythagorean theorem, the square of the hypotenuse in a right triangle is equal to the sum of the squares of the two legs, i.e., AC^2=AD^2+CD^2.

4. According to the figure: CD is the radius of circle C, so CD=6, AD=8. Substituting into the formula gives: AC^2=64+36=100, so AC=10.

6. In the figure, x+6=AC, so x=AC-6=10-6=4.

等腰三角形; 切线; 圆

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 C</note> and <note>line AB</note> have exactly one common point, <note>D</note>, which is called the <note>point of tangency</note>. Therefore, <note>line AB</note> is the <note>tangent</note> to <note>circle C</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 triangle ACD, angle ADC is a right angle (90 degrees)</note>, therefore triangle ACD is a right triangle. <note>Side CD and side AD are the legs</note>, <note>side AC is the hypotenuse</note>.

Property of the Tangent Line to a Circle

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

In <note>circle C</note>, <note>point D is the point of tangency of line AB with the circle</note>, and <note>line segment CD is the radius of the circle</note>. According to the property of the tangent line to a circle, <note>the tangent AB is perpendicular to the radius CD at the point of tangency D, i.e., ∠CDA=90 degrees</note>.

Pythagorean Theorem

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

<note>In right triangle CDA</note>, <note>angle CDA is a right angle (90 degrees)</note>, <note>sides CD and AD are the legs</note>, <note>side AC is the hypotenuse</note>, so according to the Pythagorean Theorem, <note>AC² = AD² + CD²</note>.


Question: Find x. A = 357 in^2.

Answer: 21

1. Given the area of the triangle A = 357 square inches, according to the area formula of a triangle, the area of any triangle can be expressed using its base and height as A = 1/2 * base * height.

2. Since the base of the right triangle is 34 inches, let the height be x inches, then using the area formula A = 1/2 * 34 * x, we can calculate x.

3. Substitute the given area value into the formula, i.e., 357 = 1/2 * 34 * x.

4. Simplify the equation to 357 = 17x, thus solving for x = 357/17.

5. After calculation, we get x = 21.

直角三角形; 普通三角形

Area Formula of a Triangle

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

In the <note>triangle</note>, the <note>base is 34 inches</note>, and the <note>height is x inches</note>. According to the area formula of a triangle, the area of the triangle is equal to the <note>base multiplied by the height divided by 2</note>, that is, area = <note>(34 * x) / 2</note>. <note>Given that the area A = 357 square inches</note>, the area formula is expressed as <note>357 = 1/2 * 34 * x</note>.

Definition of Altitude

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

The segment perpendicular to the opposite side from the top vertex is the altitude of that vertex. The segment forms a right angle (90 degrees) with side 34, which indicates that the segment is the perpendicular distance from the top vertex to the opposite side.


Question: Find the length of side XY in the isosceles triangle XYZ.

Answer: 7

1. Given condition: The given figure is an isosceles triangle XYZ, XY and YZ are the two equal sides of the isosceles triangle. ####XY equals YZ.

2. Based on the conclusion from step 1, we can set up the equation 2x + 3 = 4x - 1.

3. Move 4x to the left side and 3 to the right side of the equation, we get 2x - 4x = -1 - 3.

4. Simplify -2x = -4 to get x = 2.

5. Substitute x = 2 into the expression for side XY, calculate: XY = 2(2) + 3 = 4 + 3 = 7.

6. Based on the calculation in step 5, XY = 7.

7. Substitute x = 2 into the expression for side YZ, verify the calculation: YZ = 4(2) - 1 = 8 - 1 = 7.

8. The verification calculation results in YZ = 7, which meets the given condition.

Definition of Isosceles Triangle

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

<note>Side XY is equal to side YZ</note>, therefore <note>triangle XYZ is an isosceles triangle</note>.


Question: Find the length of the height to the hypotenuse.

Answer: 4 \sqrt { 3 }

1. From the figure, it can be seen that ∠GEF is a right angle, and EH is the height, i.e., EH⊥GF, then ∠GHE=∠EHF=90°. According to the definition of a right triangle, triangle EGF, triangle GHE, and triangle EHF are all right triangles. Based on the property of the height on the hypotenuse of a right triangle, the two smaller triangles formed by the height on the hypotenuse GF of the larger triangle EGF are similar to the larger triangle EGF, so triangle EGH~triangle EGF, triangle EHF~triangle EGF, and triangle EGH~triangle EHF, and all are right triangles.

####

2. According to the obtained information: triangle GEH and triangle EFH are similar, yielding a proportion: GH/EH = EH/HF.

3. Given GF = 16 and HF = 12, so GH=GF-HF=16-12=4. Substituting into the proportion gives: 4/EH=EH/12.

4. Cross-multiplying gives EH^2=4 * 12=48, then EH=√48=√(16 * 3)=4√3.

5. The length of the height EH on the hypotenuse GF of triangle EGF is 4√3.

直角三角形; 垂线; 垂直平分线

Definition of Triangle

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

In the figure of this problem, <note>triangle EGF is a geometric figure composed of three non-collinear points E, G, F and their connecting line segments EG, GF, EF</note>. <note>Points E, G, F are the three vertices of the triangle</note>, <note>and line segments EG, GF, EF are the three sides of the triangle</note>. <note>Triangle EGH is a geometric figure composed of three non-collinear points E, G, H and their connecting line segments EG, GH, EH</note>. <note>Points E, G, H are the three vertices of the triangle</note>, <note>and line segments EG, GH, EH are the three sides of the triangle</note>. <note>Triangle EHF is a geometric figure composed of three non-collinear points E, H, F and their connecting line segments EH, HF, EF</note>. <note>Points E, H, F are the three vertices of the triangle</note>, <note>and line segments EH, HF, EF are the three sides of the triangle</note>.

Definition of Right Triangle

In the figure of this problem, in triangle EGF, <note>angle GEF is a right angle (90 degrees)</note>, so triangle EGF is a right triangle. <note>Sides EG and EF are the legs</note>, <note>side GF is the hypotenuse</note>. In triangle EGH, <note>angle GHE is a right angle (90 degrees)</note>, so triangle EGH is a right triangle. <note>Sides HG and EH are the legs</note>, <note>side GE is the hypotenuse</note>. In triangle EHF, <note>angle EHF is a right angle (90 degrees)</note>, so triangle EHF is a right triangle. <note>Sides HE and HF are the legs</note>, <note>side EF is the hypotenuse</note>.

Definition of Altitude

In the figure of this problem, from vertex E perpendicular to the opposite side GF, <note>line segment GH is the altitude from vertex E</note>. <note>Line segment EH forms two right angles (90 degrees) with side GH and side FH</note>, which indicates that <note>line segment GH is the perpendicular distance from vertex E to the opposite side GF</note>.

Property of the Altitude on the Hypotenuse in a Right Triangle

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

In the figure of this problem, <note>in the right triangle EGF, angle GEF is a right angle (90 degrees)</note>, draw the altitude EH from <note>vertex E to the hypotenuse GF</note>. According to the property of the altitude on the hypotenuse in a right triangle, <note>the altitude EH divides the right triangle EGF into two new right triangles EGH and EHF</note>. <note>Triangle EGH is similar to triangle EGF</note>, and <note>triangle EHF is also similar to triangle EGF</note>. Therefore, triangle EGH ~ triangle EGF, triangle EHF ~ triangle EGF, and <note>triangle EGH ~ triangle EHF</note>.

Definition of Similar Triangles

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

<note>Triangle EGH and triangle EHF are similar triangles</note>. According to the definition of similar triangles: <note>∠EGH = ∠FEH</note>, <note>∠EHG = ∠FHE</note>, <note>∠GEH = ∠EFH</note>; <note>GH/EH = EH/HF = GE/EF</note>.


Question: A regular pentagon and a square share a vertex X. Edges XY and XZ are edges of a third regular polygon, which has a vertex at X. How many sides does this polygon have?

Answer: 20

1. Given a regular pentagon and a square sharing vertex X, lying in the same plane.

2. According to the interior angle theorem of regular polygons: the interior angle of the regular pentagon is 108 degrees, and the interior angle of the square is 90 degrees.

3. The angle at the common vertex X is 360 degrees, including the 108 degrees of the regular pentagon and the 90 degrees of the square. The interior angle of the third regular polygon is 360 degrees - 108 degrees - 90 degrees = 162 degrees.

4. According to the interior angle theorem of regular polygons, each interior angle of a regular n-gon is equal to (n-2)×180°/n, where n is the number of sides. Substituting into the formula, we get (n-2)×180/n=162, solving for n gives n = 20.

5. Finally, it is concluded that the third regular polygon has 20 sides.

正多边形; 五边形; 正方形

Definition of Regular Polygon

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

<note>Regular pentagon</note> and <note>square</note> share vertex <note>X</note>, <note>all sides of the regular pentagon are equal</note> and <note>all interior angles are equal</note>; <note>all sides of the square are equal</note>, and <note>all interior angles are equal</note>. Therefore, <note>the regular pentagon</note> and <note>the square are both regular polygons</note>.

Interior Angle Theorem of a Regular Polygon

The measure of each interior angle of a regular n-sided polygon is equal to \(\frac{(n-2) \times 180^\circ}{n}\).

<note>The central angle of the third regular polygon is 162 degrees</note>, using the equation <note>(n-2)×180°÷n=162</note> we can find <note>n = 20</note>. Therefore, <note>the third regular polygon has 20 sides</note>.


Question: ΔRSV ≅ ΔTVS. Find x.

Answer: 12

1. Given ∠RVS = 78°, ∠RSV = 90°, ##in triangle RVS, by the triangle angle sum theorem, we get ∠R = 12°##.

2. ##Since Δ RSV and Δ TVS are congruent right triangles, by the definition of congruent triangles, we know ∠R=∠T=12°##.

3. ##So X=12##.

普通三角形; 直角三角形; 反射

Definition of Right Triangle

In the figure of this problem, in triangle RSV, <note>angle RSV is a right angle (90 degrees)</note>, therefore triangle RSV is a right triangle. <note>Side RS and side SV are the legs</note>, <note>side RV is the hypotenuse</note>. In triangle TVS, <note>angle TVS is a right angle (90 degrees)</note>, therefore triangle TVS is a right triangle. <note>Side TV and side VS are the legs</note>, <note>side TS is the hypotenuse</note>.

Triangle Angle Sum Theorem

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

<note>In the figure of this problem, in triangle RSV, angle RVS, angle RSV, and angle R are the three interior angles of triangle RSV</note>, according to the Triangle Angle Sum Theorem, <note>angle RVS + angle RSV + angle R = 180°</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>Δ RSV and Δ TVS</note> are congruent triangles, <note>Δ RSV and Δ TVS corresponding angles are equal: ∠R=∠T</note>.


Question: As shown in the figure, in ⊙O, OA ⊥ BC, ∠AOB = 48°, D is a point on ⊙O, then the degree of ∠ADC is ()

Answer: 24°

1. Given OA⊥BC, and OA is the radius of ⊙O, ##points B and C are both points on ⊙O, according to the definition of a chord, BC is a chord of ⊙O.##

2. ##According to the perpendicular bisector theorem, OA bisects arc CAB, that is, arc CA = arc AB. Connect OC, according to the definition of central angles, ∠AOC and ∠AOB are central angles; then according to the central angle theorem, ∠AOC = ∠AOB.##

3. ##The central angle corresponding to arc AC is ∠AOC, and the inscribed angle is ∠ADC. According to the inscribed angle theorem, ∠ADC = 1/2 * ∠AOC.##

4. ##Substituting the given condition, we get ∠ADC = 1/2 * 48° = 24°.##

####

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

圆; 圆心角; 圆周角; 垂线; 弧

Definition of Chord

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

<note>In circle O, point C and point B are any two points on the circle</note>, <note>line segment BC connects these two points</note>, so <note>line segment BC is a chord of circle O</note>.

Definition of Central Angle

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

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

Central Angle Theorem

In the same circle or in congruent circles, arcs corresponding to equal central angles are equal, chords corresponding to equal central angles are equal, and the perpendicular distances from the center to these chords are also equal.

<note>In circle O, arc CA = arc AB</note>, according to the Central Angle Theorem: <note>central angle ∠AOC = ∠AOB</note> are equal.

Definition of Inscribed Angle

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

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

Inscribed Angle Theorem

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

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

Perpendicular Diameter Theorem

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

In the diagram of this problem, in circle O, <note>diameter OA is perpendicular to chord BC</note>, then according to the Perpendicular Diameter Theorem, <note>diameter OA bisects chord BC</note>, that is <note>OB=OC</note>, and <note>diameter OA bisects the two arcs subtended by chord BC</note>, <note>that is arc AC = arc AB</note>.


Question: ∠1 and ∠2 form a pair of linear angles, and m∠2 = 67. Find the degree of ∠1.

Answer: 113

1. According to the problem statement, ∠1 and ∠2 form a pair of linear pairs. That is, ∠1 and ∠2 are adjacent and supplementary angles. According to the definition of supplementary angles, the sum of the degrees of the two angles is equal to 180 degrees.

2. It is known that the degree of ∠2 is 67 degrees, so the degree of ∠1 can be solved using the definition of supplementary angles.

3. According to the definition of supplementary angles, ∠1 + ∠2 = 180 degrees.

4. Substituting the known condition, ∠1 + 67 degrees = 180 degrees.

5. Solving this equation, we get ∠1 = 180 degrees - 67 degrees.

6. Calculating 180 degrees - 67 degrees = 113 degrees.

7. Through the above reasoning, the final answer is 113 degrees.

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.

<note>∠1 and ∠2 have a common side</note>, their other sides <note>are extensions in opposite directions</note>, so <note>∠1 and ∠2 are adjacent supplementary angles</note>.

Definition of Supplementary Angles

Two angles are supplementary if and only if the sum of their measures is 180 degrees.

In the figure of this problem, <note>the property of a linear pair indicates that ∠1 and ∠2 are supplementary angles</note>, <note>their degree sum is 180 degrees</note>.


Question: Find x.

Answer: 4 \sqrt { 2 }

1. Given that in the upper left small triangle, one angle of the triangle is 45 degrees, and the triangle contains a right angle, therefore the triangle is a right triangle.

2. Using the sine function in the right triangle, we have: x/8 = sin(45 degrees).

3. Substituting the value of sin(45 degrees), we get x = 4 \\sqrt { 2 }.

4. Through the above reasoning, the final answer is x = 4 \\sqrt { 2 }.

直角三角形; 等腰三角形; 三角形的外角

Definition of Right Triangle

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

<note>In the figure of this problem, the triangle contains a right angle (90 degrees)</note>, so the triangle is a right triangle. <note>Side x and the unknown side are the legs, and side 8 is the hypotenuse</note>.

Definition of Sine Function

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

In the diagram of this problem, in the right triangle, <note>the 45-degree angle is an acute angle</note>, <note>side x is the side opposite the 45-degree angle</note>, and <note>side 8 is the hypotenuse</note>. According to the definition of the sine function, the sine of the 45-degree angle is equal to <note>the ratio of the opposite side x to the hypotenuse 8</note>, that is, <note>sin(45 degrees) = x / 8</note>.


Question: Quadrilateral E F G H is a rectangle. If E F = 4 x - 6 and H G = x + 3, find E F.

Answer: 6

1. Given that quadrilateral EFGH is a rectangle, according to the definition of a rectangle, the opposite sides are equal, so EF=HG.

2. According to the problem, EF = 4x - 6, HG = x + 3, and EF=HG. Therefore, 4x - 6 = x + 3.

3. Solving the equation 4x - 6 = x + 3, first subtract x from both sides: 3x - 6 = 3.

4. Next, add 6 to both sides: 3x = 9.

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

6. Substitute x = 3 into EF = 4x - 6, we get EF = 4*3 - 6 = 12 - 6 = 6.

7. Through the above reasoning, the final answer is that the length of EF is 6.

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 EFGH is a rectangle</note>, whose interior angles <note>∠EHG, ∠HGF, ∠GFE, and ∠FEH are all right angles (90 degrees)</note>, and <note>side EF is parallel and equal in length to side HG</note>, <note>side EH is parallel and equal in length to side FG</note>.


Question: Using parallelogram M N P R, find m \angle R N P.

Answer: 38

1. The given conditions are the interior angles and some algebraic expressions of the side lengths of parallelogram M N P R. According to the definition of ##parallelogram##, M N is parallel to RP, MR is parallel to NP.

####

##2. According to the information in the problem, ∠M R N = 38°##.

##3. Since the side MR of parallelogram M N P R is parallel to side NP, the two parallel lines are intersected by the diagonal RN, forming alternate interior angles ∠MRN and ∠PNR. According to the parallel postulate 2, the alternate interior angles ∠MRN and ∠PNR are equal, i.e., ∠MRN=∠PNR.##

##4. As can be seen from the figure, ∠MRN=38°, then ∠RNP=∠MRN= 38°##.

####

##5##. Through the above reasoning, the final answer is 38°.

平行四边形; 对顶角; 邻补角; 内错角

Definition of Parallelogram

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

<note>Quadrilateral M N P R is a parallelogram</note>, <note>side M N is parallel and equal to side R P</note>, <note>side M R is parallel and equal to side N P</note>.

Parallel Postulate 2 of Parallel Lines

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

In the figure of this problem, two parallel lines <note>MR and NP</note> are intersected by a third line <note>RN</note>, forming the following geometric relationships:
1. Corresponding angles: None.
2. Alternate interior angles: <note>angle MRN and angle RNP are equal</note>.
3. Consecutive interior angles: <note>angle MRP and angle RPN are supplementary, i.e., angle MRP + angle RPN = 180 degrees</note>; <note>angle RMN and angle PNM are supplementary, i.e., angle RMN + angle PNM = 180 degrees</note>.
These relationships illustrate that when two parallel lines are intersected by a third line, corresponding angles are equal, alternate interior angles are equal, and consecutive interior angles are supplementary.

Definition of Alternate Interior Angles

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

<note>Two parallel lines MR and NP are intersected by a transversal RN</note>, where <note>angle MRN and angle RNP are located between the two parallel lines and on opposite sides of the transversal RN</note>, therefore <note>angle MRN and angle RNP are alternate interior angles</note>. <note>Alternate interior angles are equal, that is, angle MRN is equal to angle RNP</note>.


Question: Find y.

Answer: 8 \sqrt { 3 }

1. Given that this is a right triangle, the sides and corresponding angles are labeled as shown in the figure. Let this right triangle be ABC, where ∠ACB is the right angle, ∠CAB is 60 degrees, and side CA=16√3.

2. Based on the complementary nature of acute angles in a right triangle, we can determine that ∠B=30°. Therefore, triangle ABC is a 30°-60°-90° triangle.

3. According to the properties of a 30°-60°-90° triangle, we know that AB=2*AC. Substituting the values, we get 2y=16√3.

直角三角形; 正弦; 余弦

Definition of Right Triangle

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

Properties of 30°-60°-90° Triangle

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

In the <note>30°-60°-90° triangle ABC</note>, <note>angle B is 30 degrees</note>, <note>angle A is 60 degrees</note>, and <note>angle C is 90 degrees</note>. <note>Side AB is the hypotenuse</note>, <note>side y is the side opposite the 30-degree angle</note>, and <note>side x is the side opposite the 60-degree angle</note>. According to the properties of a 30°-60°-90° triangle, <note>side y is equal to half of side AB</note>, and <note>side x is equal to side y times √3</note>. That is: <note>y = 1/2 * 16√3</note>, <note>x = y * √3</note>.


Question: \overrightarrow B A and \overrightarrow B C are opposite rays, \overrightarrow B D bisects \angle A B F. If m \angle F B C = 2 x + 25 and m \angle A B F = 10 x - 1, find m \angle D B F.

Answer: 64.5

1. Given that ray BA and ray BC are opposite rays, therefore ∠ABC is a straight angle. According to the definition of a straight angle, ∠ABC = 180°.

2. According to the problem, ∠FBC = 2x + 25 and ∠ABF = 10x - 1. Based on the principle of angle addition, ∠ABC can be expressed as ∠ABF + ∠FBC = 180°. Therefore, (10x - 1) + (2x + 25) = 180.

3. Solving the equation (10x - 1) + (2x + 25) = 180, we get 12x + 24 = 180. Then, by solving the linear equation, we get 12x = 156, further solving we get x = 13.

4. Since ray BD bisects ∠ABF, ∠ABD = ∠DBF = 1/2∠ABF.

5. Substituting x = 13 into ∠ABF = 10x - 1, we get ∠ABF = 10(13) - 1 = 129.

6. Since ray BD bisects ∠ABF, ∠DBF = 1/2 * 129° = 64.5°.

7. Through the above reasoning, the final answer is ∠DBF = 64.5°.

射线; 邻补角; 对顶角

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>In the figure of this problem, the vertex of angle ABF is point B. A line BD is drawn from point B, which divides angle ABF into two equal angles, namely angle ABD and angle DBF are equal.</note> Therefore, <note>line BD is the angle bisector of angle ABF</note>. This means <note>∠ABD = ∠DBF = 1/2∠ABF</note>.

Definition of an Angle

An angle is a geometric figure formed by two rays sharing a common endpoint. This common endpoint is called the vertex of the angle, and the two rays are referred to as the sides of the angle.

In the problem diagram, <note>angle ABC is composed of two rays AB and BC</note>, which share a common endpoint <note>B</note>. This common endpoint <note>B is called the vertex of angle ABC</note>, and <note>rays AB and BC are called the sides of angle ABC</note>. In the problem diagram, <note>angle FBC is composed of two rays FB and BC</note>, which share a common endpoint <note>B</note>. This common endpoint <note>B is called the vertex of angle FBC</note>, and <note>rays FB and BC are called the sides of angle FBC</note>. In the problem diagram, <note>angle ABF is composed of two rays AB and BF</note>, which share a common endpoint <note>B</note>. This common endpoint <note>B is called the vertex of angle ABF</note>, and <note>rays AB and BF are called the sides of angle ABF</note>. In the problem diagram, <note>angle ABD is composed of two rays AB and BD</note>, which share a common endpoint.

Definition of Straight Angle

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

In the figure of this problem, <note>ray BA</note> rotates around <note>endpoint B</note> to form a straight line with <note>the initial side, creating a straight angle ABC</note>. According to the definition of a straight angle, <note>the measure of a straight angle is 180 degrees, i.e., angle ABC = 180 degrees</note>.


Question: The radius of circle J is 10 units, the radius of \\odot K is 8 units, and B C = 5.4 units. Find C K.

Answer: 2.6

1. Given conditions: ##The radius of circle K is 8 units, and the length of BC is 5.4 units.##

2. ##In circle K, according to the definition of the radius, the distance from B to K is the radius of the circle, BK=8##.

3. ##According to the figure, BK=BC+CK##.

4. ##Given BK=8 and BC=5.4, then 8=5.4+CK, CK=8-5.4=2.6##.

####

5. ##Through the above reasoning, the final answer is CK = 2.6 units.##

Definition of Radius

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

<note>In the diagram of this problem, in circle K, point K is the center of the circle</note>, <note>point B and point D are any points on the circle</note>, <note>line segment BK and line segment KD are segments from the center to any point on the circle</note>, therefore <note>line segment BK and line segment KD are the radii of circle K</note>, with a length of 8 units.

Definition of Line Segment

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

<note>Line segment BC is a part of a straight line</note>, including <note>endpoint B and endpoint C</note> and all points between them. Line segment BC has two endpoints, which are <note>B and C</note>, and <note>every point on line segment BC is located between endpoint B and endpoint C</note>.


Question: Solve for x.

Answer: 70

1. Given that arc FG = 70°, FG = HJ, let O be the center of the circle.

2. According to the property of the central angle, the degree of the central angle is equal to the degree of its corresponding arc. Given arc FG = 70°, the central angle of arc FG, ∠FOG = 70°.

3. In the figure, draw a short line on each of the sides FG and HJ to indicate that these two sides are equal in length, i.e., chord FG = chord HJ. According to the central angle theorem, in the same circle or congruent circles, equal central angles correspond to equal arcs and equal chords. Since chord FG = chord HJ, arc HJ = arc FG.

4. According to the central angle theorem, in the same circle or congruent circles, equal central angles correspond to equal arcs. Since arc HJ = arc FG, the central angles corresponding to these arcs are also equal, i.e., ∠HOJ = ∠FOG = 70°.

圆; 圆心角; 切线; 等腰三角形; 圆周角

Definition of Circle

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

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

Definition of Arc

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

In the figure of this problem, there are two points F and G on circle O, <note>arc FG</note> is a segment of the curve connecting these two points. According to the definition of an arc, <note>arc FG</note> is a segment of the curve between two points F and G on the circle. There are two points H and J on circle O, <note>arc HJ</note> is a segment of the curve connecting these two points. According to the definition of an arc, <note>arc HJ</note> is a segment of the curve between two points H and J on the circle.

Definition of Chord

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

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

Definition of Central Angle

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

<note>In circle O, points F and G are two points on the circle</note>, <note>the center of the circle is point O</note>. The line segments <note>OG and OF</note> form an angle called the <note>central angle ∠FOG</note>. <note>In circle O, points H and J are two points on the circle</note>, <note>the center of the circle is point O</note>. The line segments <note>OH and OJ</note> form an angle called the <note>central angle ∠HOJ</note>.

Properties of Central Angles

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

In the figure of this problem, it is known that the <note>central angle ∠FOG</note> corresponds to the <note>arc FG</note>. According to the properties of central angles, the degree measure of a central angle is equal to the degree measure of its corresponding arc, that is, <note>the degree measure of ∠FOG = the degree measure of arc FG</note>. It is known that the <note>central angle ∠HOJ</note> corresponds to the <note>arc HJ</note>. According to the properties of central angles, the degree measure of a central angle is equal to the degree measure of its corresponding arc, that is, <note>the degree measure of ∠HOJ = the degree measure of arc HJ</note>.

Central Angle Theorem

In the same circle or in congruent circles, arcs corresponding to equal central angles are equal, chords corresponding to equal central angles are equal, and the perpendicular distances from the center to these chords are also equal.

In this problem diagram, in circle O, <note>central angle ∠FOG</note> is equal to <note>∠HOJ</note>. According to the <note>Central Angle Theorem</note>: 1. <note>The arcs corresponding to the two central angles are equal, that is, arc FG = arc HJ</note>; 2. The corresponding chords are equal, that is, <note>chord FG = chord HJ</note>.


Question: Find x.

Answer: 10 \sqrt { 3 }

1. Given a right triangle with one angle of 60°, according to the triangle angle sum theorem, the last angle is 30°.

2. In this triangle, the relationship of the sides is: side length x is the adjacent side of the 30° angle, and the known side length of 20 is the hypotenuse of the right triangle. Therefore, according to the cosine function: cos(30°) = x / 20.

3. Solving the equation, the result is x = 10√3.

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

直角三角形; 正弦; 余弦

Definition of Right Triangle

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

<note>The angle between side x and side y is a right angle (90 degrees)</note>, therefore the triangle is a right triangle. <note>Side x and side y are the legs</note>, <note>side 20 is the hypotenuse</note>.

Cosine Function

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

<note>In a right triangle, side x is the adjacent side to the 30° angle, and side 20 is the hypotenuse</note>. Therefore, according to the definition of the cosine function, <note>the cosine of the 30° angle is equal to the ratio of the adjacent side x to the hypotenuse 20</note>, that is, <note>cos(30°) = x / 20</note>.

Triangle Angle Sum Theorem

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

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


Question: Line AB is tangent to circle O. If AB = 24 and OB = 25, find DB.

Answer: 18

1. Given that line AB is tangent to circle O, point B is the point of tangency, OB = 25, AB = 24. According to the property of tangents, the tangent is perpendicular to the radius at the point of tangency, thus ∠OAB is a right angle.

2. Since ∠OAB is a right angle, according to the definition of a right triangle, triangle OAB is a right triangle. Using the Pythagorean theorem (i.e., the square of the hypotenuse of a right triangle is equal to the sum of the squares of the other two sides), we can find the length of OA.

3. The application of the Pythagorean theorem is: OB^2 = OA^2 + AB^2, where OB = 25, AB = 24

4. Substituting the given conditions, we get 25^2 = OA^2 + 24^2. Through calculation, 625 = OA^2 + 576.

5. Solving the equation 625 - 576 = OA^2, we get OA^2 = 49, thus OA = 7.

6. OA is the radius of circle O, OD is also the radius of circle O, therefore OD = OA = 7.

7. Since AO and OD are the radii of circle O, OD = OA = 7, OB is known to be 25, so DB = OB - OD.

8. Calculating DB = 25 - 7 = 18.

9. Through the above reasoning, the final answer is 18.

切线; 直角三角形; 线段; 圆

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.

Original: <note>Circle O and line AB have exactly one common point B</note>, this common point is called the point of tangency. Therefore, <note>line AB is the tangent to circle O</note>.

Definition of Right Triangle

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

Definition of Radius

<note>Point O is the center of the circle</note>, point A and point D are any points on the circle, <note>line segment OA and line segment OD are segments from the center of the circle to any point on the circle</note>, therefore <note>line segment OA and line segment OD are the radii of circle O</note>.

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

In circle O, point B is the point of tangency of line AB with the circle, line segment OB is the radius of the circle. According to the property of the tangent line to a circle, the tangent line AB is perpendicular to the radius OB at the point of tangency B, that is, ∠OAB = 90 degrees.

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, <note>in the right triangle OAB, ∠OAB is a right angle (90 degrees)</note>, <note>sides OA and AB are the legs</note>, <note>side OB is the hypotenuse</note>, so according to the Pythagorean Theorem, <note>OB^2 = OA^2 + AB^2</note>.


Question: Find the length of \widehat A B. Round the result to two decimal places.

Answer: 9.77

1. From the figure, the central angle ∠AOB = 80°, the diameter of the circle is 14 meters, according to the definition of radius, then the radius OA = 14/2 = 7 meters.

2. Given the central angle ∠AOB = 80°, OA = 7, according to the conversion formula between degrees and radians, calculate the central angle θ = 80 * (π/180) = 4π/9. Then according to the arc length formula of the sector: L = θr, (θ is expressed in radians), here r is the radius.

3. Substitute the known values into the formula to get: arc length AB = θr = 4π/9 × 7.

4. Calculate to get the arc length AB ≈ 9.772.

5. After the above calculation, round to two decimal places, so the arc length is 9.77 meters.

Definition of Radius

In the figure of this problem, in circle O, <note>point O is the center of the circle</note>, <note>point A and point B are any points on the circle</note>, <note>line segment OA and line segment OB are line segments from the center of the circle to any point on the circle</note>, therefore <note>line segment OA and line segment OB are the radii of the circle</note>, their lengths are both 7 meters, because <note>the radius is equal to half of the diameter (14 meters/2)</note>.

Definition of Central Angle

In <note>circle O</note>, <note>point A and point B are two points on the circle</note>, and <note>the center of the circle is point O</note>. The angle ∠AOB formed by the lines <note>OA and OB is called the central angle</note>, and its measure is <note>80°</note>.

Formula for the Length of an Arc of a Sector

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

In the figure of this problem, in <note>sector AOB</note>, the <note>central angle AOB is θ (expressed in radians)</note>, and the <note>radius is r</note>. According to the formula for the length of an arc of a sector, the arc length L is equal to the <note>central angle θ</note> multiplied by the <note>radius r</note>, that is, <note>L = θ * r</note>.

Formula for Conversion between Degrees and Radians

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

<note>In the figure of this problem, the angle of AOB is 80</note>, according to the formula for conversion between degrees and radians, the conversion to radians = degrees * (π/180), that is, <note>80 * (π/180) = 4π/9</note>.

Definition of Sector

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

In the figure of this problem, in <note>sector AOB</note>, <note>radius OA</note> and <note>radius OB</note> are two radii of the circle, and <note>arc AB</note> is the arc enclosed by these two radii. Therefore, according to the definition of a sector, the figure formed by these two radii and the <note>arc AB</note> they enclose is a <note>sector</note>.


Question: ΔWXY ≅ ΔWXZ. Find y.

Answer: 4

1. Given △WXY ≌ △WXZ, according to the definition of congruent triangles, corresponding sides are equal, we can obtain XY = XZ.

2. Given XY = 19, XZ = 3y + 7, according to the congruent sides XY = XZ, we can obtain 19 = 3y + 7.

3. Based on the equation 19 = 3y + 7, simplifying it gives 3y = 19 - 7.

6. Through the above derivation, we can finally obtain the value of y as 4.

等腰三角形; 直角三角形

Definition of Congruent Triangles

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

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


Question: Find x.

Answer: \sqrt { 21 }

1. Let the three vertices of the triangle be A, B, and C. From the figure, it can be seen that one of the angles is 90 degrees. According to the definition of a right triangle, this triangle is a right triangle. Also, in the figure, one of the right-angle sides has a length of 2, the other right-angle side has an unknown length x, and the hypotenuse has a length of 5.

2. According to the Pythagorean theorem, in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the two right-angle sides. Specifically, if the right-angle sides are a and b, and the hypotenuse is c, then c^2 = a^2 + b^2. In this problem, let a=2, b=x, and c=5.

3. Substitute the known side lengths into the Pythagorean theorem to get 5^2 = 2^2 + x^2.

4. Calculate 5^2 to get 25, and 2^2 to get 4. Substitute these values into the equation to get 25 = 4 + x^2.

5. Solve the equation 25 = 4 + x^2 by first moving 4 to the left side of the equation to get 25 - 4 = x^2, which means x^2 = 21.

6. By taking the square root of the equation x^2 = 21, we can get x = √21.

7. Finally, the value of x in the right triangle is √21.

直角三角形; 直线; 余弦

Definition of Right Triangle

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

Pythagorean Theorem

In this problem, the triangle is a right triangle, where <note>one angle is marked with a right angle symbol</note>. It is known that one leg of the right triangle is 2, the other leg is x, and the hypotenuse is 5. According to the Pythagorean Theorem, <note>5² = 2² + x²</note>, that is, 25 = 4 + <note>x²</note>.


Question: ABCD is a parallelogram with side lengths as shown in the figure. The perimeter of ABCD is 22. Find AB.

Answer: 7

1. Given ABCD is a parallelogram, ##according to the properties of parallelogram theorem##, we have AB = CD and AD = BC.

2. The values of these sides are AB = 2y + 1, ##CD = 3 - 4w, AD = 3x - 2, BC = x - w + 1##.

3. ##Given AB = CD and AD = BC, we have 2y + 1 = 3 - 4w, 3x - 2 = x - w + 1##.

4. ##Solving the system of equations: For the equation 2y + 1 = 3 - 4w, we get 2y + 4w = 2##.

5. ##For the equation 3x - 2 = x - w + 1, we get 3x - 2 - x + w = 1, i.e., 2x + w = 3##.

6. The problem states the perimeter of ABCD is 22, therefore we have 2(AB) + 2(AD) = 22, substituting the known conditions, ##we get 2(2y + 1) + 2(3x - 2) = 22, expanding the right side of the equation we get 4y + 2 + 6x - 4 = 22##.

7. ##Simplifying we get: 4y + 6x = 24##.

8. Solving the system of equations: Using 2y + 4w = 2 and 2x + w = 3, ##we can derive w = 3 - 2x and 2y + 4(3 - 2x) = 2, i.e., 2y + 12 - 8x = 2, we get 2y = 8x - 10##.

9. ##Substituting into 4y + 6x = 24 we get: 2(8x - 10) + 6x = 24, i.e., 16x - 20 + 6x = 24, i.e., 22x = 44, finally we get x = 2##.

10. Using x = 2 in 2x + w = 3, ##we get w = -1##.

11. Substituting y = 3 into the expression for AB 2y + 1 we get AB = 2 × 3 + 1 = 7.

12. Through the above reasoning, the final answer is the length of AB is 7.

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 parallelogram ABCD, <note>parallelogram ABCD</note>, the opposite angles <note>∠DAB and ∠BCD are equal</note>, and the opposite angles <note>∠ABC and ∠CDA are equal</note>; the sides <note>AB and CD are equal</note>, and the sides <note>AD and BC are equal</note>. Therefore, we have <note>AB = CD</note> and <note>AD = BC</note>, that is, <note>2y + 1 = 3 - 4w</note> and <note>3x - 2 = x - w + 1</note>.##


Question: In \odot B, C E = 13.5. Find B D. Round to two decimal places.

Answer: 4.29

1. Given that the radius of circle B is AB = 8, according to the definition of a chord, CE is a chord on circle O, CE = 13.5, and AF is perpendicular to CE at point D.

2. AF is perpendicular to CE and passes through the center B, according to the perpendicular bisector theorem, CE is bisected by BD, so CD = DE. According to CE = CD + DE, we get CD = 13.5 / 2 = 6.75.

3. In the right triangle BDC, BC is the radius of circle B, BC = AB = 8, according to the Pythagorean theorem, BD^2 + CD^2 = BC^2.

4. Substitute the known values CD = 6.75 and BC = 8 into the equation, we get BD^2 + 6.75^2 = 8^2.

5. Calculate 6.75^2 = 45.5625 and 8^2 = 64, substitute to get BD^2 = 64 - 45.5625.

6. Calculate 64 - 45.5625 = 18.4375.

7. Finally, BD = √18.4375.

8. Calculate BD = 4.2935, round to the nearest hundredth BD ≈ 4.29.

9. Through the above reasoning, the final answer is 4.29.

圆; 垂线; 垂直平分线; 弦

Definition of Chord

In <note>circle B</note>, <note>point C and point E</note> are any two points on the circle, and <note>line segment CE</note> connects these two points, so <note>line segment CE is a chord of circle B</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 AF is the diameter, connecting the center B and points C and E on the circumference</note>, with a length equal to twice the radius, i.e., <note>AF = 2 * AB</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.

Original text: <note>In circle B, point B is the center, and points A and F are any points on the circle</note>, <note>segments BA and BF are segments from the center to any point on the circle</note>, therefore <note>segments BA and BF are the radii of circle B</note>, with a length of 8.

Definition of Right Triangle

In the figure of this problem, in triangle BCD, <note>angle BDC is a right angle (90 degrees)</note>, therefore <note>triangle BCD is a right triangle</note>. <note>Side BD and side CD are the legs</note>, <note>side BC is the hypotenuse</note>.

Perpendicular Diameter Theorem

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

<note>In circle B, diameter AF is perpendicular to chord CE</note>, then according to the Perpendicular Diameter Theorem, <note>diameter AF bisects chord CE</note>, that is <note>CD=DE</note>, and <note>diameter AF bisects the two arcs subtended by chord CE</note>, that is <note>arc CE = arc CF</note>.

Pythagorean Theorem

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


Question: Find G I, if G H = 9, G K = 6, and K J = 4.

Answer: 15

1. Given G H = 9, G K = 6, K J = 4, ##K H ∥ J I##.

2. ##According to the parallel axiom 2 of parallel lines, the definition of parallel lines, and the definition of corresponding angles, we have ∠GKH=∠GJI. According to the similarity theorem (AA) of triangles, since ∠GKH=∠GJI and ∠KGH=∠JGI, △GKH∽△GJI##. According to the definition of similar triangles: corresponding sides are proportional, corresponding angles are equal.

3. Let the similarity ratio be k, then we have G H / G I = G K / G J.

4. According to the given conditions, G K = 6, G J = G K + K J = 6 + 4 = 10.

5. ##Substituting the given conditions into the equation, we get: 9/GI=6/10##.

7. Thus, through the above reasoning, the final answer is G I = 15.

等腰三角形; 线段; 普通三角形

Definition of Similar Triangles

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

<note>In the figure of this problem, triangle GKH and triangle GJI are similar triangles.</note> According to the definition of similar triangles: <note>∠GKH = ∠GJI, ∠GHK = ∠GIJ, ∠KGH = ∠JGI</note>; <note>GH/GI = GK/GJ = KH/JI</note>.

Definition of Parallel Lines

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

<note>Line segment KH and line segment JI are located in the same plane</note>, and they <note>do not intersect</note>, therefore according to the definition of parallel lines, <note>line segment KH and line segment JI are parallel lines</note>, represented as <note>KH ∥ JI</note>.

Parallel Postulate 2 of Parallel Lines

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

<note>Two parallel lines KH and JI</note> are intersected by <note>a third line GJ</note>, forming the following geometric relationships: 1. <note>Corresponding angles: angle GKH and angle GJI are equal</note>. These relationships indicate that when two parallel lines are intersected by a third line, <note>corresponding angles are equal</note>.

Definition of Corresponding Angles

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

<note>Two parallel lines KH and JI</note> are intersected by a <note>line GJ</note>, where <note>angle GKH and angle GJI are on the same side of the intersecting line GJ and on the same side of the two intersected lines KH and JI</note>, therefore <note>angle GKH and angle GJI are corresponding angles</note>. <note>Corresponding angles are equal, i.e., angle GKH is equal to angle GJI</note>.

Similarity Theorem for Triangles (AA)

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

<note>Angle GKH is equal to angle GJI</note>, and <note>angle KGH is equal to angle JGI</note>, so <note>triangle GKH is similar to triangle GJI</note>.


Question: Circle I is the incircle of triangle ABC, and D, E, F are the 3 points of tangency. If ∠DEF=52°, then the degree measure of ∠A is ()

Answer: 76°

1.## Given that circle I is the incircle of triangle ABC, D, E, F are the points of tangency. Connect DI, FI##.

2. ##From the figure, we know that ∠DEF conforms to the definition of a circumferential angle, which is the circumferential angle subtended by arc DF, and ∠DIF conforms to the definition of a central angle, which is the central angle subtended by arc DF##.

3. ##According to the theorem of circumferential angles, we have ∠DIF = 2∠DEF = 52° × 2 = 104°##

4.##According to the property of tangents to a circle, we know that ∠IDA = ∠IFA = 90°##

5.##Based on the theorem of the sum of interior angles of a quadrilateral, we know that the sum of the interior angles of quadrilateral ADIF is 360°, i.e., ∠A + ∠IDA + ∠IFA + ∠DIF = 360°##

6.##Substituting ∠IDA = ∠IFA = 90°, ∠DIF = 104° into the above equation and simplifying, we get ∠A = 76°##

7.##Through the above reasoning, the final answer is 76°##

圆; 切线; 内错角; 普通三角形; 圆内接四边形

Incircle

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

<note>The incircle of triangle ABC is circle I</note>, <note>with center I</note>. <note>Circle I is tangent to each side AB, BC, CA of triangle ABC</note>, <note>with the tangency points being D, E, F respectively</note>. <note>The center I is called the incenter</note>, and <note>the distances from incenter I to each side AB, BC, CA of triangle ABC are equal</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 this problem, in circle I, points D, E, F are on the circle</note>, <note>the central angle corresponding to arc DEF and arc DF is ∠DIF</note>, and the inscribed angle is <note>∠DEF</note>. According to the Inscribed Angle Theorem, <note>∠DEF is equal to half of the central angle ∠DIF corresponding to arc DEF, i.e., ∠DEF = 1/2 ∠DIF</note>.

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

<note>角IDA</note>, <note>角IFA</note>, <note>角DIF</note>, and <note>角A</note> are the four interior angles of the quadrilateral. According to the Sum of Interior Angles of a Quadrilateral Theorem, the sum of these four interior angles is 360°, that is, <note>角IDA + 角IFA + 角DIF + 角A = 360°</note>.

Definition of Inscribed Angle

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

<note>In circle I, the vertex of angle DEF (point E) is on the circumference</note>, <note>the two sides of angle DEF intersect circle I at points D and F respectively</note>. Therefore, <note>angle DEF 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.

Original text: <note>In circle I</note>, <note>point D and point F are two points on the circle</note>, <note>the center of the circle is point I</note>. <note>The angle ∠DIF formed by the lines DI and FI is called the central angle</note>.

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

In the diagram of this problem, in circle <note>I</note>, point <note>D</note> is the point of tangency between line <note>AB</note> and the circle, and segment <note>DI</note> is the radius of the circle. According to the property of the tangent line to a circle, the tangent line <note>AB</note> is perpendicular to the radius <note>DI</note> at the point of tangency <note>D</note>, i.e., <note>∠IDA=90 degrees</note>. In circle <note>I</note>, point <note>F</note> is the point of tangency between line <note>AC</note> and the circle, and segment <note>IF</note> is the radius of the circle. According to the property of the tangent line to a circle, the tangent line <note>AC</note> is perpendicular to the radius <note>IF</note> at the point of tangency <note>F</note>, i.e., <note>∠IFA=90 degrees</note>.


Question: As shown in the figure, in Rt△ABC, ∠C=90°, AC=4, BC=3, then the value of sinB is equal to ()

Answer: \frac{4}{5}

1. Given in the right triangle ABC, ##∠C##=90°, AC=4, BC=3.

2. According to the ##Pythagorean theorem##, in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the two legs, i.e., ##AB?= AC?+ BC?##.

3. Substituting the given conditions, we get ##AB? = 4? + 3?##.

4. Calculating, we get ##AB? ##= 16 + 9 = 25.

6. According to the ##definition of the sine function##, ##sin∠B ##= opposite side/hypotenuse.

8. Substituting the values, we get ##sin∠B ##= 4/5.

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

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 <note>right triangle ABC</note>, <note>angle B is an acute angle</note>, <note>side AC is the side opposite to angle B</note>, and <note>side AB is the hypotenuse</note>. According to the definition of the sine function, <note>the sine of angle 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

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

<note>Angle C is a right angle (90 degrees)</note>, sides AC and BC are the legs, side AB is the hypotenuse, so according to the Pythagorean Theorem, <note>AB² = AC² + BC²</note>.


Question: Find y.

Answer: 6 \sqrt { 2 }

1. Given that in a right triangle, the hypotenuse is 12, and one acute angle is 45°.

2. Because in a right triangle, one acute angle is 45°, this is a 45°-45°-90° triangle. For a 45°-45°-90° triangle, the two legs are equal, and the hypotenuse is √2 times the length of a leg.

3. According to the above property, side x equals side y, thus x=y=6√2.

4. After the above reasoning, the final result is y=6√2.

直角三角形; 正弦; 余弦; 正切

Definition of Right Triangle

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

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

Pythagorean Theorem

<note>In the figure of this problem</note>, in a right triangle, <note>one angle is a right angle (90 degrees)</note>, <note>sides x and y are the legs</note>, <note>side 12 is the hypotenuse</note>, so according to the Pythagorean Theorem, <note>12² = x² + y²</note>.

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

<note>In a right triangle, one angle is a right angle (90 degrees)</note>, according to the <note>Complementary Property of Acute Angles in Right Triangle</note>, the sum of the <note>other two angles is 90 degrees</note>, that is, <note>45-degree angle + unknown angle = 90°</note>.

Properties of Isosceles Triangle

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

<note>In an isosceles triangle, the two acute angles are equal</note>. Therefore, according to the properties of isosceles triangles, <note>the sides opposite the equal angles are equal, i.e., side x = side y</note>.


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

Answer: 30

1. Given the parallelogram J K L M, according to ##the definition of a parallelogram, it can be concluded that J K is parallel to ML##.

2. According to ##the parallel axiom 2 of parallel lines, alternate interior angles are equal, therefore ∠JLM=∠KJL##.

3. ##Given ∠JLM=30°, so ∠KJL=30°##.

####

##4##. Through the above reasoning, the final answer is ∠K J L = 30°.

平行四边形; 对顶角; 邻补角; 内错角

Definition of Parallelogram

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

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

Parallel Postulate 2 of Parallel Lines

<note>In the diagram of this problem, two parallel lines JK and ML</note> are intersected by a third line <note>JL</note>, forming the following geometric relationship: alternate interior angles: <note>angle JLM and angle KJL are equal</note>. This demonstrates that when two parallel lines are intersected by a third line, the alternate interior angles are equal.

Definition of Alternate Interior Angles

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

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


Question: Find x.

Answer: 24 \sqrt { 3 }

1. In a right triangle, it is known that it contains a 30° angle, the ratio of the side opposite to the 30° angle to the hypotenuse is 1:2, i.e., sin 30° = 1/2, at this time the hypotenuse y is 24 * 2 = 48.

2. Using the cosine formula cos(30°) = adjacent side / hypotenuse, we have cos(30°) = x / 48.

4. Solving the above equation, x = 48 * (√3)/2, i.e., x = 24√3.

5. Through the above reasoning, the final answer is x = 24√3.

Definition of Right Triangle

In the figure of this problem, in the triangle, <note>the right angle is located at the upper left corner</note>, therefore this triangle is a right triangle. <note>Side 24 and side x are the legs</note>, <note>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.

In the figure of this problem, in the 30°-60°-90° triangle, <note>let angle C be 90 degrees, AC=24, AB=y, BC=x</note>. According to the properties of the 30°-60°-90° triangle, <note>side AC is equal to half of side AB</note>, <note>side BC is equal to √3 times side AC</note>. That is: <note>AC = 1/2 * AB, BC = AC * √3</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.

<note>In the diagram of this problem, in the right-angled triangle, side x is the adjacent side, and side y is the hypotenuse</note>. Therefore, according to the definition of the cosine function, <note>the cosine of the angle 30° is equal to the ratio of the adjacent side x to the hypotenuse y, that is, cos(30°) = x/y</note>.


Question: As shown in the figure, given that in ⊙O, the central angle ∠AOB=100°, then the inscribed angle ∠ACB is equal to ().

Answer: 130°

1. Given the minor angle ∠AOB = 100°, we know the major angle ∠AOB = 360° - 100° = 260° .

2. The central angle corresponding to the major arc AOB is the major angle ∠AOB = 260°, and the inscribed angle corresponding to it is ∠ACB .

3. According to the inscribed angle theorem, ∠ACB = major angle ∠AOB / 2 = 130° .

4. Through the above reasoning, the final answer is ∠ACB = 130° .

圆; 圆心角; 圆周角; 弧; 弦

Definition of Central Angle

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

Definition of Inscribed Angle

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

Inscribed Angle Theorem

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


Question: Find the ratio from B to B'.

Answer: \frac { 4 } { 3 }

1. ##Let the triangles in the figure be triangle QAB and triangle QA'B', where point Q, point B, and point B' are on the same line segment##. We need to determine the ratio from B to B'.

2. ##Given QB = 6, QB' = 8., so QB'/QB=8/6=4/3##.

####

##3##. Therefore, the ratio from B to B' is 4 : 3.

Definition of Line Segment

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

In the figure of this problem, <note>line segment QB' is a part of a straight line</note>, including <note>endpoint Q and endpoint B'</note> and all points between them. <note>Line segment QB' has two endpoints, which are Q and B'</note>, and every point on line segment QB' lies <note>between endpoint Q and endpoint B'</note>.


Question: Find x.

Answer: 90

1. Given that the angle in the figure is a right angle, according to the definition of a right angle, the measure of a right angle is 90°.

2. Based on the information provided in the figure, angle x is another angle on the straight line, adjacent to the right angle.

3. According to the definition of supplementary angles, the sum of two supplementary angles is 180°.

4. Therefore, angle x + 90° = 180°.

5. Solving this equation, angle x = 180° - 90°.

6. After calculation, angle x = 90°.

7. So, the final answer is 90°.

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.

<note>Angle x and the right angle share a common side</note>, their <note>other sides are extensions in opposite directions</note>, so <note>angle x and the right angle are adjacent supplementary angles</note>. According to the definition of adjacent supplementary angles, <note>angle x + 90° = 180°</note>.


Question: Find T X, if E X = 24 and D E = 7.

Answer: 32

1. Observing the figure, we can see that the red markings indicate right angle symbols, thus ∠DEX=90°. According to the definition of a right triangle, triangle DEX is a right triangle.

2. Given EX = 24 and DE = 7, according to the Pythagorean theorem, we know that in the right triangle DEX, the square of the hypotenuse equals the sum of the squares of the two legs, i.e., XD? = EX? + DE? = 24? + 7? = 625.

3. Based on the above, we get XD? = 625, i.e., XD = √625 = √25 * √25 = 25.

4. According to the problem statement, DE is the radius. Based on the definition of the radius, DE = DT = 7.

5. From the figure, we can see that TX involves two parts: DT and XD. Given DT = 7 and XD = 25, we can conclude TX = DT + XD = 7 + 25 = 32.

6. Through the above reasoning, the final answer is 32.

直线; 圆; 弦; 垂线; 圆内接四边形

Definition of Radius

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

<note>In the diagram of this problem, in the circle, point D is the center, points E, A, and T are any points on the circle, and line segments DE, DA, and DT are segments from the center to any point on the circle, therefore line segments DE, DA, and DT are the radii of the circle</note>, their lengths are equal, all being 7.

Definition of Right Triangle

In the figure of this problem, in triangle DXE, <note>angle DEX is a right angle (90 degrees)</note>, therefore <note>triangle DXE is a right triangle</note>. <note>Side DE and side EX are the legs</note>, <note>side DX is the hypotenuse</note>.

Definition of Line Segment

<note>Line segment TX</note> is a part of a straight line, including <note>endpoint T</note> and <note>endpoint X</note> and all points between them.

Pythagorean Theorem

<note>In the figure of this problem, in the right triangle DXE, ∠DEX is a right angle (90 degrees)</note>, <note>sides DE and EX are the legs, and side DX is the hypotenuse</note>, so according to the Pythagorean Theorem, ##DX² = EX² + DE²##.


Question: Find x.

Answer: 9

1. Let the triangle in the figure be ABC, the vertex of the 60° angle be point A, the vertex of the right angle be point C, and the vertex of the third angle be point B. It is known that in the right triangle ∠ACB = 90°, ∠BAC = 60°.

2. According to the triangle angle sum theorem, the sum of the interior angles of a triangle is 180°, so ∠CAB + ∠CBA + ∠ACB = 180°.

3. Given ∠ACB = 90° and ∠BAC = 60°, substituting in we get 60° + ∠CBA + 90° = 180°.

4. Solving the equation we get ∠CBA = 30°.

5. According to the sine theorem, in a right triangle, sin∠CBA = opposite side AC/hypotenuse AB, in this problem it is x/18.

6. sin(30°) = 1/2, substituting into the equation we get 1/2 = x/18.

7. Using cross multiplication to solve the equation we get x = 18 * 1/2.

8. Through the above reasoning, x = 9.

9. Finally, the answer is x = 9.

直角三角形; 正弦; 余弦

Definition of Sine Function

<note>In a right triangle, the 60° angle is an acute angle</note>, <note>side x is the side opposite the 60° angle</note>, <note>side 18 is the hypotenuse</note>. According to the definition of the sine function, the sine of the 60° angle is equal to <note>the ratio of the opposite side x to the hypotenuse 18</note>, that is, <note>sin(60°) = x / 18</note>.

Triangle Angle Sum Theorem

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

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


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

Answer: 95

1. The given conditions of the problem are: the upper base of a trapezoid is 8 km, the lower base is 11 km, and the height is 10 km.

2. The formula for calculating the area of a trapezoid is: Area = (upper base + lower base) * height / 2.

3. Substituting the given conditions, upper base = 8 km, lower base = 11 km, height = 10 km, the calculation gives the area of the trapezoid = (8 + 11) * 10 / 2.

4. The calculation steps are as follows:

   a. 8 + 11 = 19.

   b. 19 * 10 = 190.

   c. 190 / 2 = 95.

5. Therefore, the area of the trapezoid is 95 square kilometers.

梯形; 垂线; 普通四边形

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 trapezoid is composed of two parallel sides, namely <note>upper base (8 kilometers)</note> and <note>lower base (11 kilometers)</note>, as well as two non-parallel sides. Therefore, according to the <note>definition of trapezoid</note>, this figure is a <note>trapezoid</note> because it <note>has only one pair of parallel sides</note>.

Definition of Altitude

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

<note>The vertical distance from the upper base to the lower base of 10 kilometers is the height of the trapezoid</note>.<note>This vertical line forms a right angle (90 degrees) with the upper and lower bases</note>, indicating that this vertical line is the perpendicular distance from the upper base to the lower base.

Trapezoid Area Formula

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

<note>The upper base of the trapezoid is 8 kilometers, the lower base is 11 kilometers, and the height is 10 kilometers</note>, so the area of the trapezoid is (8 + 11) * 10 / 2.


Question: A sector of paper with a central angle of 120° and a radius of 6cm is rolled into a bottomless conical paper hat (as shown in the figure). What is the circumference of the base of this paper hat?

Answer: 4πcm

1. Given that the central angle of the sector paper is 120°, the radius is 6 cm, ##let the three vertices of the sector in the figure be A, O, B##.

2. ##According to the development diagram of the cone##, the circumference of the base of the bottomless conical paper hat is equal to the arc length of the sector.

3. ##According to the conversion formula between degrees and radians, convert the central angle to radians to get θ = 120° × (π/180°) = (2π/3)##.

4. ##According to the formula for the arc length of a sector L = θr (where L is the arc length, θ is the central angle in radians, r is the radius), the arc length L = (2π/3) × 6 = 4π## cm.

5. ##It is calculated that the circumference of the base of the paper hat is## 4π cm.

6. In summary, the circumference of the base of the paper hat is 4π cm.

圆心角; 扇形; 圆锥; 圆

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 circle O, points A and B are two points on the circle, and the center of the circle is point O</note>. <note>The angle ∠AOB formed by the lines OA and OB is called the central angle</note>, and the central angle measures 120°.

Definition of Radius

<note>The length of the line segment from the center of the circle to any point on the circumference is 6 centimeters</note>, therefore this line segment is the radius of the sector.

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

In the figure of this problem, <note>the circumference of the base of the bottomless conical paper hat is equal to the arc length of the sector</note>. <note>The central angle of the sector is 120°</note>, <note>and the radius is 6 cm</note>. According to the formula for the length of an arc of a sector, <note>the arc length L is equal to the central angle θ (expressed in radians) multiplied by the radius r, i.e., L = θ * r</note>. Therefore, <note>the arc length L = (2π/3) × 6</note>.

Development of a Cone

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

<note>The development of a cone is a sector</note>. The radius of the sector is <note>the slant height OA of the cone</note>, and the arc length of the sector is <note>the circumference of the base circle of the cone</note>.

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

<note>In figure OAB</note>, <note>the angle AOB is 120°</note>, according to the <note>Formula for Conversion between Degrees and Radians</note>, conversion to radians = degrees * (π/180), that is <note>120° * (π/180)</note>.

Definition of Sector

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

<note>In sector OAB, radius OA and radius OB are two radii of the circle, and the arc AB is the arc enclosed by these two radii</note>, so according to the definition of a sector, the figure formed by these two radii and the <note>arc AB</note> they enclose is a <note>sector</note>.

Cone

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

<note>The base of the cone is a circle</note>. <note>The vertex of the cone is point O</note>, the distance between vertex O and the center of the circle is the height of the cone. The lateral surface of the cone is a curved surface, <note>the distance from vertex O to any point on the circumference is 6 cm</note>.


Question: Find m \angle Z.

Answer: 110

1. Given quadrilateral WXYZ is a quadrilateral, the measures of the four interior angles are ∠X = a°, ∠Y = (a-28)°, ∠Z = (a+2)°, and ∠W = (1/2a+8)°.

2. According to the ##quadrilateral interior angle sum theorem##, the sum of the four interior angles of a quadrilateral is 360°. Therefore, the equation a + (a - 28) + (a + 2) + (1/2a + 8) = 360 can be set up.

3. Solve the equation: first combine like terms to get 3.5a - 18 = 360.

4. Add 18 to both sides of the equation to get 3.5a = 378.

5. Next, divide both sides of the equation by 3.5 to solve for a, i.e., a = 378 / 3.5.

6. Thus, the calculation yields a = 108.

7. Substitute the value of a into the expression for ∠Z (a + 2) to get ∠Z = (108 + 2)°.

8. The calculation results in ∠Z = 110°.

9. Through the above reasoning, the final answer is ∠Z = 110°.

平行四边形; 三角形的外角; 邻补角

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 WXYZ</note> is composed of four line segments <note>WX, XY, YZ, and WZ</note>, which are called the sides of the quadrilateral. <note>Quadrilateral WXYZ</note> has four vertices, namely <note>point W, point X, point Y, and point Z</note>, and it has four interior angles, namely <note>angle ∠W, angle ∠X, angle ∠Y, and angle ∠Z</note>.

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

In <note>quadrilateral WXYZ</note>, <note>angles ∠X, ∠Y, ∠Z, and ∠W are the four interior angles of the quadrilateral</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>∠X + ∠Y + ∠Z + ∠W = 360°</note>.