Question: The distances from chords AC and DF to the center of the circle are equal. If the radius of \odot G is 26, find AC.

Answer: 48

1. Given that the center of the circle is G and the radius GA = 26, #### it is known from the figure that GE = 10 and GB = 10. ## Because the distances from the center of the circle to the chords AC and DF are equal, GB = GE = 10. And based on the distance from a point to a line, we have: BG⊥AC, GE⊥DF. ##

##2. According to the perpendicular bisector theorem, we have AB = BC = ?AC. In △ABG, ∠ABG = 90°, and based on the definition of a right triangle, △ABG is a right triangle. According to the Pythagorean theorem, AG? = GB? + AB?. ##

##3. Since GA = 26 and GB = 10, substituting into the equation we get 26? = 10? + AB?. Further solving the equation, we get AB? = 26? - 10? => AB = √(26? - 10?) = √(676 - 100) = √576 = 24. ##

##4. Since AB = ?AC, we have AC = 2AB = 2*24 = 48. ##

5. Through the above reasoning, #### we finally conclude that the length of the chord AC is 48.

圆; 弦; 对称; 垂直平分线; 直角三角形

Definition of Radius

In the figure of this problem, in circle G, <note>point G is the center of the circle</note>, <note>points A and D are any two points on the circle</note>, <note>segments GA and GD are the segments from the center to any point on the circle</note>, therefore <note>segments GA and GD are the radii of the circle</note>.

Perpendicular Diameter Theorem

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

In <note>circle G</note>, <note>draw GB perpendicular to AC at point B through the center G</note>, and draw <note>GE perpendicular to DF at point E</note>. Then, according to the Perpendicular Diameter Theorem, <note>GB bisects chord AC, that is AB=BC</note>.

Distance from a Point to a Line

The distance from a point to a line is defined as the shortest distance from a point not on the line to the line, which is the perpendicular distance.

In the diagram of this problem, <note>point G is a point outside line AC</note>, draw <note>the perpendicular segment GB from point G to line AC</note>. According to the definition of the distance from a point to a line, <note>the length of the perpendicular segment GB is the shortest distance from point G to line AC</note>, i.e., <note>d=GB</note>. <note>Point G is a point outside line DF</note>, draw <note>the perpendicular segment GE from point G to line DF</note>. According to the definition of the distance from a point to a line, <note>the length of the perpendicular segment GE is the shortest distance from point G to line DF</note>, i.e., <note>d=GE</note>.

Definition of Right Triangle

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

<note>GB⊥AC gives ∠ABG=90°</note>, so <note>△ABG is a right triangle</note>, <note>AB and BG are the legs</note>, <note>AG is the hypotenuse</note>.

Pythagorean Theorem

<note>In the figure of this problem, in the right triangle ABG, ∠ABG=90°</note>, AB and BG are the legs, <note>AG is the hypotenuse</note>, so according to the Pythagorean Theorem, <note>AG²=BG²+AB²</note>.


Question: Find x in the parallelogram.

Answer: 37

1. Given that the quadrilateral shown in the problem is a parallelogram, let the 115° angle be angle A, and label the parallelogram's four angles in counterclockwise order from point A, obtaining a parallelogram ABCD.

2. According to the properties of a parallelogram, opposite angles are equal, ∠A=∠C, i.e., 115°=(2x+41)°.

3. Solve the equation: 2x+41=115, yielding 2x=74, i.e., x=37.

4. Through the above reasoning, the final answer is 37.

平行四边形; 同旁内角; 邻补角

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 ABCD is a parallelogram</note>, in which <note>AB ∥ CD and AB = CD</note>, <note>AD ∥ BC and AD = BC</note>.

Properties of Parallelogram Theorem

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

In the problem figure, <note>in parallelogram ABCD</note>, <note>angles A and C are equal</note>, <note>angles B and D are equal</note>; <note>sides AB and CD are equal</note>, <note>sides AD and BC are equal</note>; <note>diagonals AC and BD bisect each other</note>, that is, <note>the intersection point O divides diagonal AC into two equal segments AO and CO</note>, <note>and divides diagonal BD into two equal segments BO and DO</note>.


Question: Find the area of the figure.

Answer: 60

1. Given that the lengths of the two legs of the triangle are 13, and the length of the base is 10. ##According to the definition of an isosceles triangle, the triangle in the problem is an isosceles triangle##.

2. ##We draw a height from the vertex to the base, let this height be segment AD, and the base be BC. According to the theorem that the height, median, and angle bisector coincide in an isosceles triangle, segment AD perpendicularly bisects the base BC. Given BC=10, then BD=5##.

3. ##Since height AD perpendicularly bisects the base BC, angle ADB is 90 degrees. According to the definition of a right triangle, triangle ABD is a right triangle##. In right triangle ABD, using the Pythagorean theorem, i.e., ##c² = a² + b²##, where c is the hypotenuse AB, a is BD, and b is AD.

4. According to the problem, AB = 13, BD = 5, substituting in we get ##13² = 5² + AD²##. Solving, 169 = 25 + ##AD²##.

5. Thus ##AD²## = 144, hence AD = ##√144## = 12.

6. The area formula for a triangle: Area = 1/2 * base * height, substituting the given conditions we get Area = 1/2 * 10 * 12.

7. After calculation, the area is 60.

##8.## Through the above reasoning, the final answer is 60.

等腰三角形; 直角三角形; 线段

Definition of Isosceles Triangle

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

In <note>triangle ABC</note>, <note>sides AB and AC are equal</note>, therefore <note>triangle ABC is an isosceles triangle</note>.

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 AD is a part of a straight line</note>, containing <note>endpoint A and endpoint D and all points between them</note>. <note>Line segment AD has two endpoints, which are A and D respectively</note>, and <note>every point on line segment AD is located between endpoint A and endpoint D</note>.

Definition of Right Triangle

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

Pythagorean Theorem

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

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

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

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

In the diagram of this problem, <note>in isosceles triangle ABC, the vertex angle is ∠BAC</note>, and the base is side BC. The <note>angle bisector AD of the vertex angle not only bisects the vertex angle ∠BAC, but also bisects the base BC, making BD = DC = 5, and is perpendicular to the base BC, forming a right angle ∠ADB (90 degrees)</note>. Therefore, the segment <note>AD is both the angle bisector of the vertex angle, the median of the base, and the altitude</note>.

Area Formula of a Triangle

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

In <note>triangle ABC</note>, <note>side BC is the base</note>, and <note>segment AD is the height</note>. According to the area formula of a triangle, <note>the area of triangle ABC is equal to the base BC multiplied by the height AD and then divided by 2</note>, i.e., area = (10 * 12) / 2.

Definition of Median of a Triangle

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

In the figure of this problem, in triangle ABC, <note>vertex A</note> is a vertex of the triangle, <note>opposite side BC</note> is the side opposite to vertex A. <note>Point D is the midpoint of side BC</note>, <note>segment AD</note> is the segment from vertex A to midpoint D of opposite side BC, therefore <note>AD is a median of triangle ABC</note>.

Definition of Altitude

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

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


Question: As shown in the figure, lines AB and CD intersect at point O, EO⊥AB, perpendicular at point O, ∠BOD=50°, then ∠COE=()

Answer: 140°

1. Given EO is perpendicular to line AB, according to the definition of perpendicularity, ∠AOE = 90°.

2. Since lines AB and CD intersect at point O, it is known from the problem that ∠BOD = 50°.

3. By the definition of vertical angles, which states that vertical angles formed by intersecting lines are equal, ∠AOC = ∠BOD = 50°.

4. Since ∠COE = ∠AOC + ∠AOE, substituting the known values, we get ∠COE = 50° + 90° = 140°.

5. Through the above reasoning, the final answer is 140°.

对顶角; 垂线; 直线; 射线

Definition of Perpendicular Lines

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

The lines <note>EO and AB</note> intersect to form the <note>angle ∠AOE which is 90 degrees</note>, therefore, according to the definition of perpendicular lines, <note>lines EO and AB are perpendicular to each other</note>.

Definition of Vertical Angles

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

In the diagram of this problem, <note>line AB and CD intersect at point O</note>, forming four angles: <note>∠AOC</note>, <note>∠BOD</note>, <note>∠AOD</note>, and <note>∠BOC</note>. According to the definition of vertical angles, <note>∠AOC and ∠BOD are vertical angles</note>, <note>∠AOD and ∠BOC are vertical angles</note>. Since the angles of vertical angles are equal, <note>∠AOC = ∠BOD = 50°</note>.


Question: E B \parallel D C. Find x.

Answer: 9.5

1. Given that EB is parallel to DC, according to the ##theorem of proportional segments divided by parallel lines##, the corresponding parts of the triangle's sides divided by the parallel lines are proportional.

2. From the ##theorem of proportional segments divided by parallel lines##, it follows that ##DE/AE = BC/AB##.

3. From the given information, DE = 4, ##BC## = 5.

4. The value of ##AE## is equal to ##AD## - ED, so ##AE = 6##.

5. The value of AB is equal to x - 2.

6. From the determined ratio ##4/6 = 5/(x - 2)##.

7. Performing a simple cross multiplication: ##(4 * (x - 2)) = 5 * 6##.

8. Further simplification yields ##4x - 8 = 30##.

9. Solving the equation, moving ##8## to the other side and simplifying, we get 4x = 38.

10. Continuing to simplify, we get x = 38/4 = 9.5.

11. Through the above reasoning, the final answer for the value of x is 9.5.

Proportional Segments Theorem

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

In the figure of this problem, in triangle <note>ADC</note>, the line <note>EB</note> is parallel to side <note>DC</note>, and intersects the other two sides <note>AD</note> and <note>AC</note> at points <note>E</note> and <note>B</note>. According to the Proportional Segments Theorem, we have: <note>DE/AE = BC/AB</note>. That is, the segments intercepted are proportional to the corresponding segments of the original triangle.

Definition of Triangle

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

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


Question: If AB \parallel DC, find x.

Answer: 115

1. Given that AB is parallel to DC, according to the parallel line postulate 2, the interior angles on the same side are supplementary, therefore x° + 65° = 180°.

####

##2##. Solving the equation gives x = 180° - 65°.

##3##. x = 115°.

##4##. Based on the above reasoning, the final answer is 115°.

平行线; 同位角; 对顶角

Parallel Postulate 2 of Parallel Lines

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

<note>Two parallel lines AB and CD are intersected by a third line AD</note>, forming the following geometric relationship: Co-interior angles: <note>Angle A and Angle D are supplementary, i.e., x° + 65° = 180°</note>.

Definition of Consecutive Interior Angles

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

<note>Two lines AB and CD</note> are intersected by <note>a third line AD</note>, <note>angles A and D are on the same side of the intersecting line AD and within the intersected lines AB and CD</note>, thus <note>angles A and D are consecutive interior angles</note>. <note>Consecutive interior angles A and D are supplementary</note>, that is, <note>angle A + angle D = 180 degrees</note>.


Question: As shown in the figure, if ∠1 = ∠3, ∠2 = 60°, then the degree of ∠4 is ()

Answer: 120°

1. Let the two horizontal lines from top to bottom be line a and line b, and the other two lines from left to right be line c and line d. Given ∠1=∠3, according to a sufficient condition for two lines to be parallel (alternate interior angles are equal), we can conclude that line a ∥ line b.

2. Since line a ∥ line b and ∠2=60°, let ∠4's adjacent supplementary angle on line d be ∠5. Because ∠4 and ∠6 are adjacent supplementary angles on line d, according to the definition of a straight angle, we have ∠4+∠5=180°.

3. According to the parallel postulate 2 and the definition of alternate interior angles, ∠2=∠5=60° (alternate interior angles are equal).

4. Since ∠5=60°, we have ∠4=180°-∠5=180°-60°=120°.

5. Finally, the answer is 120°.

内错角; 同位角; 平行线

The original Chinese enclosed in <note>两条平行线分别为直线a和直线b</note> remains enclosed after translation, intersected by a third <note>line c</note>, forming the following geometric relationships:
1. Corresponding angles: ##None##.
2. Alternate interior angles: <note>Angle 2 and angle 5 are equal, i.e., ∠2=∠5</note>.
3. Consecutive interior angles: ##None##.

Definition of Alternate Interior Angles

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

<note>In the figure of this problem, two parallel lines a and b are intersected by a line c</note>, <note>where angles 1 and 3 are located between the two parallel lines and on opposite sides of the transversal c</note>, therefore <note>angles 1 and 3 are alternate interior angles</note>. <note>Alternate interior angles are equal, that is, angle 1 is equal to angle 3</note>. <note>Two parallel lines a and b are intersected by a line d</note>, <note>where angles 2 and 5 are located between the two parallel lines and on opposite sides of the transversal d</note>, therefore <note>angles 2 and 5 are alternate interior angles</note>. <note>Alternate interior angles are equal, that is, angle 2 is equal to angle 5</note>.

Definition of Linear Pair of Angles

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

In the figure of this problem, <note>∠4 and ∠5 have a common side</note>, and <note>their other sides are extensions in opposite directions</note>, so <note>∠4 and ∠5 are adjacent supplementary angles</note>, <note>∠4 + ∠5 = 180°</note>.

Definition of Straight Angle

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

<note>Ray d rotates around the endpoint until it forms a straight line with the initial side, forming straight angle x</note>. According to the definition of a straight angle, the measure of a straight angle is 180 degrees, i.e., <note>angle 4 + angle 5 = 180 degrees</note>.


Question: What is the value of x?

Answer: 10

1. Let the vertex of the triangle with 6X° be A, the vertex with 4X° be B, and the vertex with 8X° be C.

2. In △ABC, it is known that ∠BAC = 6x°, ∠ABC = 4x°, ∠BCA = 8x°.

3. According to the triangle angle sum theorem, the sum of the three interior angles of a triangle is equal to 180°, that is, ∠BAC + ∠ABC + ∠BCA = 180°.

4. Substitute the specific angle expressions into the equation to get 6x + 4x + 8x = 180°.

5. Combine like terms to get 18x = 180°.

6. Divide both sides by 18 to get x = 10°.

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

Triangle Angle Sum Theorem

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

<note>The three interior angles of the triangle are 6x°, 4x°, and 8x°</note>. According to the Triangle Angle Sum Theorem, <note>the sum of these three interior angles should be equal to 180°</note>. We express this mathematically as: <note>6x° + 4x° + 8x° = 180°</note>.


Question: Find m \angle K.

Answer: 100

1. Given quadrilateral JKLM, JK∥ML and sides JM = 6, KL = 6. According to the definition of an isosceles trapezoid, quadrilateral JKLM is an isosceles trapezoid.

2. The figure marks ∠M as 80°, based on the properties of an isosceles trapezoid, in an isosceles trapezoid, the base angles are equal, i.e., ∠M = ∠L, ∠J = ∠K.

3. Substituting ∠M = 80°, we get ∠L = 80°.

4. Since JK∥ML, according to the parallel line axiom 2, ∠L + ∠K = 180°.

6. Through the above reasoning, the final answer is 100°.

Definition of Isosceles Trapezoid

A trapezoid is isosceles if and only if its non-parallel sides (legs) are congruent (∅).

In the figure of this problem, in the isosceles trapezoid JKLM, <note>side JK and side ML are parallel</note> (i.e., the two parallel sides of the trapezoid), <note>side JM and side KL are the legs of the trapezoid</note>, and <note>side JM is equal to side KL</note> (i.e., the legs are equal). Therefore, JKLM is an isosceles trapezoid.

Parallel Postulate 2 of Parallel Lines

<note>The upper base JK and the lower base ML of trapezoid JKLM are parallel</note>. According to the properties of parallel lines, if two parallel lines are intersected by a third line, corresponding angles are equal. Here, <note>lines JM and KL are the intersecting lines</note>, <note>the interior angles on the same side ∠K and ∠L are supplementary</note>. Therefore, <note>∠K = 180° - ∠L</note>. Hence, <note>∠K = 100°</note>.

In the figure of this problem, <note>two lines JK and ML are intersected by the third line KL</note>, <note>angles K and L are on the same side of the intersecting line KL and within the intersected lines JK and ML</note>, so <note>angles K and L are consecutive interior angles</note>. <note>Consecutive interior angles K and L are supplementary, that is, angle K + angle L = 180 degrees</note>.


Question: Solve for x.

Answer: 18 \sqrt { 6 }

1. Given conditions: In a right triangle, the length of the side opposite the hypotenuse is 18√3, and the included angle is 45°.

 2. According to the 45°-45°-90° triangle theorem, the two legs of a 45°-45°-90° triangle are equal, and any leg multiplied by √2 equals the hypotenuse.

 3. The two legs are equal to 18√3, based on the properties of an isosceles right triangle, we can get the hypotenuse x = 18√3 * √2.

 4. Calculating, we get x = 18√3 * √2 = 18 * √3 * √2 = 18 * √6.

 5. Therefore, through the above reasoning, the final answer is 18√6.

直角三角形; 等腰三角形

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</note>, in the triangle, <note>one angle is a right angle (90 degrees)</note>, therefore the triangle is a <note>right triangle</note>. <note>The two sides with a length of 18√3 are the legs</note>, <note>the side with a length of x is the hypotenuse</note>.

Properties of 45°-45°-90° Triangle

In a 45°-45°-90° triangle, the two legs opposite the 45-degree angles are congruent, and the hypotenuse is √2 times the length of either leg.

<note>In the figure of this problem, in a 45°-45°-90° triangle, the two acute angles are 45 degrees, and the right angle is 90 degrees</note>.<note>Side x is the hypotenuse, and the two sides of length 18√3 are the legs opposite the 45-degree angles</note>. According to the properties of a 45°-45°-90° triangle, <note>side x is equal to 18√3 times √2</note>. That is: <note>x = 18√3 * √2</note>.

Definition of Isosceles Right Triangle

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

<note>The triangle is an isosceles right triangle</note>, <note>one of its angles is a right angle (90 degrees)</note>, with a length of <note>18√3 for the equal right-angle sides</note>.


Question: ΔLMN ≅ ΔQRS. Find x.

Answer: 20

1. Given that △LMN is congruent to △QRS, according to the definition of congruent triangles, the corresponding sides and angles of △LMN and △QRS are equal, i.e., ∠M = ∠R, LM = QR.

2. Substitute LM = 2x + 11 and QR = 3x - 9 into the equation, we have: 2x + 11 = 3x - 9.

3. Solve this equation: first, rearrange the terms to get: 2x - 3x = -9 - 11.

4. Further simplify to obtain: -x = -20.

5. Solve for x: x = 20.

6. Through the above reasoning, the final answer is x = 20.

普通三角形; 对称; 平移

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 LMN and triangle QRS are congruent triangles</note>, <note>The corresponding sides and corresponding angles of triangle LMN are equal to those of triangle QRS</note>, that is: <note>side LM = side QR</note>, <note>side LN = side QS</note>, <note>side MN = side RS</note>; at the same time, <note>The corresponding angles are also equal</note>: <note>angle L = angle Q</note>, <note>angle M = angle R</note>, <note>angle N = angle S</note>. Given in the problem <note>LM = 2x + 11</note>, <note>QR = 3x - 9</note>, therefore <note>2x + 11 = 3x - 9</note>.


Question: In H, the diameter is 18, L M = 12, and m \widehat L M = 84. Find H P. Round the result to two decimal places.

Answer: 6.71

1. Given that the diameter of the circle is 18, the radius is 9. Connect HL, it is known that HL meets the definition of radius, HL is the radius of circle H and is 9, JK is the diameter of circle H and is 18.

2. Given that LM is a chord on circle H, the diameter JK is perpendicular to LM. According to the perpendicular bisector theorem, JK is the perpendicular bisector of LM, i.e., ∠LPH = 90°, LP = PM = 1/2 * LM. Given LM = 12, so LP = 1/2 * 12 = 6.

3. In triangle LHP, ∠LPH is 90°. According to the definition of a right triangle, this triangle is a right triangle.

4. In right triangle LHP, according to the Pythagorean theorem: LH^2 = LP^2 + HP^2. Substituting LH = 9 and LP = 6, we get HP = √(LH^2 - LP^2) = √(9^2 - 6^2) = 3√5.

5. Taking the approximate value of √5 as 2.236, the result is 6.708. Keeping the result to two decimal places and rounding, we get 6.71.

6. Through the above reasoning, the final answer is that the length of HP is 6.71 (rounded to two decimal places).

圆; 垂线; 直角三角形; 圆周角; 弧

Definition of Circle

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

<note>In circle H, point H is the center of the circle</note>, with a radius of 9. <note>All points in the diagram that are at a distance of 9 from point H are on circle H</note>.

Definition of Diameter

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

In the figure of this problem, <note>line segment JK is the diameter</note>, connecting <note>the center H and points J and K on the circumference</note>, with a length of 2 times the radius, that is, <note>JK = 18</note>.

Definition of Radius

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

In the diagram of this problem, in circle H, <note>point H is the center of the circle</note>, <note>points J and K are any points on the circle</note>, <note>line segments HJ and HK are segments from the center to any point on the circle</note>, therefore <note>line segments HJ and HK are the radii of the circle</note>, and their length is 18/2 = 9. <note>Point L is any point on the circle</note>, <note>line segment HL is a segment from the center to any point on the circle</note>, therefore <note>line segment HL is the radius of the circle</note>, and its length is 18/2 = 9.

Pythagorean Theorem

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

<note>In the figure of this problem, in the right triangle LHP, angle LPH is a right angle (90 degrees), sides PL and PH are the legs, and side LH is the hypotenuse</note>, so according to the Pythagorean Theorem, <note>LH^2 = LP^2 + HP^2</note>.

Definition of Chord

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

In the figure of this problem, in circle H, <note>point L and point M are any two points on the circle</note>, <note>line segment LM connects these two points</note>, so <note>line segment LM is a chord of circle H</note>.

Definition of Right Triangle

In the figure of this problem, in triangle LHP, <note>angle LPH is a right angle (90 degrees)</note>, so triangle LHP is a right triangle. <note>Sides LP and HP are the legs</note>, <note>side LH 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.

In <note>circle H</note>, <note>diameter JK is perpendicular to chord LM</note>, then according to the <note>Perpendicular Diameter Theorem</note>, <note>diameter JK bisects chord LM</note>, that is, <note>LP=PM</note>, and <note>diameter JK bisects the arcs subtended by chord LM</note>, that is, <note>arc LK=arc KM</note>.


Question: Find the area of the rhombus.

Answer: 132

1. Given that one diagonal of the rhombus is 12 mm, ##the other diagonal is 22 mm##.

##2. Given that the area of the rhombus is calculated as: Area = (d1 * d2 / 2). d1 and d2 refer to the two diagonals of the rhombus.##

##3##. Substituting the values into the formula, we get: Area = (12 * 22) / 2.

##4. Calculating the area, we get: Area = 264 / 2 = 132 mm^2.##

##5. Through the above reasoning, the final answer is 132 mm^2.##

Definition of Rhombus

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

<note>In a quadrilateral, all sides are equal</note>, therefore the quadrilateral is a rhombus. Additionally, <note>the diagonals of the quadrilateral bisect each other at right angles</note>.

Definition of Diagonal

A diagonal is a line segment connecting one vertex of a polygon to another vertex that is not adjacent to it.

<note>The diagonals of a rhombus have one length of 12 mm and the other length of 22 mm</note>. According to <note>the definition of diagonal, a diagonal is a line segment connecting a vertex of a polygon to a non-adjacent vertex</note>. Therefore, these two line segments are <note>the diagonals of the rhombus</note>.

Rhombus Area Formula

The area of a rhombus is equal to half the product of its diagonals.

The original text: <note>The diagonals of the rhombus are 12 mm and 22 mm respectively</note>. According to the rhombus area formula, the area of the rhombus is equal to half the product of the two diagonals, that is, Area = (<note>12 mm * 22 mm</note>) / 2 = <note>132 mm²</note>.


Question: In the figure, a regular polygon is inscribed in a circle. Find the degree measure of a central angle.

Answer: 60

1. The given condition is the regular polygon shown in the figure and it is inscribed in a circle. Observing the figure, ##regular hexagon PNMLKJ## each side corresponds to an equal arc on the circumference.

2. ##The definition of the central angle refers to the angle between the lines connecting two adjacent vertices of a regular polygon to its center. The central angle can also be expressed as the angle at the center of the circumscribed circle corresponding to each side of the regular polygon##.

3. ##Use the formula for calculating the central angle##.

4. ##The central angle of a regular polygon is equal to 360° /n, where n is the number of sides of the regular polygon##.

5. ##Given n=6, so the central angle of this regular polygon is equal to 360° /6=60°##.

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

正多边形; 圆; 圆心角

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.

Original text: <note>Regular hexagon PJKLMN</note>. All six sides of this regular polygon (<note>PJ</note>, <note>JK</note>, <note>KL</note>, <note>LM</note>, <note>MN</note>, <note>NP</note>) are of equal length, and all six interior angles (<note>∠PJK</note>, <note>∠JKL</note>, <note>∠KLM</note>, <note>∠LMN</note>, <note>∠MNP</note>, <note>∠NPJ</note>) are equal.

Inscribed Polygon

A polygon is called an inscribed polygon if all its vertices lie on the circumference of a circle.

All vertices of the <note>regular hexagon PJKLMN</note>, <note>P, J, K, L, M, N</note>, lie on <note>circle O</note>, so this polygon is an <note>inscribed polygon 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 a circle, point K and point L are two points on the circle, and the center of the circle is point R</note>. The angle <note>∠KRL formed by the lines RK and RL is called the central angle</note>.

A central angle is an angle formed by two radii connecting the center of a regular polygon to two of its adjacent vertices. A central angle can also be described as the angle subtended at the center of a circumscribed circle by any side of the regular polygon.

In the figure of this problem, <note>angle KRL is a central angle</note>, <note>formed by the lines connecting the center R of the regular polygon with two adjacent vertices K and L</note>. According to the <note>definition of central angle</note>, angle KRL is the central angle of the regular polygon, and <note>its measure is equal to the central angle of the circumscribed circle corresponding to each side of the regular polygon</note>.

Formula for Central Angle

The central angle of a regular polygon equals 360° / n, where n is the number of sides of the regular polygon.

<note>PJKLMN is a regular polygon with 6 sides</note>, where 6 represents the number of sides of the regular polygon. According to the formula for the central angle, <note>the degree of each central angle of the regular polygon is equal to 360° / 6</note>, that is, <note>central angle = 360° / 6</note>.


Question: As shown in the figure, in ⊙O, chord AC ∥ radius OB, ∠BOC=50°, then the degree of ∠OAB is ()

Answer: 25°

1. Given ∠BOC=50°, according to the inscribed angle theorem, we know that the inscribed angle is equal to half of the central angle corresponding to its arc, thus ∠BAC = 1/2∠BOC = 25°.

2. According to the problem, chord AC is parallel to radius OB, based on the parallel lines axiom 2, alternate interior angles are equal. Therefore, ∠BAC=∠OBA=25°.

3. Since OA and OB are both radii of circle O, triangle AOB is an isosceles triangle.

4. According to steps 1 and 2, we know ∠OBA=25°, thus ∠OAB=25°.

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

圆; 弧; 圆心角; 平行线

Definition of Inscribed Angle

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

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

Definition of Central Angle

In the figure of this problem, <note>circle O</note>, <note>points B and C are two points on the circle</note>, <note>the center of the circle is point O</note>. <note>The angle ∠BOC formed by the lines OB and OC is called the central 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 <note>circle O</note>, <note>points A, B, and C are on the circle</note>, <note>the central angle corresponding to arc BC is ∠BOC = 50°</note>, <note>the inscribed angle is ∠BAC</note>. According to the Inscribed Angle Theorem, <note>∠BAC is equal to half of the central angle ∠BOC corresponding to arc BC</note>, that is, <note>∠BAC = 1/2 ∠BOC = 25°</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>Chord AC is parallel to radius OB</note>, forming the following geometric relationship: 1. <note>Alternate interior angles: ∠BAC and ∠OBAC are equal, i.e., ∠BAC=∠OBA=25°</note>. According to Parallel Postulate 2 of Parallel Lines, <note>if two parallel lines are intersected by a third line, the alternate interior angles are equal</note>.

Properties of Isosceles Triangle

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

In the figure of this problem, <note>in the isosceles triangle OAB, sides OA and OB are equal</note>. Therefore, according to the properties of isosceles triangles, <note>the angles opposite the equal sides are equal, i.e., ∠OBA = ∠OAB</note>.

Definition of Isosceles Triangle

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

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


Question: Find the value of the variable x in the figure.

Answer: 125

1. Given that two parallel lines are intersected by a transversal at two points, forming an angle of 55° on the transversal, let the angle x° be ∠X and the angle y° be ∠Y. According to the definition of alternate interior angles, ∠X and ∠Y are alternate interior angles.

2. According to the parallel postulate 2, alternate interior angles are equal, therefore ∠X = ∠Y.

3. According to the definition of adjacent supplementary angles, ∠X and ∠55° are adjacent supplementary angles. Then, according to the definition of a straight angle, ∠X + 55° = 180°.

5. Through the above reasoning, the final answer is x = 125°.

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 problem diagram, <note>two parallel lines are intersected by a transversal</note>, where <note>∠X and ∠Y are located between the two parallel lines and on opposite sides of the transversal</note>, therefore <note>∠X and ∠Y are alternate interior angles</note>.

Parallel Postulate 2 of Parallel Lines

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

Definition of Linear Pair of Angles

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

In the figure of this problem, <note>∠X and ∠55° share a common side</note>, <note>their other sides are extensions in opposite directions</note>, so <note>∠X and ∠55° are adjacent supplementary angles</note>.

Definition of Straight Angle

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

<note>The ray rotates around endpoint X to form a straight line with the initial side</note>, forming a straight angle. The measure of a straight angle is 180 degrees, that is, <note>∠X+55°=180°</note>.


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

Answer: 90

1. In the figure, a right angle mark is drawn at the intersection of radius AO and radius CO, indicating that these two lines are perpendicular to each other, so ∠AOC is a right angle, i.e., ∠AOC=90°.

2. By the property of the central angle, since O is the center of the circle, ∠AOC is a central angle. Also, because the degree of the central angle is equal to the degree of the arc it subtends, ∠AOC subtends arc AC, so arc AC=90°.

3. Through the above reasoning, the final answer is that the degree of arc AC is 90°.

Definition of Central Angle

<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 AO and CO is called the central angle</note>.

Properties of Central Angles

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

<note>In the figure of this problem</note>, it is known that <note>the central angle ∠AOC corresponds to the arc AC</note>. According to the properties of central angles, <note>the degree measure of the central angle ∠AOC is equal to the degree measure of the arc AC</note>, that is, <note>the degree measure of ∠AOC = the degree measure of the arc AC</note>.


Question: As shown in the figure, four small squares with side length 1 form a large square. A, B, O are vertices of the small squares. The radius of ⊙O is 1. P is a point on ⊙O and is located inside the small square at the upper right. Then sin∠APB equals ()

Answer: \frac{√{2}}{2}

1. Given four small squares with side length 1 forming a larger square, A, B, O are the vertices of the small squares.

2. Given the central angle AOB=90°, according to the inscribed angle theorem, ∠APB=45°.

4. Through the above reasoning, the final answer is √{2}/2.

圆; 正弦; 直角三角形; 正方形; 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.

In the figure of this problem, <note>in circle ⊙O, point O is the center of the circle</note>, <note>the radius is 1</note>. <note>All points in the figure that are at a distance of 1 from point O are on circle ⊙O</note>.

Definition of Central Angle

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

Definition of Inscribed Angle

In <note>circle O</note>, <note>the vertex of angle APB (point P) is on the circumference</note>, <note>the two sides of angle APB intersect circle O at points A and B</note>. Therefore, <note>angle APB is an inscribed angle</note>.


Question: Find h in the triangle.

Answer: 8

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

2. From the figure, it is known that ∠A=60°, ∠B=90°. According to the triangle angle sum theorem, ∠A+∠B+∠C=60°+90°+∠C=180°, that is, ∠C=180°-60°-90°=30°.

3. According to the properties of a 30°-60°-90° triangle, in a 30°-60°-90° triangle, the side opposite the 30-degree angle is half of the hypotenuse.

4. Given that the short side AB=4 in the figure, and the hypotenuse=h, according to the above property, h/2=4, thus h=2*4=8.

5. After the above reasoning, the final answer is 8.

Definition of Right Triangle

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

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

Properties of 30°-60°-90° Triangle

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

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

Triangle Angle Sum Theorem

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

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


Question: In the figure, m \angle 3 = 110 and m \angle 12 = 55. Find the measure of \angle 1.

Answer: 110

1. Given that the measure of ##∠ 3## is 110°.

2. According to the definition of corresponding angles, when two parallel lines are intersected by a third line, ##∠1## and ##∠3## are on the same side of the transversal and on the same side of the two intersected lines, so ##∠1## and ##∠3## are corresponding angles.

3. According to Parallel Postulate 2, if two parallel lines are intersected by a third line, corresponding angles are equal, thus ##∠1 = ∠3 = 110°##.

4. Based on the above reasoning, ##∠1 = 110°##.

内错角; 同位角; 平行线

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>∠1 and ∠3 are on the same side of the transversal and on the same side of the two lines being intersected</note>, therefore <note>∠1 and ∠3 are corresponding angles</note>.

Parallel Postulate 2 of Parallel Lines

<note>In the diagram of this problem, two parallel lines are intersected by a third line, forming corresponding angles: ∠1 and ∠3 are equal.</note>


Question: In \odot P, the radius is 2 inches, find the length of \widehat R S. Round to two decimal places.

Answer: 4.54

1. The given condition is that the radius of circle P is 2 inches, and angle RPS is 130°.

2. ##From the figure, it is known that the angle corresponding to arc RS within the central angle of the sector is 130°. According to the formula for the arc length of a sector##, the formula for the length of arc RS is: Arc length = Radius × Central angle in radians.

3. Therefore, it is necessary to convert the angle 130° to radians. ##According to the conversion formula between degrees and radians, using the formula: Radians = Degrees × (π/180). Calculating the radians = 130 × (π/180) = (13/18) π##.

4. Using the above formula to calculate the length of arc RS, Arc RS = Radius × Central angle in radians = 2 × (13/18) π.

5. Calculating this expression specifically: 2 × (13/18) π = (26/18) π = (13/9) π. ##π is approximately 3.14159##.

6. ##(13/9) × 3.14159## calculation results in approximately 4.5378.

7. Rounding to ##two decimal places##, the length of arc RS is 4.54 inches.

8. The final answer is 4.54 inches.

Definition of Arc

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

On <note>circle P</note>, there are two points <note>R and S</note>. <note>Arc RS</note> is a segment of a curve connecting these two points. According to the definition of an arc, <note>Arc RS</note> is <note>a segment of a curve between two points R and S on the circle</note>.

Definition of Central Angle

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

In the diagram of this problem, <note>∠RPS is the central angle</note>, its <note>vertex P is the center of the circle</note>, and the sides <note>PR and PS are two radii of the circle</note>.

Formula for Conversion between Degrees and Radians

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

In the figure of this problem, in the sector <note>RPS</note> of circle P, the angle of sector <note>RPS is 130°</note>. According to the formula for conversion between degrees and radians, the conversion to radians is angle * (π/180), which is <note>130° * (π/180)</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 \( r \): \( L = \theta r \).

The formula for calculating the length of <note>arc RS</note> is: <note>arc RS = radius × central angle in radians = 2 × {130 × (π / 180)}</note>.


Question: Find y.

Answer: 4 \sqrt { 2 }

1. In ΔABC, it is known that ∠ACB is 90 degrees, AC = z, AB = 12, CB = x. Let segment y intersect AB at point P. From the diagram, it can be seen that ∠CPA = 90°.

2. Because in the right triangle ACB, ∠ACB = 90°, and ∠A is the common angle of ΔACP and ΔACB, therefore, according to the similarity theorem (AA), ΔACP and ΔACB are similar.

3. According to the definition of similar triangles, we have AP / AC = AC / AB, and we can find the length of AC. Given AP = AB - PB = 12 - 8 = 4, substituting the known conditions, we get 4 / AC = AC / 12, therefore AC = 4 √3.

4. Then, according to the Pythagorean theorem, AC^2 = AP^2 + CP^2, we get (4 √3)^2 = 4^2 + CP^2.

5. Solving the above equation, we get CP = y = 4 √2.

6. Finally, through the above steps, we determine y = 4 √2.

直角三角形; 中点; 内错角

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 ACB and triangle ACP are similar triangles</note>. According to the definition of similar triangles: <note>∠ABC = ∠ACP</note>, <note>∠BAC = ∠CAP</note>, <note>∠BCA = ∠CPA</note>; <note>AP/AC = 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.

In the <note>right triangle ACP</note>, <note>angle APC is a right angle (90 degrees)</note>, <note>side AP and side PC are the legs</note>, <note>side AC is the hypotenuse</note>, so according to the Pythagorean Theorem, <note>AC^2 = AP^2 + CP^2</note>.

Definition of Right Triangle

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

Similarity Theorem for Triangles (AA)

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

In <note>triangle ACP and triangle ACB</note>, if <note>angle APC is equal to angle ACB</note>, and <note>angle A is equal to angle A</note>, then <note>triangle ACP is similar to triangle ACB</note>.


Question: Find the area of the triangle. Round the result to two decimal places.

Answer: 7.51

1. Let the three vertices of the triangle be A, B, and C, where ∠ACB is a right angle. According to the definition of a right triangle, △ABC is a right triangle. Also, ∠ABC equals 59°. The length of side AC is 5 inches.

2. In a right triangle, using the definition of the sine function: sin(∠ABC) = opposite side / hypotenuse. Here, ∠ABC = 59°, side AC is the opposite side, and side AB is the hypotenuse. Therefore, sin(59°) = AC / AB.

3. Calculate sin(59°), obtaining sin(59°) ≈ 0.8572. Therefore, 0.8572 = 5 / AB, solving for AB = 5 / 0.8572 ≈ 5.83.

4. In a right triangle, using the definition of the cosine function: cos(∠ABC) = adjacent side / hypotenuse. Here, ∠ABC = 59°, side BC is the adjacent side, and side AB is the hypotenuse. Therefore, cos(59°) = BC / AB.

5. Calculate cos(59°), obtaining cos(59°) ≈ 0.5150. Therefore, 0.5150 = BC / 5.83, solving for BC = 0.5150 * 5.83 ≈ 3.00245.

6. Calculate the area of the triangle. The area formula for right triangle ABC is: area = 1/2 * base * height. Here, the base is BC (approximately 3.00245), and the height is AC (5).

7. Substitute into the area formula, obtaining area = 1/2 * 3.00245 * 5 ≈ 7.506.

8. Round the result to two decimal places, the final area is 7.51.

直角三角形; 普通三角形; 正弦; 余弦

Definition of Sine Function

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

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

Cosine Function

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

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

Area of Right Triangle

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

In the figure of this problem, in the right triangle ABC, <note>angle ACB is a right angle (90 degrees)</note>, sides AC and BC are the legs of the right triangle, <note>with one leg AC as the height and the other leg BC as the base</note>, so the area of the right triangle is equal to half the product of these two legs, that is, Area = 1/2 * side AC * side BC.

Definition of Right Triangle

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


Question: Find x.

Answer: 30

1. Given ΔABC, ∠CAB=2x°, ∠CBA=2x°, ##side AB = side AC, thus the triangle is an isosceles triangle##.

2. ##According to the properties of an isosceles triangle, ∠ACB = ∠CBA = 2x°##.

3. According to the ##triangle angle sum theorem, ∠CAB + ∠CBA + ∠ACB = 180°, solving the equation gives 6x = 180, thus x = 30##.

####

##4##. Finally, through the above reasoning, the answer is: ##x = 30##.

等腰三角形; 三角形的外角

Definition of Triangle

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

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

Definition of Isosceles Triangle

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

<note>Side AB and side AC are equal</note>, therefore <note>triangle ABC is an isosceles triangle</note>.

Triangle Angle Sum Theorem

In <note>triangle ABC</note>, <note>angle CAB</note>, <note>angle CBA</note>, and <note>angle BCA</note> are <note>the three interior angles of triangle ABC</note>. According to the <note>Triangle Angle Sum Theorem</note>, <note>angle CAB + angle CBA + angle BCA = 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 ABC</note>, <note>sides AB and AC are equal</note>. Therefore, according to the properties of isosceles triangles, <note>the angles opposite the equal sides are equal</note>, i.e., <note>∠ACB = ∠CBA</note>.


Question: As shown in the figure, given that the radius of ⊙O is 5, the central angles corresponding to chords AB and CD are ∠AOB and ∠COD respectively, and ∠AOB and ∠COD are supplementary. If chord CD = 8, then the length of chord AB is ()

Answer: 6

1. Given that the radius of ⊙O is 5, chords AB and CD subtend central angles ∠AOB and ∠COD respectively, and ∠AOB and ∠COD are supplementary.

2. ##Extend AO in the opposite direction to intersect ⊙O at point E##, connect BE.

3. Since ∠AOB and ∠COD are supplementary, we have ∠AOB + ∠COD = 180°. According to the definition of a straight angle, ##the straight angle AOE is 180°. According to the definition of adjacent supplementary angles, ∠AOB + ∠EOB = 180°##.

4. ##Since ∠AOB + ∠EOB = 180° and ∠AOB + ∠COD = 180°, we have ∠BOE = ∠COD##.

5. ##According to the congruent triangles theorem (SAS), OD = OE, OC = OB, ∠BOE = ∠COD, so △OCD ≅ △OBE. According to the definition of congruent triangles, CD = BE = 8##.

6. ##According to the definition of the inscribed angle and the corollary of the inscribed angle theorem (2), the inscribed angle subtended by the diameter is a right angle, ∠ABE = 90°. According to the definition of a right triangle, triangle ABE is a right triangle##.

7. ##In the right triangle ABE##, we can use the Pythagorean theorem to find the length of AB.

8. The Pythagorean theorem is ##a^2 + b^2 = c^2##, where a and b are the legs, and c is the hypotenuse. In this problem, AE is the hypotenuse, and AB and BE are the legs.

9. ##Substituting the given conditions, AE is the diameter, so AE = 10, we get AB = √(AE^2 - BE^2) = √(10^2 - 8^2) = √(100 - 64) = √36 = 6##.

10. Through the above reasoning, the final answer is that the length of chord AB is 6.

圆; 弦; 圆心角; 邻补角; 余弦

Definition of Circle

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

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

Definition of Central Angle

<note>In the diagram of this problem, 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 ∠AOB formed by the lines <note>OA and OB is called the central angle.</note> Similarly, <note>points C and D are two points on the circle, and the angle ∠COD formed by the lines OC and OD is called the central angle.</note>

Definition of Chord

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

In the figure of this problem, <note>in circle O, points A and B are any two points on the circle, and segment AB</note> connects these two points, so <note>segment AB is a chord of circle O</note>. Similarly, <note>points C and D are any two points on the circle, and segment CD</note> connects these two points, so <note>segment CD is also a chord of circle O</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>AE is the diameter</note>, connecting <note>the center O</note> and <note>points A and E on the circumference</note>, with a length equal to 2 times the radius, that is, <note>AE = 2 * 5=10</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 ABE, angle ∠ABE is a right angle (90 degrees)</note>, therefore triangle ABE is a right triangle. <note>Sides AB and BE are the legs</note>, <note>side AE is the hypotenuse</note>.

Definition of Straight Angle

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

<note>Ray AO</note> rotates around the endpoint <note>O</note> to form a straight line with the initial side, creating <note>straight angle AOE</note>. According to the definition of a straight angle, <note>a straight angle measures 180 degrees, i.e., angle AOE = 180 degrees</note>.

Definition of Linear Pair of Angles

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

<note>Angle AOB and angle BOE share a common side OB</note>, their <note>other sides AO and OE are extensions of each other in opposite directions</note>, so <note>angle AOB and angle BOE are adjacent supplementary angles</note>.

Triangular Congruence Theorem (SAS)

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

<note>Side OD is equal to side OE</note>, <note>side CO is equal to side BO</note>, and <note>angle DOC is equal to angle BOE</note>, therefore according to the Triangular Congruence Theorem (SAS), <note>these two triangles are congruent</note>.

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

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

<note>In the figure of this problem, in circle O, the inscribed angle ∠ABE subtended by the diameter AE is a right angle (90 degrees).</note>

Pythagorean Theorem

In the figure of this problem, <note>△ABE is a right triangle</note>, <note>∠ABE is a right angle (90 degrees)</note>, <note>sides AB and BE are the legs</note>, <note>side AE is the hypotenuse</note>, so according to the Pythagorean Theorem, <note>AE^2 = AB^2 + BE^2</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>Triangle DOC and triangle BOE are congruent triangles</note>, the <note>corresponding sides and corresponding angles of triangle DOC are equal to those of triangle BOE</note>, that is: <note>side OD = side OE</note> <note>side OC = side OB</note> <note>side BD = side CE</note>, at the same time, <note>the corresponding angles are also equal</note>: <note>angle DOC = angle BOE</note> <note>angle ODC = angle OEB</note> <note>angle OCE = angle OBE</note>.


Question: Find the area of the triangle.

Answer: 285

1. From the figure, it can be seen that the triangle whose area is to be calculated is an obtuse triangle. In the figure, draw a dashed auxiliary line to the right along the short side of 19 inches, and draw a dashed auxiliary line upwards from the vertex of the hypotenuse of 41 inches, which is perpendicular to the dashed auxiliary line along the short side of the triangle. One dashed line has a length of 27 inches, and the other dashed line has a length of 30 inches.

2. According to the area formula for a triangle, Area = 1/2 * base * height. In the figure, the base of the obtuse triangle is the short side of 19 inches, and the height is the dashed auxiliary line of 30 inches. Substituting the actual values, we calculate: Area = 1/2 * 19 inches * 30 inches.

3. After calculation, the area is: 1/2 * 570 square inches = 285 square inches.

4. Based on the above calculations and given conditions, the final confirmed area of the triangle is 285 square inches.

直角三角形; 普通三角形; 线段

Definition of Triangle

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

Area Formula of a Triangle

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

The base of the triangle is <note>19 inches</note>, and the height is <note>30 inches</note>. According to the area formula of a triangle, the area of the triangle is equal to the base multiplied by the height and then divided by 2, i.e., area = (<note>19 inches</note> * <note>30 inches</note>) / 2.

Definition of Altitude

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

<note>In the figure of this problem</note>, the segment <note>30 perpendicular to the extension line of side 19</note> from the <note>vertex at the bottom of the triangle</note> is the <note>altitude</note> of that vertex. <note>The segment 30 forms a right angle (90 degrees) with the extension line of side 19</note>, which indicates that <note>segment 30 is the vertical distance from the bottom vertex to the extension line of side 19</note>.


Question: Use a calculator to find \angle J in degrees, accurate to the nearest degree.

Answer: 40

1. Given ∠JLK=90°, according to the definition of a right triangle, triangle JLK is a right triangle, and \\overline{JK} = 14 and \\overline{KL} = 9. The task is to calculate the degree measure of \\angle J.

2. According to the Pythagorean theorem, in a right triangle, the sum of the squares of the two legs is equal to the square of the hypotenuse, i.e., \\overline{JL}^2 + \\overline{KL}^2 = \\overline{JK}^2. In this problem, we need to find \\overline{JL}.

3. Substitute the given sides into the Pythagorean theorem: \\overline{JL}^2 + 9^2 = 14^2, i.e., \\overline{JL}^2 + 81 = 196.

4. Solve the above equation to get: \\overline{JL}^2 = 196 - 81 = 115. Therefore, \\overline{JL} = \\sqrt{115}.

5. To solve for \\angle J, we can use the definition of the sine function, i.e., in a right triangle, sin(\\angle J) = \\frac{opposite}{hypotenuse} = \\frac{KL}{JK} = \\frac{9}{14}.

6. Use a calculator to find the value of \\angle J as sin^{-1}(\\frac{9}{14}), thus obtaining \\angle J \\approx 39.77^\\circ.

7. Round \\angle J to the nearest integer degree, resulting in 40 degrees.

8. Through the above reasoning, the final answer is 40 degrees.

直角三角形; 正弦; 余弦; 正切

Definition of Right Triangle

In the figure of this problem, in triangle JKL, <note>angle JLK is a right angle (90 degrees)</note>, thus triangle JKL is a right triangle. <note>Side JL and side KL are the legs</note>, <note>side JK is the hypotenuse</note>.

Definition of Sine Function

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

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

Definition of Triangle

In the figure of this problem, <note>triangle JKL is composed of three non-collinear points J, K, L</note> and their <note>connecting line segments JL, LK, JK</note>. <note>Points J, K, L are the three vertices of the triangle</note>, and <note>line segments JL, LK, JK are the three sides of the triangle</note>.

Pythagorean Theorem

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

In the right triangle JKL, angle L is a right angle (90 degrees), sides JL and KL are the legs, and side JK is the hypotenuse, so according to the Pythagorean Theorem, JK^2 = JL^2 + KL^2.


Question: As shown in the figure, in the right triangle △ABC, ∠C=90°, AC=4, tanA=rac{1}{2}, then the length of BC is ()

Answer: 2

1. Given that in Rt△ABC, ∠C=90°, according to the definition of the tangent function, the tangent value of an acute angle in a right triangle is equal to the length of the opposite side divided by the length of the adjacent side.

2. According to the problem, tanA=1/2, using three letters to represent the angle, we get tan∠BAC=BC/AC.

3. Given AC=4, substitute into the above formula, 1/2=BC/AC.

4. Solve the equation, we get 1/2=BC/4, then BC=2.

5. Through the above reasoning, the final answer is that the length of BC is 2.

直角三角形; 正切; 线段

Definition of Right Triangle

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

Definition of Tangent Function

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

In this problem, <note>tan∠BAC</note> = <note>BC / AC</note>. According to the problem statement, <note>tan∠BAC = 1/2</note>, indicating that <note>the ratio of BC to AC is 1:2</note>.


Question: If C E = t - 2, E B = t + 1, C D = 2, and C A = 10, find C E.

Answer: 1

1. The given conditions are CE = t - 2, EB = t + 1, CD = 2, CA = 10, CB = CE + EB = (t - 2) + (t + 1) = 2t - 1.

2. Given that AB ∥ DE in the figure, according to the definition of corresponding angles, ∠A and ∠EDC are corresponding angles, ∠B and ∠DEC are corresponding angles; according to the parallel postulate 2, ∠A = ∠EDC, ∠B = ∠DEC. According to the AA similarity theorem, triangles CDE and CAB are similar.

3. According to the definition of similar figures, CD:CA = CE:CB, substituting the given conditions, we get 2:10 = (t - 2):(2t - 1).

4. Solving the equation, we get 10*(t - 2) = 2*(2t - 1)

6. Substituting t into CE = t - 2, we get CE = 1

7. Through the above steps, we conclude CE = 1.

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 AB and DE</note> are intersected by a <note>line AC</note>, where <note>∠A and ∠EDC are on the same side of the intersecting line AC and on the same side of the two intersected lines AB and DE, thus ∠A and ∠EDC are corresponding angles</note>. Corresponding angles are equal, that is, <note>∠A=∠EDC</note>; in this problem's diagram, two parallel lines AB and DE are intersected by a <note>line CB</note>, where <note>∠B and ∠DEC are on the same side of the intersecting line CB and on the same side of the two intersected lines AB and DE, thus ∠B and ∠DEC are corresponding angles</note>. Corresponding angles are equal, <note>∠B=∠DEC</note>.

Parallel Postulate 2 of Parallel Lines

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

Original text: <note>Two parallel lines AB and DE</note> are intersected by <note>a third line AC</note>, forming the following geometric relationship: <note>corresponding angles: ∠A and ∠EDC are equal</note>; in the figure of this problem, <note>two parallel lines AB and DE</note> are intersected by <note>a third line CB</note>, forming the following geometric relationship: <note>corresponding angles: ∠B and ∠DEC are equal</note>.

Similarity Theorem for Triangles (AA)

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

<note>In the figure of this problem, in triangles CDE and CAB</note>, if <note>∠A=∠EDC</note> and <note>∠B=∠DEC</note>, then <note>triangle CDE is similar to triangle CAB</note>.

Definition of Similar Figures

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

In the figure of this problem, <note>figure CDE and figure CAB are similar figures</note>. According to the definition of similar figures, the ratios of their corresponding sides are equal, that is, <note>the ratio of side CD to side CA is equal to the ratio of side CE to side CD, and is also equal to the ratio of side DE to side AB</note>.


Question: As shown in the figure, AB∥CD, ray AE intersects CD at point F. If ∠1=115°, then the degree of ∠2 is ()

Answer: 65°

1. Given AB∥CD, ##∠BAE and ∠AFD are same-side interior angles. According to Parallel Postulate 2##, we can conclude that ∠BAE + ∠AFD = 180°.

2. From the problem statement, we know ∠1=115°, i.e., ∠BAE=115°. Therefore, based on the previous reasoning, ∠AFD = 65°.

3. Since ∠2 and ∠AFD are vertical angles, according to ##the definition of vertical angles##, we can conclude that ∠2 = ∠AFD.

4. Therefore, based on the aforementioned reasoning, we can conclude that ∠2 = 65°.

Definition of Vertical Angles

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

In the figure of this problem, point F is the intersection of line CD and ray AE, forming four angles: ∠AFD, ∠CFE, <note>∠AFC</note>, and ∠EFD. According to the definition of vertical angles, ∠AFD and ∠CFE are vertical angles, and <note>∠AFC</note> and ∠EFD are vertical angles. Since the angles of vertical angles are equal, ∠AFD = ∠CFE, <note>∠AFC</note> = ∠EFD.

Parallel Postulate 2 of Parallel Lines

Original text: <note>AB∥CD</note>, and there is a transversal <note>AE</note>, so according to the consecutive interior angles theorem of parallel lines, <note>∠BAE + ∠AFD = 180°</note>.

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

<note>Two lines AB and CD are intersected by a third line AE</note>, <note>angles ∠1 and ∠AFD are on the same side of the intersecting line AE and within the intersected lines AB and CD</note>, so <note>angles ∠1 and ∠AFD are consecutive interior angles</note>. <note>Consecutive interior angles ∠1 and ∠AFD are supplementary</note>, that is, <note>angle 1 + angle AFD = 180 degrees</note>.


Question: The layout of Elm, Plum, and Oak streets is shown in the figure. Find the value of x.

Answer: 125

1. The arrows on the two lines in the figure indicate that these two lines are parallel, i.e., Elm∥Plum, and the lines Elm and Plum are intersected by the third line Oak.

2. According to the parallel lines axiom 2, the interior angles on the same side are supplementary. From the figure, it is known that the 55° angle and the x° angle are interior angles on the same side, so the 55° angle and the x° angle are supplementary, i.e., 55° + x° = 180°.

3. Solving for x° gives x° = 180° - 55° = 125°, so x = 125.

4. Through the above reasoning, it is finally concluded that x = 125.

同位角; 平行线; 内错角

In the diagram of this problem, <note>Elm and Plum are parallel lines</note>, intersected by <note>Oak Street</note>, forming the following geometric relationship: consecutive interior angles: <note>the 55° angle formed by the intersection of Elm Street and Oak Street and the consecutive interior angle formed by the intersection of Plum Street and Oak Street are supplementary</note>, that is, <note>55° + x° = 180°</note>.

<note>Two lines Elm and Plum are intersected by a third line Oak</note>, <note>the angles 55° and x° are on the same side of the intersecting line Oak and within the intersected lines Elm and Plum</note>, so <note>the angles 55° and x° are consecutive interior angles</note>. <note>Consecutive interior angles 55° and x° are supplementary, i.e., 55° + x° = 180°</note>.


Question: Given that the area of the parallelogram is 2000 square units, find its height.

Answer: 40

1. The given condition is that the area of the parallelogram is 2000 square units, the length is x + 10, and the height is x.

2. According to the area formula of a parallelogram, Area = base * height, we get 2000 = (x + 10) * x.

3. Expand the formula 2000 = x^2 + 10x.

4. Convert the formula into the standard quadratic equation x^2 + 10x - 2000 = 0.

5. According to the quadratic equation formula ax^2 + bx + c = 0, the root formula is x = [-b ± sqrt(b^2 - 4ac)] / 2a, where a = 1, b = 10, c = -2000.

6. Calculate b^2 - 4ac, we get 10^2 - 4 * 1 * (-2000) = 100 + 8000 = 8100.

7. Calculate sqrt(8100) = 90.

8. Substitute into the root formula x = [-10 ± 90] / 2, where x1 = (-10 + 90) / 2 = 40, x2 = (-10 - 90) / 2 = -50.

9. Since the side length cannot be negative, exclude x2, thus x = 40.

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

平行四边形; 平行线; 垂线

Area Formula of a Parallelogram

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

<note>Side x+10 is the base</note>, and the corresponding height is the <note>vertical distance from the base x+10 to the opposite side</note>, denoted as <note>x</note>. Therefore, according to the area formula of a parallelogram, the area of the parallelogram is equal to the base length <note>x+10</note> multiplied by the corresponding height <note>x</note>, i.e., <note>A = x(x+10)</note> = 2000.

Definition of Altitude

<note>The line segment x perpendicular to the base from the top left vertex of the parallelogram</note> is the altitude of that vertex. <note>The line segment x forms a right angle (90 degrees) with the base</note>, which indicates that <note>the line segment x is the vertical distance from the top left vertex of the parallelogram to the base</note>.


Question: As shown in the figure, in ⊙O, chords AB and CD intersect at point E, BE = DE, ∠B = 40°, then the degree of ∠A is ()

Answer: 40°

1. Given BE=DE, ∠B=40°, according to the definition of an isosceles triangle, triangle BED is an isosceles triangle; according to the properties of an isosceles triangle, we get ∠DBE=∠EDB=40°.

2. According to the definition of the inscribed angle, ∠CDB and ∠CAB are inscribed angles; and because arc CB corresponds to the inscribed angles ∠CDB and ∠CAB, according to the inscribed angle theorem inference 1, we can get ∠CDB=∠CAB.

3. From the above inference, we can get ∠CDB=∠CAB=40°.

圆; 弧; 圆周角; 弦; 等腰三角形

Definition of Isosceles Triangle

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

In <note>triangle BED</note>, <note>BE = DE</note>, therefore <note>triangle BED is an isosceles triangle</note>.

Properties of Isosceles Triangle

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

In the isosceles triangle <note>BED</note>, <note>sides BE and DE are equal</note>. Therefore, according to the properties of isosceles triangles, the angles opposite the equal sides are equal, <note>i.e., angle EDB = angle DBE</note>.

Definition of Arc

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

<note>In the figure of this problem, there are two points B and C on circle O</note>, <note>arc BC is a segment of a curve connecting these two points</note>. According to the definition of arc, <note>arc BC is a segment of a curve between two points B and C on the circle</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 O, the vertex of angle BAC (point A) is on the circumference, and the two sides of angle BAC intersect circle O at points B and C respectively</note>. Therefore, angle BAC is an inscribed angle; <note>In circle O, the vertex of angle BDC (point D) is on the circumference, and the two sides of angle BDC intersect circle O at points B and C respectively</note>. Therefore, angle BDC is an inscribed angle.

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

<note>In circle O, arc BC</note> corresponds to <note>the inscribed angles ∠CDB and ∠CAB</note>. According to <note>Corollary 1 of the Inscribed Angle Theorem</note>, the <note>inscribed angles ∠CDB and ∠CAB</note> corresponding to <note>the same arc AB</note> are equal, that is, <note>∠CDB=∠CAB</note>.


Question: The positions of lines a, b, c, d are as shown in the figure. If ∠1=100°, ∠2=100°, ∠3=125°, then ∠4 equals ()

Answer: 55°

1. Let the vertical angle of ∠1 be ∠5, the supplementary angle of ∠2 be ∠6, the vertical angle of ∠3 be ∠7, and the quadrilateral formed by the intersection of four lines be quadrilateral ABCD. Given ∠1=100°, ∠2=100°, according to the definition of vertical angles, we get ∠1=∠5=100°, ∠3=∠7=125°.

2. According to the definition of a straight angle, let the two ends of line b be points E and F. Since ∠2 and ∠6 form the straight angle EDF, ∠2 and ∠6 are supplementary, that is, ∠2+∠6=180°. Given ∠2=100°, so ∠6=180°-∠2=180°-100°=80°.

3. According to the theorem of the sum of interior angles of a quadrilateral, in any quadrilateral, the sum of the four interior angles is 360°. Therefore, ∠5+∠6+∠7+∠4=360°. Given ∠5=100°, ∠7=125°, ∠6=80°, so ∠4=360°-∠5-∠6-∠7=360°-100°-125°-80°=55°.

4. Based on the above reasoning, the answer is ∠4=55°.

同旁内角; 平行线; 内错角; 对顶角

Definition of Vertical Angles

<note>Line c and line a intersect at point O</note>, forming four angles: <note>∠1, ∠2, ∠AOB, and ∠BOA</note>. According to the <note>definition of vertical angles</note>, <note>∠1 and ∠2 are vertical angles</note>, <note>∠AOB and ∠BOA are vertical angles</note>. Since <note>vertical angles are equal</note>, therefore <note>∠1=∠2=100°</note>.

Definition of Straight Angle

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

<note>In the figure of this problem, ray FD rotates around endpoint D to form a straight line with the initial side, creating straight angle EDF</note>. According to the definition of a straight angle, <note>a straight angle measures 180 degrees</note>, that is, <note>angle EDF = 180 degrees</note>.

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

In the figure of this problem, <note>in quadrilateral ABCD, angles 5, 6, 7, and 4 are the four interior angles of the quadrilateral</note>. According to the Sum of Interior Angles of a Quadrilateral Theorem, the sum of these four interior angles is 360°, that is, <note>angle 5 + angle 6 + angle 7 + angle 4 = 360°</note>.


Question: Given: As shown in the figure, in ⊙O, OA⊥BC, ∠AOB=70°, then the degree of ∠ADC is ()

Answer: 35°

1. Given OA is perpendicular to BC, ##according to the perpendicular bisector theorem, arc CA = arc AB, so the angle subtended by arc CA at the circumference is equal to the angle subtended by arc AB at the circumference##.

2. ##According to the inscribed angle theorem, and the conclusion from step one, ∠AOB = 2∠ADC##.

####

##3##. Since ∠AOB = 70°, ∠ADC = ∠AOB/2 = 35°.

圆; 垂线; 圆心角; 圆周角; 圆内接四边形

Definition of Inscribed Angle

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

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>radius OA bisects chord BC</note>, and <note>radius OA bisects the two arcs subtended by chord BC</note>, that is, <note>arc CA = arc AB</note>, so <note>the angle subtended by arc CA at the circumference is equal to the angle subtended by arc AB at the circumference</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 figure of this problem, <note>∠ADC is the inscribed angle that subtends arc AC</note>, <note>∠AOB is the central angle that subtends arc AB</note>, because <note>arc CA = arc AB</note>. Therefore, according to the Inscribed Angle Theorem, <note>∠ADC = ∠AOB / 2 = 70° / 2 = 35°</note>.

Definition of Central Angle

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

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


Question: As shown in the figure, in △ABC, ∠C=90°, AC=BC, AD bisects ∠CAB and intersects BC at D, DE⊥AB at E. If AB=6cm, then the perimeter of △DBE is ()

Answer: 6cm

1. Given △ABC, ∠ACB=90°. Because DE⊥AB, ∠AED is a right angle, i.e., ∠AED=90°.

2. Since AD bisects ∠CAB, ∠CAD=∠EAD.

3. According to the congruence theorem (AAS), because ∠ACB=90°=∠AED, ∠CAD=∠EAD, and AD=AD, △ACD≌△AED.

4. Since △ACD≌△AED, we get AC=AE, CD=ED.

5. Since AC=BC, from the above conclusion we can deduce BC=AE.

6. From the figure, the perimeter of △DBE is DE+BD+EB.

7. Since DE=CD, and DE+CD=BC, while BC is AE, adding EB we get the perimeter of △DBE as AE+EB.

8. Thus, we can obtain the perimeter of ΔDBE as AB=6cm.

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

直角三角形; 等腰三角形; 垂线; 垂直平分线; 线段

Definition of Angle Bisector

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

<note>The vertex of angle CAB is point A</note>, from <note>point A a line AD is drawn</note>, this line divides angle CAB into <note>two equal angles, i.e., ∠CAD and ∠BAD are equal</note>. Therefore, <note>line AD is the angle bisector of angle CAB</note>.

Congruence Theorem for Triangles (AAS)

Two triangles are congruent if two angles and the included side of one triangle are respectively equal to two angles and the included side of the other triangle.

In the diagram of this problem, in triangles <note>ADC</note> and <note>ADE</note>, <note>∠CAD equals ∠EAD, ∠ACD equals ∠AED, and side AD equals AD</note>. Since two angles and the corresponding side of these two triangles are equal, according to the Angle-Angle-Side criterion of the Congruence Theorem for Triangles <note>AAS</note>, it can be concluded that <note>triangle ADC is congruent to triangle ADE</note>.

Definition of Congruent Triangles

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

In the figure of this problem, <note>triangle ACD and triangle ADE are congruent triangles</note>, <note>the corresponding sides and corresponding angles of triangle ACD are equal to those of triangle ADE</note>, namely:
<note>side AC = side AE</note>
<note>side AD = side AD</note>
<note>side CD = side ED</note>,
at the same time, the corresponding angles are also equal:
<note>angle CAD = angle EAD</note>
<note>angle DAC = angle EAC</note>
<note>angle ADC = angle ADE</note>.


Question: Find x.

Answer: 3 \sqrt { 35 }

1. Let the triangle be ABC. According to the definition of a right triangle, ΔABC is a right triangle where ∠BAC is a right angle, BC is the hypotenuse, and AB and AC are the legs.

3. According to the Pythagorean theorem, we have: AB^2 + AC^2 = BC^2, i.e., x^2 + 13^2 = 22^2.

5. Substitute the values into the equation to get: x^2 + 169 = 484.

6. Solve this equation: x^2 = 484 - 169, to get x^2 = 315.

8. Simplify √315 to get x = √(9*35) = 3√35.

9. Through the above reasoning, the final answer is 3√35.

Definition of Right Triangle

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

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

Pythagorean Theorem

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

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


Question: As shown in the figure, Xiaoming wants to measure the height AB (the distance from the top A to the horizontal ground BD) of a pavilion. He places a step DE (DE=BC=0.6 meters, ensuring points A, B, and C are collinear) next to the pavilion step BC, and places a mirror horizontally at point G on the platform, measuring CG=12 meters. Then he steps back along the line CG to point E, where he can just see the top A of the pavilion in the mirror, measuring GE=2 meters. Xiaoming's height EF=1.6 meters. What is the approximate height AB of the pavilion?

Answer: 10.2米

1. Given that ∠AGC and ∠FGE are equal angles corresponding to mirror imaging, we obtain ∠AGC=∠FGE from the problem statement.

2. According to the given conditions, ∠ACG and ∠FEG are both right angles, i.e., ∠ACG=∠FEG=90°.

3. ##According to the AA criterion for similar triangles##, we can conclude that △ACG and △FEG are similar triangles.

4. Based on the ##definition of similar triangles##, we have AC:EF=CG:GE.

5. Substituting the known values into the proportion of similar triangles, we get the proportion AC:1.6=12:2.

6. Solving the proportion, we find AC=9.6 meters.

7. According to the given conditions, BC=0.6 meters, and AB=AC+BC.

8. Therefore, AB=9.6+0.6=10.2 meters.

9. Through the above reasoning, we finally obtain the answer as 10.2 meters.

直角三角形; 反射; 对称

Similarity Theorem for Triangles (AA)

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

<note>Angle AGC is equal to angle FGE</note>, and <note>angle ACG is equal to angle FEG</note>, so <note>triangle ACG is similar to triangle FEG</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>△ACG and △FEG are similar triangles</note>. According to the definition of similar triangles: <note>∠ACG = ∠FEG</note>, <note>∠CAG = ∠EFG</note>, <note>∠CGA = ∠FGE</note>; <note>AC/EF = CG/GE = AG/FG</note>.


Question: Find the area of the figure. Round the result to the nearest tenth.

Answer: 28.3

1. Given that the radius of the circle is 3 inches, according to the ##circle area formula##, the area of the circle can be calculated using the formula A = πr^2.

2. Substituting the given radius r = 3 inches, we get the area A = π * (3)^2.

3. Calculating, we get A = 9π square inches.

5. Calculating, we get A ≈ 28.274333877.

6. Rounding the result to one decimal place, we get A ≈ 28.3.

7. Through the above reasoning, the final answer is 28.3.

Definition of Radius

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

In the diagram of this problem, in the circle, <note>the center of the circle is a black dot</note>, <note>any point on the circle is any point on the circumference</note>, <note>the line segment from the center of the circle to any point on the circumference is 3 inches</note>, therefore <note>the line segment is the radius of the circle</note>. Symbolically, <note>the radius is r = 3 inches</note>.

Area Formula of a Circle

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

<note>The radius of the circle is 3 inches</note>. According to the area formula of a circle, the area A of the circle equals pi times <note>the square of the radius 3</note>, that is <note>A = π * 3^2</note>.

Definition of Circle

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

<note>In a circle, the black dot is the center</note>, <note>the radius is 3 inches</note>. All points in the figure that are at a distance of <note>3 inches from the black dot are on the circle</note>.


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

Answer: 76

1. Given that the opposite sides of a parallelogram are equal, one pair of sides is 20 ft and 18 ft.

2. Therefore, the other pair of opposite sides of the parallelogram are also 20 ft and 18 ft respectively.

3. According to the ##perimeter of the parallelogram##, the perimeter formula is: 2 × (long side + short side).

4. Substitute the long side and short side into the formula: Perimeter = 2 × (20 + 18) ft.

5. Calculate to get: Perimeter = 2 × 38 ft.

6. That is, Perimeter = 76 ft.

7. Therefore, the final perimeter of the parallelogram is 76 ft.

Definition of Parallelogram

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

In the diagram of this problem, the two pairs of opposite sides of the quadrilateral are 20 feet and 20 feet, parallel and equal, and 18 feet and 18 feet, parallel and equal.

Perimeter of a Parallelogram

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

In the figure of this problem, the parallelogram's <note>two pairs of adjacent sides are 20 feet and 18 feet respectively</note>. According to the formula for the perimeter of a parallelogram, the perimeter of the parallelogram is equal to twice the sum of its two pairs of adjacent sides, that is, <note>Perimeter P = 2(a + b) = 2(20 + 18) = 76 feet</note>.


Question: Find a, if F G = 18, G H = 42, and F K = 15.

Answer: 57

1. Let the center of the circle be O, given that the segment FG = 18, GH = 42, and the segment FK = 15. According to the secant theorem, if two secants start from an external point and intersect the circle at two segments respectively, then the product of the external segment and the entire secant are equal: FK × (FK + KJ) = FG × (FG + GH).

2. Given GH = 42, calculate: FG + GH = 18 + 42 = 60.

3. From the expression of the secant theorem FK × (FK + KJ) = FG × (FG + GH), we have: 15 × (15 + a) = 18 × 60.

4. Calculate the equation: 1080 = 15 × (15 + a).

5. Divide both sides by 15, we get the equation: 15 + a = 1080 / 15.

7. After calculation, we finally get a = 57.

8. After the above reasoning, the final answer is a = 57.

圆; 弦; 弧; 圆周角

Secant Line

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

In the figure of this problem, <note>line FH intersects circle O at two points, namely point G and point H</note>. According to the definition of a secant line, <note>line FH intersects circle O at two distinct points</note>, so <note>line FH is a secant line of circle O</note>. <note>Line FJ intersects circle O at two points, namely point K and point J</note>. According to the definition of a secant line, <note>line FJ intersects circle O at two distinct points</note>, so <note>line FJ is a secant line of circle O</note>.

Secant Line Theorem

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

<note>There are two secant lines FH and FJ in circle O, intersecting at point F outside the circle</note>. The external part of secant line FH is FG, and the internal part is GH; the external part of secant line FJ is FK, and the internal part is KJ. According to the Secant Line Theorem, <note>the product of the external part of secant line FH and the entire secant line is equal to the product of the external part of secant line FK and the entire secant line, i.e., FG * FH = FK * FJ</note>.


Question: Quadrilateral ABCD is a rectangle. If m \angle 1 = 38, find m \angle 2.

Answer: 52

1. Given quadrilateral ABCD is a rectangle, ∠1 = 38°.

2. According to the definition of a rectangle, ∠CAB = 90°.

4. Substituting ∠1 = 38°, we solve for ∠2 = 52°.

矩形; 邻补角; 对顶角

Definition of Rectangle

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

<note>Quadrilateral ABCD is a rectangle</note>, with its interior angles <note>∠ABD, ∠BDC, ∠DCA, ∠CAB all being right angles (90 degrees)</note>, <note>sides AB and CD are parallel and equal in length</note>, <note>sides AC and BD are parallel and equal in length</note>.