Question:
|  | | | | | OQ | |
and

| RT |
are parallel lines. Which angles are corresponding angles?

-  $\angle $ QPN and  $\angle $ OPS
-  $\angle $ QPN and  $\angle $ QPS
-  $\angle $ QPN and  $\angle $ TSU
-  $\angle $ QPN and  $\angle $ TSP

Answer: -  \$\angle \$ QPN and  \$\angle \$ TSP

1. Given that line OQ and line RT are parallel, and line NU is their transversal.

2. According to the definition of corresponding angles, this definition states: if two lines are cut by a transversal, then a pair of angles located on the same side of the transversal but on different lines are corresponding angles.

3. Observing the figure, we look for the angle corresponding to angle QPN.

4. Angle QPN is located on one side of the transversal NU, on line OQ, as an upper interior angle.

5. On the other set of parallel lines RT, the interior angle on the same side of the transversal NU corresponding to angle QPN is angle TSP.

6. According to the definition of corresponding angles, angle QPN and angle TSP are corresponding angles.

7. Check other alternative answer options:

8. Angle QPN and angle OPS are located on different sides of the transversal, not fitting the definition.

9. Angle QPN and angle QPS are in different locations, not the same angle.

10. Angle QPN and angle TSU are in different interior angle positions, not fitting the definition.

11. Through the above reasoning, the final answer is that angle QPN and angle TSP are corresponding angles.

Definition of Parallel Lines

In the figure of this problem, <note>line OQ and line RT lie in the same plane</note>, and <note>they do not intersect</note>, therefore according to the definition of parallel lines, <note>line OQ and line RT are parallel lines</note>.

Definition of Corresponding Angles

In the figure of this problem, <note>angle QPN and angle TSP are corresponding angles</note>, because they are located on <note>the same side of the transversal NU</note> but on <note>different lines OQ and RT</note>. According to the <note>definition of corresponding angles</note>, these two <note>angles are equal</note>.


Question:
|  | | | | | OQ | |
and

| RT |
are parallel lines. Which angles are supplementary angles?

-  $\angle $ TSU and  $\angle $ TSP
-  $\angle $ TSU and  $\angle $ OPN
-  $\angle $ TSU and  $\angle $ RSP
-  $\angle $ TSU and  $\angle $ QPS

Answer: -  \$\angle \$ TSU and  \$\angle \$ TSP

1. According to the figure in the problem, line RT is parallel to line OQ, and line SU intersects the two parallel lines. According to the theorem of the intercepted line of parallel lines (also known as the corresponding angles theorem), the corresponding angles between the intersection points T and Q on the parallel lines RT and OQ with the intercepted line SU are equal, i.e., ∠TSU and ∠PQO are corresponding angles, so ∠TSU = ∠PQO.

2. To determine whether two angles are supplementary, their degree sum should be 180°.

3. First, check ∠TSU and ∠TSP. Point S is common between ∠TSU and ∠TSP, and note that ∠TSP is the exterior angle extended from line ST to line SU. According to the linear pair theorem, the two angles in a linear pair are supplementary (the sum of their degrees is 180°). Therefore, ∠TSU and ∠TSP are supplementary angles.

4. Check ∠TSU and ∠OPN. According to the vertical angles theorem, ∠TSU and ∠OPN are not vertical angles and do not form a linear pair, so their sum is not 180°, thus ∠TSU and ∠OPN are not supplementary angles.

5. Check ∠TSU and ∠RSP. Point S is common between ∠TSU and ∠RSP, but these two angles do not belong to the same linear pair because the other ray of ∠RSP passes through points R and P. Therefore, ∠TSU and ∠RSP are not supplementary angles.

6. Check ∠TSU and ∠QPS. According to the figure, ∠TSU includes ∠QPS, so they cannot be supplementary angles because there is no straight path of 180°.

7. After the above reasoning, the final answer is that ∠TSU and ∠TSP are supplementary angles.

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

In the figure of this problem, <note>two parallel lines RT and OQ</note> are intersected by <note>a line SU</note>, where <note>angle TSU and angle PQO are on the same side of the intersecting line SU and on the same side of the intersected lines RT and OQ</note>, therefore <note>angle TSU and angle PQO are corresponding angles. Corresponding angles are equal, i.e., angle TSU is equal to angle PQO</note>.

Definition of Vertical Angles

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

In the figure of this problem, line SU intersects with line OQ at point P, forming four angles: <note>∠SPU</note>, <note>∠QPN</note>, <note>∠SPQ</note>, and <note>∠UPN</note>. According to the definition of vertical angles, <note>∠SPU and ∠QPN are vertical angles</note>, <note>∠SPQ and ∠UPN are vertical angles</note>. Since the angles of vertical angles are equal, <note>∠SPU=∠QPN</note>, <note>∠SPQ=∠UPN</note>.


Question:
Find m $\angle $ B.

m $\angle $ B= $\Box$ °

Answer: m \$\angle \$ B=45°

1. Given △BCD, BC is a right angle, ∠BCD is 90°.

2. △BCD is a right triangle. According to the definition of a right triangle, trigonometric ratios can be used to find the measure of angle B.

3. By the definition of the sine function, sin(∠DBC) = opposite side/hypotenuse. In this triangle, the opposite side is CD and the hypotenuse is BD.

4. Given CD = 5 and BD = 5√2, substituting into the sine function definition, we get sin(∠DBC) = CD/BD = 5/(5√2) = 1/√2.

5. Rationalizing 1/√2, we get √2/2.

6. In the range of angles from 0° to 90°, the sine value of a standard angle is sin(45°) = √2/2.

7. Therefore, combining the results from steps three and five with the known sine value, we get ∠DBC = 45°.

8. Through the above reasoning, the final answer is m ∠B = 45°.

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, triangle <note>BCD</note> has <note>angle BCD as a right angle (90 degrees)</note>, therefore <note>triangle BCD is a right triangle</note>. <note>Sides BC and CD are the legs</note>, <note>side BD 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 BCD, <note>angle DBC is an acute angle</note>, <note>side CD is the opposite side of angle DBC</note>, <note>side BD is the hypotenuse</note>. According to the <note>definition of the sine function</note>, <note>the sine value of angle DBC is equal to the ratio of the opposite side CD to the hypotenuse BD</note>, that is, <note>sin(∠DBC) = CD / BD = 5 / 5√2 = 1/√2</note>.


Question:
|  | | | | | LN | |
and

| OQ |
are parallel lines. Which angles are alternate interior angles?

-  $\angle $ LMP and  $\angle $ OPM
-  $\angle $ LMP and  $\angle $ QPR
-  $\angle $ LMP and  $\angle $ NMK
-  $\angle $ LMP and  $\angle $ QPM

Answer: -  \$\angle \$ LMP and  \$\angle \$ QPM

1. Given that line LN is parallel to line OQ, it is known from the problem statement that line KR is the transversal of these lines.

2. According to the definition of alternate interior angles, we need to find the angles on both sides of the transversal KR and between the segments LN and OQ.

3. From the diagram, angle LMP is located on one side of KR and between the two parallel lines.

4. Check the provided pairs of angles to find which angle is on the other side of KR and between the two parallel lines.

5. First, consider angle LMP and angle OPM. Angle OPM is on the same side of the transversal, so it is not an alternate interior angle.

6. Then, consider angle LMP and angle QPR. Angle QPR is outside the transversal, so it is not an alternate interior angle.

7. Next, consider angle LMP and angle NMK. Angle NMK is outside the transversal, so it is not an alternate interior angle.

8. Finally, consider angle LMP and angle QPM. Angle QPM is on the other side of KR and between the two parallel lines.

9. Therefore, angle LMP and angle QPM meet the conditions for being alternate interior angles formed by a transversal intersecting parallel lines, i.e., they are on both sides of KR and between the two parallel lines.

10. Through the above reasoning, the final answer is that angle LMP and angle QPM are alternate interior angles.

Definition of Parallel Lines

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

<note>Line LN and line OQ lie in the same plane</note>, and <note>they have no intersection points</note>, so according to the definition of parallel lines, line LN and line OQ are parallel lines.

Definition of Alternate Interior Angles

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

<note>Two parallel lines LN and OQ</note> are intersected by a <note>line KR</note>, where <note>angle LMP</note> and <note>angle QPM</note> are between the two parallel lines and on opposite sides of the intersecting line KR. Therefore, <note>angle LMP</note> and <note>angle QPM are alternate interior angles</note>. Alternate interior angles are equal, that is, <note>angle LMP is equal to angle QPM</note>.


Question:
|  | | | | | PR | |
and

| SU |
are parallel lines. Which angles are alternate interior angles?

-  $\angle $ PQT and  $\angle $ UTQ
-  $\angle $ PQT and  $\angle $ STV
-  $\angle $ PQT and  $\angle $ RQO
-  $\angle $ PQT and  $\angle $ UTV

Answer: -  \$\angle \$ PQT and  \$\angle \$ UTQ

1. According to the problem statement, line PR and line SU are parallel, and line OV is the transversal, forming multiple alternate interior angles.

2. Based on the definition of alternate interior angles, two angles formed by a transversal intersecting two parallel lines, where these angles are on opposite sides of the transversal and between the parallel lines.

3. Consider ∠PQT, which is between line PR and line SU and located on the left side of the transversal OV.

4. Examine ∠UTQ, which is between line PR and line SU and located on the right side of the transversal OV.

5. ∠PQT and ∠UTQ satisfy the definition of alternate interior angles because ∠PQT and ∠UTQ are between the same pair of parallel lines and on opposite sides of the transversal.

6. Examine other options: ∠PQT and ∠STV are on the same side of the transversal OV and are not alternate interior angles; ∠PQT and ∠RQO, where ∠RQO is not between the two parallel lines and is not an alternate interior angle; ∠PQT and ∠UTV, where ∠UTV is not between the two parallel lines and is not an alternate interior angle.

7. Through the above reasoning, the final answer is ∠PQT and ∠UTQ are alternate interior angles.

<note>Line PR and line SU are parallel lines</note>, <note>line OV is a transversal</note>. <note>Angle PQT and angle UTQ are located between the two parallel lines PR and SU, and on opposite sides of the transversal OV</note>, therefore <note>angle PQT and angle UTQ are alternate interior angles</note>. Alternate interior angles are equal, i.e., <note>angle PQT is equal to angle UTQ</note>.


Question:
|  | | | | | EG | |
and

| HJ |
are parallel lines. Which angles are corresponding angles?

-  $\angle $ JIK and  $\angle $ HIK
-  $\angle $ JIK and  $\angle $ GFI
-  $\angle $ JIK and  $\angle $ EFI
-  $\angle $ JIK and  $\angle $ HIF

Answer: -  \$\angle \$ JIK and  \$\angle \$ GFI

1. Given that line EG and line HJ are parallel, line DK is their transversal.

2. According to the properties of transversals of parallel lines, if parallel lines are intersected by a transversal, then corresponding angles are equal, and alternate interior angles are equal.

3. First, observe the options given in the problem to determine the respective positions of each angle.

4. Examine the corresponding angle of ∠JIK:

   - ∠JIK is located between ∠HJI and ∠KJI, clearly ∠JIK is at the intersection point I of line DK and line HJ, and adjacent to ∠KJI.

5. Check the corresponding relationship of ∠JIK with other angles in the options:

   - ∠HIK: It is on the same side of line HJ as ∠JIK but not in the same position, not a corresponding angle.

   - ∠GFI: It is at the intersection point F of line EG and transversal DK, and ∠JIK is on opposite sides of transversal DK and in the same position relative to the parallel lines, making it a corresponding angle.

   - ∠EFI: It is on the other side of transversal DK, not in the same position, not a corresponding angle.

   - ∠HIF: It is on the other side of transversal DK, not in the same position, not a corresponding angle.

6. In summary, by checking the positional relationships of each pair of angles, it is concluded that ∠JIK and ∠GFI are corresponding angles because they are in the same relative position, i.e., on opposite sides of transversal DK and respectively on parallel lines EG and HJ.

7. Through the above reasoning, the final answer is that ∠JIK and ∠GFI are corresponding angles.

Definition of Parallel Lines

In the figure of this problem, <note>line EG and line HJ are in the same plane</note>, and <note>they do not intersect</note>, therefore according to the definition of parallel lines, <note>line EG and line HJ are parallel lines</note>.

Definition of Corresponding Angles

<note>Line EG and line HJ are parallel lines</note>, and <note>line DK is the transversal</note>. According to the theorem of the transversal of parallel lines, <note>∠JIK and ∠GFI are corresponding angles</note>, because they are located on opposite sides of the transversal DK and lie on the parallel lines EG and HJ respectively.


Question:
Find m $\angle $ B.

m $\angle $ B= $\Box$ °

Answer: m \$\angle \$ B=45°

1. Observe the given triangle BDC. According to the definition of a right triangle, this triangle is a right triangle, where ∠BDC is a right angle.

2. Determine the side lengths of triangle BDC, where BC = 6√14 and CD = 6√7.

3. Based on the given conditions, use the side lengths of the triangle to calculate the angles in the triangle, choosing the appropriate trigonometric function to determine the degree of one of the angles.

4. According to the definition of the sine function, sin(∠B) = opposite side/hypotenuse.

5. ∠B is the angle we need to find. Based on the right triangle where ∠BDC is located, let ∠B be the angle we need to find. Then the opposite side is CD and the hypotenuse is BC.

6. Calculate sin(∠B) = CD/BC = (6√7)/(6√14) = √7/√14.

7. Simplify the fraction, sin(∠B) = √7/√14 = √(1/2) = 1/√2.

8. Compare the sine values of known special angles, remembering that sin(45°) = 1/√2.

9. Therefore, based on sin(∠B) = 1/√2, we can determine that ∠B = 45°.

10. Through the above reasoning, the final answer is ∠B = 45°.

Definition of Right Triangle

<note>Angle ∠BDC is a right angle (90 degrees)</note>, therefore triangle BDC is a right triangle. <note>Side BD and side CD are the legs</note>, <note>side BC is the hypotenuse</note>.

Definition of Sine Function

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


Question:
Find m $\angle $ C.

m $\angle $ C= $\Box$ °

Answer: m \$\angle \$ C=60°

1. Given ∠E is a right angle, according to the definition of a right triangle, triangle CDE is a right triangle. DE is the opposite side of ∠C, and CD is the hypotenuse.

2. According to the definition of the sine function, sin(∠C) = opposite side/hypotenuse = DE/CD.

3. Substituting the given conditions DE = 2√3, CD = 4, therefore sin(∠C) = (2√3)/4 = √3/2.

4. Calculating, we get sin(60°) = √3/2.

5. Therefore, the sine value of ∠C is equal to the sine value of 60°. Thus, we conclude ∠C = 60°.

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

Definition of Right Triangle

<note>Angle EDC is a right angle (90 degrees)</note>, therefore triangle CDE is a right triangle. Side DE and side CE are the legs, and side CD is the hypotenuse.

Definition of Sine Function

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


Question:
Find m $\angle $ V.

m $\angle $ V= $\Box$ °

Answer: m \$\angle \$ V=30°

1. Given triangle VWX, side VW is the right-angle side, side VX is the hypotenuse, m ∠ VWX = 90°.

2. According to the definition of a right triangle, the right-angle side is the height on the hypotenuse, thus angle ∠ VWX is a right angle.

3. According to the cosine function, the cosine value is equal to the ratio of the adjacent right-angle side to the hypotenuse, i.e., cos(∠ XVW) = VW/VX.

4. According to the values given in the problem, VW = 8√42, VX = 16√14.

5. Substitute the above values: cos(∠ XVW) = (8√42) / (16√14).

6. Simplify the ratio: cos(∠ XVW) = (8/16) × (√42/√14) = √3/2.

7. Known trigonometric values for common angles, cos(30°) = √3/2.

8. Therefore, we conclude m ∠ XVW = 30°.

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

Definition of Right Triangle

<note>In triangle VWX, angle ∠VWX is a right angle (90 degrees)</note>, therefore <note>triangle VWX is a right triangle</note>. <note>Side VW and side WX are the legs</note>, <note>side VX is the hypotenuse</note>.

Cosine Function

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

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


Question:
|  | | | | | RT | |
and

| UW |
are parallel lines. Which angles are vertical angles?

-  $\angle $ UVX and  $\angle $ UVS
-  $\angle $ UVX and  $\angle $ WVX
-  $\angle $ UVX and  $\angle $ TSV
-  $\angle $ UVX and  $\angle $ WVS

Answer: -  \$\angle \$ UVX and  \$\angle \$ WVS

1. Given that line UW and line RT are parallel, and line QX is a transversal between them. The relationship of the angles will depend on how these lines intersect.

2. According to the definition of vertical angles, when two lines intersect, the non-adjacent angles formed by the intersecting lines are equal; these angles are called vertical angles.

3. First, analyze angle UVX: this is an angle formed by line QX and a line passing through point V (such as UW and WV).

4. For option (1) angle UVX and angle UVS: both share side UV, are adjacent angles, and are not vertical angles.

5. For option (2) angle UVX and angle WVX: both share side VX, and since they are on line UW, they are adjacent angles, not vertical angles.

6. For option (3) angle UVX and angle TSV: angle UVX extends outward from point V, while angle TSV is an angle at point S, and they are not part of the same pair of intersecting lines, so they are not vertical angles.

7. Finally, for option (4) angle UVX and angle WVS: these two angles both involve line QX and its intersecting line UW. Angle UVX is on one side of UW, while angle WVS is on the opposite side, and they do not share any sides, making them vertical angles.

8. Based on the above analysis and reasoning, it is determined that angle UVX and angle WVS are vertical angles.

Definition of Parallel Lines

<note>Straight line UW and straight line RT are located in the same plane</note>, and <note>they have no intersection points</note>, so according to the definition of parallel lines, <note>straight line UW and straight line RT are parallel lines</note>.

Definition of an Angle

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

<note>Angle UVX is formed by rays UV and VX with a common vertex V</note>, <note>Angle WVS is formed by rays WV and VS with a common vertex V</note>. <note>Angle UVX and angle WVS are vertical angles formed by two intersecting lines QX and UW</note>.

Definition of Vertical Angles

In the diagram of this problem, line <note>QX</note> and line <note>UW</note> intersect at point <note>V</note>, forming four angles: angle <note>UVX</note>, angle <note>WVX</note>, angle <note>UVS</note>, and angle <note>WVS</note>. According to the definition of vertical angles, angle <note>UVX</note> and angle <note>WVS</note> are vertical angles, angle <note>WVX</note> and angle <note>UVS</note> are vertical angles. Since the angles of vertical angles are equal, angle <note>UVX = angle WVS</note>, angle <note>WVX = angle UVS</note>.


Question:
Find m $\angle $ U.

m $\angle $ U= $\Box$ °

Answer: m \$\angle \$ U=45°

1. Given ΔUST is a right triangle, ∠T is a right angle, ST=1, SU=√2.

2. According to the Pythagorean theorem, in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the two legs.

3. Set TU = x, according to the Pythagorean theorem: SU^2 = ST^2 + TU^2.

4. Substitute SU=√2 and ST=1, we get (√2)^2 = 1^2 + x^2.

5. Calculating, we get 2 = 1 + x^2, thus x^2 = 1, therefore TU = 1.

6. ΔUST is an isosceles right triangle because ST=TU=1, and ∠STU is a right angle.

7. In an isosceles right triangle, the two acute angles are equal, each angle is (180°-90°)÷2=45° (triangle angle sum theorem).

8. Therefore, ∠U = 45°.

9. Through the above reasoning, the final answer is 45°.

Definition of Right Triangle

In the figure of this problem, in triangle UST, <note>angle UTS is a right angle (90 degrees)</note>, therefore <note>triangle UST is a right triangle</note>. <note>Side ST and side TU are the legs</note>, <note>side SU 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 UST, ∠UTS is a right angle (90 degrees)</note>, <note>sides ST and TU are the legs, and side SU is the hypotenuse</note>, so according to the Pythagorean Theorem, <note>SU² = ST² + TU²</note>.

Triangle Angle Sum Theorem

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

<note>In triangle TUS, angles T, U, and S are the three interior angles of triangle TUS</note>, according to the Triangle Angle Sum Theorem, <note>angle T + angle U + angle S = 180°</note>.

Definition of Isosceles Right Triangle

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

<note>Triangle TUS is an isosceles right triangle</note>, in which <note>angle UTS is a right angle (90 degrees)</note>, <note>sides TU and ST are equal right-angle sides</note>.


Question:
KM is tangent to ⨀J. What is JM?

JM= $\Box$  yd

Answer: JM=10 yd

1. Given point K is a point on ⨀J, KM is the tangent to ⨀J. According to the property of the tangent to a circle, the tangent is perpendicular to the radius passing through the point of tangency, so ∠JKM=90°.

2. In the right triangle △JKM, using the Pythagorean theorem, which states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides, expressed as: if in a right triangle, the hypotenuse is c, and the other two sides are a and b, then a^2 + b^2 = c^2.

3. In this problem, JK is one leg, KM is the other leg, and JM is the hypotenuse. According to the given information, JK=6, KM=8.

4. Substitute the given lengths into the Pythagorean theorem formula, we get: 6^2 + 8^2 = JM^2.

5. Calculate each term, 36 + 64 = JM^2.

6. Further calculation, the equation is 100 = JM^2.

7. Take the arithmetic square root of both sides of the equation, we get JM = √100.

8. Solve to get JM = 10.

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

Definition of Tangent to a Circle

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

<note>Circle ⨀J</note> and <note>line KM</note> have exactly one common point <note>K</note>, this common point is called the <note>point of tangency</note>. Therefore, <note>line KM</note> is the <note>tangent to circle ⨀J</note>.

Definition of Radius

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

<note>In circle ⨀J, point J is the center of the circle, point K is a point on the circle, and line segment JK is a line segment from the center to any point on the circle</note>, therefore <note>line segment JK is the radius of the circle</note>.

Definition of Right Triangle

In the figure of this problem, in triangle JKM, <note>angle JKM is a right angle (90 degrees)</note>, therefore <note>triangle JKM is a right triangle</note>. <note>Side JK and side KM are the legs</note>, <note>side JM is the hypotenuse</note>.

Property of the Tangent Line to a Circle

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

<note>In circle ⨀J, point K is the point of tangency of line KM with the circle</note>, <note>segment JK is the radius of the circle</note>. According to the property of the tangent line to a circle, <note>the tangent line KM is perpendicular to the radius JK at the point of tangency K</note>, that is, <note>∠JKM=90 degrees</note>.

Pythagorean Theorem

In this problem diagram, <note>△JKM is a right triangle</note>, <note>∠JKM=90°</note>, <note>sides JK and KM are the legs</note>, <note>side JM is the hypotenuse</note>, so according to the Pythagorean Theorem, <note>JM² = JK² + KM²</note>. According to the given conditions, <note>JK=6 yd</note>, <note>KM=8 yd</note>, therefore <note>JM is calculated as follows:
6² + 8² = JM²
36 + 64 = JM²
100 = JM².
Solving for JM, we get JM = √100 = 10 yd</note>.


Question:
|  | | | | | FH | |
and

| IK |
are parallel lines. Which angles are adjacent angles?

-  $\angle $ HGJ and  $\angle $ FGE
-  $\angle $ HGJ and  $\angle $ IJG
-  $\angle $ HGJ and  $\angle $ HGE
-  $\angle $ HGJ and  $\angle $ KJL

Answer: -  \$\angle \$ HGJ and  \$\angle \$ HGE

1. According to the given conditions, line FH and line IK are parallel lines, and the figure is provided.

2. Determine the position of angle ∠HGJ, which is from point H to point G, then to point J, located on the part of line FH up to the intersection point G, and extends from G to the segment J.

3. Analyze the relative positions of each angle in the options:

4. Consider angle ∠HGJ and angle ∠FGE in option 1, and find that HGJ and FGE do not share a common side, so they are not adjacent supplementary angles.

5. Consider angle ∠HGJ and angle ∠IJG in option 2, and find that HGJ and IJG do not share a common side, so they are not adjacent supplementary angles.

6. Consider angle ∠HGJ and angle ∠HGE in option 3, and find that HGJ and HGE share ray GH, so they are adjacent supplementary angles.

7. Consider angle ∠HGJ and angle ∠KJL in option 4, and find that HGJ and KJL do not share a common side, so they are not adjacent supplementary angles.

8. Through the above reasoning, the final answer is option 3: angle ∠HGJ and angle ∠HGE are adjacent supplementary angles.

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 HGJ and angle HGE have a common side GH</note>, <note>their other sides EG and GJ are extensions in opposite directions</note>, so angle HGJ and angle HGE are adjacent supplementary angles.


Question:
|  | | | | | FH | |
and

| IK |
are parallel lines. Which angles are corresponding angles?

-  $\angle $ KJL and  $\angle $ HGE
-  $\angle $ KJL and  $\angle $ FGE
-  $\angle $ KJL and  $\angle $ HGJ
-  $\angle $ KJL and  $\angle $ IJL

Answer: -  \$\angle \$ KJL and  \$\angle \$ HGJ

1. According to the conditions in the problem, line FH and line IK are parallel lines, and line EL is their transversal.

2. According to the definition of corresponding angles, if two lines are cut by a transversal and these two lines are parallel, then the corresponding angles formed by the transversal are equal. In this problem, line FH and line IK are parallel, so the corresponding angles formed by transversal EL are equal.

3. Determine the definition of corresponding angles: Corresponding angles refer to the four pairs of angles formed by the intersection of parallel lines and a transversal, located on the same side of the transversal and outside the two parallel lines.

4. Confirm the positions of angle KJL and angle HGJ. Angle KJL is located below the intersection of line EL and line IK, outside the parallel lines, while angle HGJ is located below the intersection of line EL and line FH, outside the parallel lines.

5. Since angle KJL and angle HGJ are located outside the two parallel lines on the same side of the transversal, according to the definition of corresponding angles, they are corresponding angles.

6. Check other options:

6.1 Angle KJL and angle HGE are located on different sides of transversal EL, which does not meet the definition of corresponding angles, so they are not corresponding angles.

6.2 Angle KJL and angle FGE are also located on different sides of transversal EL, so they are not corresponding angles.

6.3 Angle KJL and angle IJL are located inside the same parallel line, which does not meet the definition of corresponding angles, so they are not corresponding angles.

7. After the above reasoning, the final answer is that angle KJL and angle HGJ are corresponding angles.

Definition of Parallel Lines

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

Definition of an Angle

In the diagram of this problem, <note>angle KJL is a geometric figure formed by two rays JK and JL</note>, these two rays have a <note>common endpoint J</note>. This <note>common endpoint J is called the vertex of angle KJL</note>, and the rays <note>JK and JL are called the sides of angle KJL</note>. <note>Angle HGJ is a geometric figure formed by two rays HG and GJ</note>, these two rays have a <note>common endpoint G</note>. This <note>common endpoint G is called the vertex of angle HGJ</note>, and the rays <note>HG and GJ are called the sides of angle HGJ</note>. <note>Angle FGE is a geometric figure formed by two rays FG and GE</note>, these two rays have a <note>common endpoint G</note>. This <note>common endpoint G is called the vertex of angle FGE</note>, and the rays <note>FG and GE are called the sides of angle FGE</note>. <note>Angle IJL is a geometric figure formed by two rays IJ and JL</note>, these two rays have a <note>common

Definition of Corresponding Angles

In the figure of this problem, <note>two parallel lines FH and IK</note> are intersected by a <note>line EL</note>, where <note>angle KJL and angle HGJ are on the same side of the intersecting line EL</note>, on the same side of the intersected lines FH and IK, therefore <note>angle KJL and angle HGJ are corresponding angles</note>. Corresponding angles are equal, that is, <note>angle KJL is equal to angle HGJ</note>.


Question:
Look at this diagram: If

| $\overleftrightarrow{CE}$ |

and

| $\overleftrightarrow{FH}$ |
are parallel lines and m $\angle $ CDG = 43°, what is m $\angle $ FGI? $\Box$ °

Answer: 43°

1. Given that line CE and line FH are parallel lines, and line BI is the transversal that crosses these two parallel lines.

2. According to the definition of corresponding angles, ∠CDG and ∠FGI are corresponding angles. Based on Parallel Postulate 2, ∠CDG and ∠FGI are equal.

3. It is given that the measure of ∠CDG is 43°.

4. According to the above theorem, ∠FGI = ∠CDG.

5. Therefore, ∠FGI is also equal to 43°.

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

Definition of Parallel Lines

In the figure of this problem, <note>line CE and line FH</note> are located in the same plane, and they do not intersect. Therefore, according to the <note>definition of parallel lines</note>, <note>line CE and line FH are parallel lines</note>.

Definition of Corresponding Angles

In the figure of this problem, <note>line CE and line FH are parallel lines</note>, <note>line BI is the transversal</note>, where <note>angle CDG and angle FGI are on the same side of transversal BI</note>, on the same side of the <note>two lines CE and FH intersected by the transversal</note>, therefore <note>angle CDG and angle FGI are corresponding angles</note>. Corresponding angles are equal, that is, <note>angle CDG is equal to angle FGI</note>.

Parallel Postulate 2 of Parallel Lines

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

In this problem diagram, <note>two parallel lines CE and FH</note> are intersected by <note>the third line BI</note>, forming the following geometric relationships: 1. Corresponding angles: <note>angle CDG and angle FGI are equal</note>. These relationships indicate that when two parallel lines are intersected by a third line, the corresponding angles are equal.


Question:
Find m $\angle $ F.

m $\angle $ F= $\Box$ °

Answer: m \$\angle \$ F=60°

1. Given that triangle EFG is a right triangle, and side GF = 8√10, side EF = 16√10.

2. According to the Pythagorean theorem, which states that in a right triangle, the sum of the squares of the two legs is equal to the square of the hypotenuse, we can set up the equation: (EG)^2 + (GF)^2 = (EF)^2.

3. Substituting the given values, we get (EG)^2 + (8√10)^2 = (16√10)^2.

4. Calculating, we get (EG)^2 + 640 = 2560.

5. Solving for (EG)^2, we get (EG)^2 = 1920.

6. From (EG)^2 = 1920, we find EG = √1920 = 8√30.

7. Calculating the ratio of sides GF and EF, cos(∠F) = GF/EF = (8√10)/(16√10) = 1/2.

8. Since the cosine function in a right triangle, cos(∠F) = 1/2 means that ∠F is 60°, because cos(60°) = 1/2.

9. Confirm that triangle EFG is a special 30°-60°-90° right triangle, inserting this property: in a 30°-60°-90° triangle, the hypotenuse is twice the length of the shorter leg.

10. Verify that the side length corresponding to ∠F matches the property of the special right triangle, further confirming that m∠F = 60°.

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

Definition of Right Triangle

In the figure of this problem, in triangle EFG, <note>angle ∠EGF is a right angle (90 degrees)</note>, therefore triangle EFG is a right triangle. <note>Side EG and side GF are the legs</note>, <note>side EF is the hypotenuse</note>.

Cosine Function

<note>In the right triangle EGF</note>, <note>side GF is the adjacent side of angle F</note>, <note>side EF is the hypotenuse</note>. Therefore, according to the definition of the cosine function, <note>the cosine of angle F is equal to the ratio of the adjacent side GF to the hypotenuse EF</note>, that is, <note>cos(F) = GF/ EF</note>.

Properties of 30°-60°-90° Triangle

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

In the given problem diagram, confirm that triangle EFG is a <note>30°-60°-90° triangle</note>, <note>angle EGF is 30 degrees</note>, <note>angle EFG is 60 degrees</note>, <note>angle EGF is 90 degrees</note>. <note>Side EF is the hypotenuse</note>, <note>side EG is opposite the 60-degree angle</note>, <note>side GF is opposite the 30-degree angle</note>. According to the properties of a 30°-60°-90° triangle, <note>the shorter leg GF is equal to half the hypotenuse EF</note>, <note>the hypotenuse EF is equal to √3 times the longer leg EG</note>. That is: <note>EF = 2 * GF</note>, <note>EF = EG * √3</note>.


Question:
Find m $\angle $ S.

m $\angle $ S= $\Box$ °

Answer: m \$\angle \$ S=45°

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

2. In Rt△STR, ST is the hypotenuse with a length of 8√2, and RS is a leg with a length of 8.

3. We need to find ∠S using the definition of the cosine function. The cosine function states that in a right triangle, the cosine of an acute angle is equal to the ratio of the length of the adjacent side to the length of the hypotenuse.

4. For this problem, the cosine function can be expressed as: cos∠S = length of the adjacent side RS / length of the hypotenuse ST.

5. Substitute the known values: cos∠S = 8 / (8√2).

6. Further simplify: cos∠S = 1/√2.

7. Rationalize the denominator to simplify 1/√2 to √2/2.

8. Since the trigonometric identity cos45° = √2/2, we find that ∠S = 45°.

9. Based on the above reasoning, the final answer is 45°.

Definition of Right Triangle

In the diagram of this problem, in triangle STR, <note>angle SRT is a right angle (90 degrees)</note>, thus triangle STR is a right triangle. <note>Side RS and side RT are the legs</note>, <note>side ST is the hypotenuse</note>.

Cosine Function

In the figure of this problem, in the right triangle STR, <note>side RS is the adjacent side of angle ∠S</note>, <note>side ST is the hypotenuse</note>. Therefore, according to the definition of the cosine function, <note>the cosine value of angle ∠S is equal to the ratio of the adjacent side RS to the hypotenuse ST, that is, cos(∠S) = RS / ST</note>.


Question:
In circle V, $\overset{\frown}{WYX}$ is highlighted. $\overset{\frown}{WX}$ measures 45°.

What fraction of the circle is highlighted?
Simplify your answer.

 $\Box$
Which expression represents the length of $\overset{\frown}{WYX}$ in centimeters?

- 7/8(2𝜋)
- 7/8(𝜋)
- 13/16(𝜋)
- 13/16(2𝜋)

Answer:
7/8

 7/8*(2𝜋)

1. Given that in circle V, the measure of arc WX is 45°, which is the part outside of arc WYX. The total degree of the circumference is 360°.

2. According to the relationship between arc and central angle, the degree of central angle ∠WVX corresponding to arc WX is equal to the degree of the central angle, which is 45°.

3. The degree of arc WYX is the total degree of the circle minus the degree of arc WX, which is 360° - 45° = 315°.

4. Calculate the proportion of arc WYX in the circle. The proportion between the degree of arc WYX and the degree of the circumference is 315°/360°.

5. Simplify 315/360 to get 315 ÷ 45 / 360 ÷ 45 = 7/8.

6. Therefore, arc WYX occupies 7/8 of the entire circle.

7. The circumference of the circle C = 2πr, where r is the radius, and the radius r = 1 cm in the problem.

8. Substitute r = 1 to get C = 2π × 1 = 2π.

9. The length of arc WYX is 7/8 of the circumference C, so the length of arc WYX is 7/8 × 2π.

10. Through the above reasoning, the final answer is that the length of arc WYX is 7/8 × 2π, i.e., choose the option 7/8 × (2π).

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 <note>circle V</note>, <note>point V is the center</note>, <note>with a radius of 1 cm</note>. All points in the figure that are <note>1 cm away from point V</note> are on <note>circle V</note>.

Definition of Arc

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

In the diagram of this problem, there are two points <note>W and X</note> on <note>circle V</note>, and <note>arc WX</note> is the curve connecting these two points, corresponding to the central angle <note>∠WVX</note>, measured at <note>45°</note>. <note>Arc WYX</note> is the remaining part of the circumference after subtracting arc WX.

Circumference Formula of Circle

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

<note>In circle V, point V is the center of the circle</note>, <note>line segments VW and VX are the radius r</note>. According to the circumference formula of the circle, <note>the circumference C of the circle is equal to 2π multiplied by the radius r</note>, i.e., C = 2πr. Since r = 1 cm, <note>the circumference C of the circle is 2π × 1 = 2π cm</note>.


Question:
Find m $\angle $ W.

m $\angle $ W= $\Box$ °

Answer: m \$\angle \$ W=60°

1. According to the definition of the sine function, in a right triangle, sin(∠XWV) is equal to the ratio of the opposite side VX to the hypotenuse VW, i.e., sin(∠XWV) = VX/VW.

2. The problem provides VX = √30 and VW = 2√10. Substituting these values, we get sin(∠XWV) = √30/(2√10).

3. By rationalizing the denominator, we factor out √10 to the numerator to get sin(∠XWV) = (√30 * √10)/(2 * 10) = √300/20.

4. Calculating √300 gives 10√3, thus sin(∠XWV) = 10√3/20 = √3/2.

5. According to the trigonometric values table, sin(60°) = √3/2, which indicates that ∠XWV is 60°.

6. The problem requires finding ∠W, therefore ∠W is also ∠XWV, which is 60°.

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

Definition of Sine Function

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

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


Question:
What is the volume of this figure?

 $\Box$ cubic centimeters

Answer: 284cubic centimeters

1. First, observe that the figure is composed of two rectangular prisms, so it can be divided into two rectangular prisms to solve for the volume.

2. Identify one of the rectangular prisms (the back rectangular prism), which has a length of 11 cm, a width of 4 cm, and a height of 6 cm.

3. According to the rectangular prism volume formula 'V = length × width × height', calculate the volume of the back rectangular prism: V = 11 × 4 × 6 = 264 cubic cm.

4. Next, identify the other rectangular prism (the front rectangular prism), which has a length of 2 cm, a width of 2 cm, and a height of 5 cm.

5. According to the rectangular prism volume formula 'V = length × width × height', calculate the volume of the front rectangular prism: V = 2 × 2 × 5 = 20 cubic cm.

6. Now, add the volumes of the two rectangular prisms to find the total volume of the entire figure: 264 + 20 = 284 cubic cm.

7. Through the above reasoning, the final answer is 284 cubic cm.

Definition of Rectangular Prism

A rectangular prism is defined as a three-dimensional geometric figure formed by six rectangular faces, where three mutually adjacent faces respectively define the length, width, and height.

<note>The length of the rectangular prism at the back is 11 cm</note>, <note>the width is 4 cm</note>, <note>and the height is 6 cm</note>; <note>The length of the rectangular prism at the front is 2 cm</note>, <note>the width is 2 cm</note>, <note>and the height is 5 cm</note>.

Volume Formula of Rectangular Prism

The volume of a rectangular prism is equal to the product of its length, width, and height.

In this problem, the calculation of <note>the volume of the rectangular prism</note> is: 264 cubic centimeters, the specific formula applied is <note>V = 11 × 4 × 6</note>; the calculation of <note>the volume of the rectangular prism</note> in front is: 20 cubic centimeters, the specific formula applied is <note>V = 2 × 2 × 5</note>. The final figure volume calculation is: 284 cubic centimeters, the specific formula applied is <note>264 + 20</note>.


Question:
In circle S, $\overset{\frown}{TVU}$ is highlighted. $\overset{\frown}{TU}$ measures 90°.

What fraction of the circle is highlighted?
Simplify your answer.

 $\Box$
Which expression represents the length of $\overset{\frown}{TVU}$ in inches?

- 5/6(6𝜋)
- 3/4(9𝜋)
- 5/6(9𝜋)
- 3/4(6𝜋)

Answer:
3/4

- 3/4(6𝜋)

1. Given ∮jTU = 90°, and φTSU is the central angle of the minor arc. Therefore, ∮jTU is intercepted by the central angle φTSU, and their degrees are equal, so mφTSU = 90°.

2. The total degrees of the circle is 360°. Since mφTSU = 90°, and the arc ∮jTU it intercepts is not highlighted, ∮jTU accounts for 90/360 of the total degrees of the circle.

3. Calculate the proportion of ∮jTU in the circle: 90/360 = 1/4. This indicates that the non-highlighted part of the circle accounts for 1/4 of the circle.

4. Therefore, the highlighted part ∮jTVU accounts for (1 - 1/4) = 3/4 of the entire circle.

5. The length of the highlighted arc ∮jTVU is represented as 3/4 of the circumference of the circle. Therefore, the length of ∮jTVU is 3/4 · C, where C is the circumference of the circle.

6. According to the formula for the circumference of a circle C = 2πr and the diameter length is 6 inches, the diameter d = 6 inches, so C = π · 6.

7. Substitute into the above formula: the length of ∮jTVU is 3/4 · π · 6 = 3/4 · 6π.

8. Choose from the options: 3/4(6π) (the most matching option is '3/4(6π)').

9. Through the above reasoning, the final answer is that the highlighted part of the circle ∮jTVU accounts for 3/4 of the entire circle, and its length is 3/4(6π).

Circumference Formula of Circle

In the figure of this problem, <note>in circle S, point S is the center</note>, <note>segment ST is the radius r</note>. According to the <note>Circumference Formula of Circle</note>, the circumference C of the circle is equal to 2π times <note>radius r</note>, that is, C=2πr. In this problem, <note>diameter d=6 inches</note>, therefore <note>the circumference of the circle C=π×6=6π</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>∠TSU is a central angle</note>, its vertex is at the center of the circle <note>S</note>, and its two sides pass through points on the circumference <note>T and U</note> respectively, that is, <note>the arc intercepted by ∠TSU is arc TU</note>.

Definition of Arc

<note>There are two points T and U on circle S</note>, <note>arc TU is a segment of the curve connecting these two points</note>. According to the definition of arc, <note>arc TU is a segment of the curve between the two points T and U on the circle</note>. A minor arc is an arc with a length less than 180°, while a major arc exceeds 180°.

Arc Length Formula of a Circle

The arc length refers to the length of a segment of the circumference of a circle. It can be calculated using the formula 𝓁 = C × (m/360), where m is the measure of the central angle in degrees.

<note>Arc ∮jTVU</note> is highlighted, occupying <note>3/4</note> of the entire circle. Therefore, the length of <note>Arc ∮jTVU</note> is <note>3/4 multiplied by the circumference π×6</note>, which is <note>Arc ∮jTVU's length is 3/4×6π=3/4(6π)</note>.


Question:
Find m $\angle $ Y.

m $\angle $ Y= $\Box$ °

Answer: m \$\angle \$ Y=45°

1. According to the given figure, triangle XYZ is a right triangle, and ∠XZY is 90°.

2. According to the Pythagorean theorem, for a right triangle, we have XYZ in the form of XY^2 = XZ^2 + YZ^2. Given in the problem XZ = √(10) and XY = 2√(5), then (2√5)^2 = (√10)^2 + YZ^2.

3. Calculate the side length: 4×5 = 10 + YZ^2, obtaining YZ^2 = 10. Thus, YZ = √(10).

4. Since triangle XYZ is a right triangle, and given the adjacent sides and hypotenuse, the angle should be determined according to the cosine value in trigonometric functions. The cosine function is: cos(∠Y) = ratio of the adjacent side to the hypotenuse, i.e., cos(∠Y) = YZ / XY.

5. Substitute to get the calculation: cos(∠Y) = √10 / 2√5.

6. Simplify the above expression: since √10 / 2√5 = (√10/√5) × (1/2), further simplified to √2 / 2. That is, cos(∠Y) = √2 / 2.

7. According to the known trigonometric function value, cos(45°) = √2 / 2. Thus, ∠Y = 45°.

8. Through the above reasoning, the final answer is 45°.

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 XYZ, angle ∠XZY is a right angle (90 degrees)</note>, so triangle XYZ is a right triangle. <note>Side XZ and side YZ are the legs</note>, <note>side XY is the hypotenuse</note>.

Cosine Function

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

In the figure of this problem, in the right triangle XYZ, side <note>YZ is the adjacent side of angle ∠XYZ</note>, and side <note>XY is the hypotenuse</note>. Therefore, according to the definition of the cosine function, <note>the cosine value of angle ∠XYZ is equal to the ratio of the adjacent side YZ to the hypotenuse XY</note>, that is, <note>cos(∠XYZ) = YZ / XY</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</note>, <note>in the right triangle XYZ</note>, <note>angle XZY is a right angle (90 degrees)</note>, <note>sides XZ and YZ are the legs</note>, <note>side XY is the hypotenuse</note>, so according to the Pythagorean Theorem, <note>XY² = XZ² + YZ²</note>.


Question:
In circle J, $\overset{\frown}{KML}$ is highlighted. $\overset{\frown}{KL}$ measures 135°.

What fraction of the circle is highlighted?
Simplify your answer.

 $\Box$
Which expression represents the length of $\overset{\frown}{KML}$ in inches?

- 3/4(225𝜋)
- 5/8(225𝜋)
- 3/4(30𝜋)
- 5/8(30𝜋)

Answer:
5/8

- 5/8(30𝜋)

1. Given that the degree measure of arc KL in circle J is 135°, this means it is intercepted by the central angle ∠KJL. According to the properties of central angles, the central angle is equal to the degree measure of the arc it intercepts.

2. The total circumference of the circle is 360°, so arc KL occupies a proportion of 135°/360° of the entire circle.

3. Calculating the ratio of 135°/360°, it can be simplified to 3/8.

4. Since arc KL occupies 3/8 of the circle, the remaining part of the circumference, which is arc KML, should be 1 - 3/8 = 5/8.

5. Therefore, arc KML occupies 5/8 of the entire circle, which is the highlighted portion of the circle.

6. Next, calculate the actual length of arc KML. The radius r of the circle is 15 inches. The circumference C of the circle is 2πr = 2π(15) = 30π inches.

7. Therefore, the length of arc KML is 5/8 multiplied by the circumference of the circle, which is 5/8 × 30π = 5/8(30π) inches.

8. Summarizing the above steps, arc KML occupies 5/8 of the circle, and its length is 5/8(30π) inches.

9. Through the above reasoning, the final answer is 5/8.

Definition of Central Angle

<note>∠KJL is a central angle</note>, with its vertex at the <note>center J</note>, and its sides are the radii <note>JK and JL</note>.

Definition of Arc

In the diagram of this problem, there are three points K, L, and M on circle J, <note>arc KL is a segment of the curve connecting point K and point L</note>, <note>arc KML is a segment of the curve connecting point K, point M, and point L</note>. According to the definition of an arc, <note>arc KL is a segment of the curve between two points K and L on the circle</note>, <note>arc KML is a segment of the curve between two points K and L on the circle passing through point M</note>.

Circumference Formula of Circle

In the diagram of this problem, <note>in circle J, point J is the center</note>, <note>line segments JK and JL are the radius r</note>. According to the circumference formula of a circle, the circumference C is equal to 2π times <note>the radius r</note>, that is, <note>C = 2πr</note>. In this problem, <note>r = 15 inches</note>, therefore <note>C = 2π(15) = 30π inches</note>.

Arc Length Formula of a Circle

Arc KML occupies 5/8 of the circle, so the length of <note>arc KML</note> is 5/8 × 30π = 5/8(30π) inches.

Property of Central Angle

The degree measure of an arc is equal to the degree measure of the central angle that subtends the arc.

The arc <note>KL</note> corresponds to the <note>central angle KJL</note>, and the degree measure of the arc <note>KL</note> is equal to the degree measure of the angle <note>KJL</note>.


Question:
In circle J, $\overset{\frown}{KL}$ is highlighted.  $\angle $ KJL measures 45°.

What fraction of the circle is highlighted?
Simplify your answer.

 $\Box$
Which expression represents the length of $\overset{\frown}{KL}$ in inches?

- 1/4(8𝜋)
- 1/8(16𝜋)
- 1/4(16𝜋)
- 1/8(8𝜋)

Answer:
1/8

- 1/8(8𝜋)

1. Given in circle J, arc KL is intercepted by central angle ∠KJL. Therefore, the degree measure of central angle ∠KJL of arc KL is 45°.

2. The total angle measure around a circle is 360°. Since the degree measure of the central angle of arc KL is 45°, KL represents the fraction 45°/360° of the entire circle.

3. Calculating the fraction: 45°/360° = 1/8. Therefore, in circle J, the highlighted arc KL represents 1/8 of the entire circle.

4. Hence, the length of arc KL is equal to 1/8 of the circumference of circle J.

5. The formula for the circumference of a circle is C = 2πr, where r is the radius of the circle. In this problem, the radius r of circle J is 4 inches.

6. Calculating the circumference of circle J: C = 2π × 4 = 8π.

7. Therefore, the length of arc KL is 1/8 of the circumference, which is (1/8) × 8π = π.

8. Through the above reasoning, the length of arc KL is 8π multiplied by 1/8, resulting in π, which is the answer choice of 1/8(8π).

Definition of Circle

In circle J, point J is the center, the radius is 4 inches. All points in the figure that are at a distance of 4 inches from point J are on circle J.

Definition of Arc

<note>In the diagram of this problem, there are two points K and L on circle J, and arc KL is a segment of the curve connecting these two points</note>. According to the definition of an arc, arc KL is a segment of the curve between points K and L on the circle.

Definition of Central Angle

In the diagram of this problem, in circle J, point K and point L are two points on the circle, and the center of the circle is point J. The angle ∠KJL formed by the lines JK and JL is called the central angle, and its angle is 45°.

Arc Length Formula of a Circle

The length of <note>arc KL</note> is equal to <note>1/8</note> of the circumference of <note>circle J</note>, because the degree measure of <note>∠KJL</note> is <note>45°</note>, that is, <note>45°/360°=1/8</note>.

Circumference Formula of Circle

In the figure of this problem, <note>in circle J, point J is the center of the circle</note>, <note>line segments JK and JL are the radii r</note>. According to the circumference formula of the circle, <note>the circumference C of the circle is equal to 2π multiplied by the radius r, i.e., C=2πr</note>, where <note>r=4 inches</note>, therefore <note>C=2π×4=8π</note>.


Question:
What is the volume of this figure?

 $\Box$ cubic meters

Answer: 174cubic meters

1. Decompose the given solid figure into two cuboids: the top cuboid and the bottom cuboid.

2. Calculate the volume of the top cuboid. The length of the top cuboid is 7 meters, the width is 2 meters, and the height is 3 meters.

3. According to the cuboid volume formula V = ℓ × w × h, where ℓ is the length, w is the width, and h is the height, the volume can be calculated.

4. For the top cuboid, substitute the dimensions into the formula to get V = 7 × 2 × 3 = 42 cubic meters.

5. Then, calculate the volume of the bottom cuboid. The length of the bottom cuboid is 11 meters, the width is 3 meters, and the height is 4 meters.

6. Using the same cuboid volume formula, substitute the dimensions of the bottom cuboid to get V = 11 × 3 × 4 = 132 cubic meters.

7. Add the volumes of the two cuboids to obtain the total volume of the solid figure, i.e., 42 + 132 = 174 cubic meters.

8. Through the above reasoning, the final answer is 174 cubic meters.

Volume Formula of Rectangular Prism

<note>Side 7m represents length, side 2m represents width, side 3m represents height</note>, so the volume of the rectangular prism is equal to the product of length, width, and height, i.e., Volume = 7 * 2 * 3 = 42 cubic meters. <note>Side 11m represents length, side 3m represents width, side 4m represents height</note>, so the volume of the rectangular prism is equal to the product of length, width, and height, i.e., Volume = 11 * 3 * 4 = 132 cubic meters.


Question:
What is the volume of this figure?

 $\Box$ cubic yards

Answer: 192cubic yards

1. First, decompose the three-dimensional figure into two rectangular prism parts, labeled as the upper rectangular prism and the lower rectangular prism.

2. For the upper rectangular prism, according to the given dimensions, it is known that the length of the upper rectangular prism = 12 yd, width = 2 yd, and height = 4 yd.

3. Using the rectangular prism volume formula V = length × width × height, calculate the volume of the upper rectangular prism V1 = 12 × 2 × 4 = 96 cubic yards.

4. Next, consider the lower rectangular prism, which has a length = 12 yd, width = 2 yd, and height = 4 yd (note that the lower rectangular prism has the same dimensions as the upper rectangular prism, which is a characteristic of the given figure).

5. Similarly, using the rectangular prism volume formula V = length × width × height, calculate the volume of the lower rectangular prism V2 = 12 × 2 × 4 = 96 cubic yards.

6. Finally, add the volumes of the upper and lower rectangular prisms to obtain the total volume of the entire figure: V_total = V1 + V2 = 96 + 96 = 192 cubic yards.

7. Through the above reasoning, the final answer is 192 cubic yards.

Definition of Rectangular Prism

The upper rectangular prism and the lower rectangular prism are both rectangular prisms, each having the properties of <note>length = 12 yards, width = 2 yards, and height = 4 yards</note>. <note>The upper rectangular prism is represented by 12x2x4, and the lower rectangular prism is also represented by 12x2x4</note>.

Volume Formula of Rectangular Prism

In the figure of this problem, use the volume formula of a rectangular prism to calculate the volumes of the two rectangular prisms. <note>The volume of the upper rectangular prism V1 = 12 yd × 2 yd × 4 yd = 96 cubic yards</note>. Similarly, <note>the volume of the lower rectangular prism V2 = 12 yd × 2 yd × 4 yd = 96 cubic yards</note>.


Question:
Find m $\angle $ Q.

m $\angle $ Q= $\Box$ °

Answer: m \$\angle \$ Q=45°

1. In the right triangle SQR, ∠QRS is a right angle, so trigonometric functions in right triangles can be used to find the angle.

2. According to the cosine function, cos(∠Q) = QR/QS, where QR and QS are the adjacent side and hypotenuse of ∠Q in the right triangle SQR, respectively.

3. Substitute the given side values, QR = 10√33, QS = 10√66.

4. Therefore, cos(∠Q) = QR/QS = 10√33/10√66.

5. Further simplify the ratio: cos(∠Q) = √33/√66.

6. Multiply both the numerator and denominator by √2 to rationalize: cos(∠Q) = (√33/√66) = √33/(√33 * √2) = 1/√2 = √2/2.

7. Thus, cos(∠Q) = √2/2.

8. It is known that in a right triangle, the cosine value of ∠45° is √2/2, and among common angles, only this angle meets the condition, therefore, m∠Q = 45°.

9. Through the above reasoning, the final answer is 45°.

Definition of Right Triangle

In the figure of this problem, in triangle SQR, <note>angle QRS is a right angle (90 degrees)</note>, therefore <note>triangle SQR is a right triangle</note>. <note>Side QR and side SR are the legs</note>, <note>side QS is the hypotenuse</note>.

Cosine Function

In <note>right triangle SQR</note>, <note>side QR is the adjacent side to angle ∠Q</note>, <note>side QS is the hypotenuse</note>. Therefore, according to the definition of the cosine function, <note>the cosine value of angle ∠Q is equal to the ratio of the adjacent side QR to the hypotenuse QS</note>, that is, <note>cos(∠Q) = QR / QS</note>.


Question:
What is the volume of this figure?

 $\Box$ cubic centimeters

Answer: 162cubic centimeters

1. First, we observe that the given figure can be decomposed into two rectangular prisms: a small rectangular prism at the top and a large rectangular prism at the bottom.

2. For the small rectangular prism at the top, we already know its length (l) is 3 cm, width (w) is 2 cm, and height (h) is 2 cm.

3. According to the rectangular prism volume formula V = l × w × h, the volume of the small rectangular prism at the top is calculated as V = 3 × 2 × 2 = 12 cubic cm.

4. Next, we consider the rectangular prism at the bottom, whose known length (l) is 10 cm, width (w) is 3 cm, and height (h) is 5 cm.

5. Similarly, based on the rectangular prism volume formula V = l × w × h, the volume of the rectangular prism at the bottom is calculated as V = 10 × 3 × 5 = 150 cubic cm.

6. By adding the volumes of the two rectangular prisms, we obtain the total volume of the figure, total volume = 12 + 150 = 162 cubic cm.

7. Through the above reasoning, the final answer is that the volume of the figure is 162 cubic cm.

Definition of Rectangular Prism

<note>The dimensions of the small rectangular prism at the top are length l = 3 cm, width w = 2 cm, height h = 2 cm</note>; <note>the dimensions of the large rectangular prism at the bottom are length l = 10 cm, width w = 3 cm, height h = 5 cm</note>. In the problem, <note>each rectangular prism is composed of six rectangular faces</note>.

Volume Formula of Rectangular Prism

In this problem diagram, we calculate the volume of the <note>small rectangular prism at the top as V = 3 × 2 × 2 = 12 cubic centimeters</note> using the volume formula of a rectangular prism, and the volume of the <note>large rectangular prism at the bottom as V = 10 × 3 × 5 = 150 cubic centimeters</note>. Finally, we calculate the total volume by adding the volumes of the two rectangular prisms, <note>total volume = 12 + 150 = 162 cubic centimeters</note>.


Question:
What is the volume of this figure?

 $\Box$ cubic centimeters

Answer: 88cubic centimeters

1. Observe the solid figure and find that it can be split into two rectangular prisms. The dimensions of the larger rectangular prism on the left are: length 2 cm, width 8 cm, height 5 cm.

2. Calculate the volume of the larger rectangular prism on the left. According to the formula for the volume of a rectangular prism V = l · w · h, where l represents length, w represents width, and h represents height. Substitute the specific values: V_left = 2 cm · 8 cm · 5 cm.

3. Perform the calculation to find the volume of the larger rectangular prism on the left: V_left = 80 cubic cm.

4. The dimensions of the smaller rectangular prism on the right are: length 1 cm, width 4 cm, height 2 cm.

5. Calculate the volume of the smaller rectangular prism on the right. According to the formula for the volume of a rectangular prism V = l · w · h, substitute the specific values: V_right = 1 cm · 4 cm · 2 cm.

6. Perform the calculation to find the volume of the smaller rectangular prism on the right: V_right = 8 cubic cm.

7. Solve for the total volume of the entire solid figure. Add the volumes of the two rectangular prisms: V_total = V_left + V_right.

8. Perform the calculation to find the total volume: V_total = 80 cubic cm + 8 cubic cm = 88 cubic cm.

9. Through the above reasoning, the final answer is 88 cubic cm.

Definition of Rectangular Prism

The large cube on the left and the small cube on the right both belong to rectangular prisms. The dimensions of the large rectangular prism are <note>length 2 cm, width 8 cm, height 5 cm</note>, and the dimensions of the small rectangular prism are <note>length 1 cm, width 4 cm, height 2 cm</note>.

Volume Formula of Rectangular Prism

<note>V_left represents the volume of the left large rectangular prism</note>. According to the formula, <note>V_left = 2 cm · 8 cm · 5 cm</note>. The result is 80 cubic centimeters. Similarly, <note>V_right represents the volume of the right small rectangular prism</note>, according to the formula, <note>V_right = 1 cm · 4 cm · 2 cm</note>. The result is 8 cubic centimeters.


Question: What is the volume? $\Box$  cubic yards

Answer: 648 cubic yards

1. First, identify the length, width, and height of the rectangular prism in the figure. The length of the rectangular prism is 9 yd, the width is also 9 yd, and the height is 8 yd.

2. According to the definition of the volume of a rectangular prism, the volume is equal to the product of the length, width, and height. Therefore, use the formula: Volume = Length × Width × Height.

3. Substitute the known values into the formula: Volume = 9 × 9 × 8.

4. Calculate the product: 9 × 9 = 81.

5. Then multiply the result 81 by 8 to get: 81 × 8 = 648.

6. Therefore, the volume of the rectangular prism is 648 cubic yards.

7. Through the above reasoning, the final answer is 648 cubic yards.

Definition of Rectangular Prism

Original text: <note>The length of the rectangular prism is 9 yards, the width is 9 yards, and the height is 8 yards</note>. According to the definition of a rectangular prism, these three dimensions represent the sizes in three directions in the figure, <note>mutually perpendicular</note>.

Volume Formula of Rectangular Prism

<note>The length of the rectangular prism is 9 yards, the width is 9 yards</note>, <note>and the height is 8 yards</note>. Therefore, using the volume formula, we can calculate: Volume = <note>9 yards × 9 yards × 8 yards</note> = <note>648 cubic yards</note>.


Question:
What is the volume of this figure?

 $\Box$ cubic inches

Answer: 174cubic inches

1. First, analyze the figure and decompose it into two rectangular prisms: the upper rectangular prism and the lower rectangular prism.

2. Calculate the volume of the upper rectangular prism. According to the volume formula of a rectangular prism V = l × w × h, we need to know its length, width, and height.

3. From the figure in the problem, it is known that the length of the upper rectangular prism l = 9 inches, width w = 3 inches, height h = 6 inches.

4. Substitute these values into the volume formula: V = 9 × 3 × 6 = 162 cubic inches. Therefore, the volume of the upper rectangular prism is 162 cubic inches.

5. Next, calculate the volume of the lower rectangular prism. The length, width, and height of the lower rectangular prism are l = 6 inches, w = 1 inch, h = 2 inches.

6. Substitute these values into the volume formula: V = 6 × 1 × 2 = 12 cubic inches. Therefore, the volume of the lower rectangular prism is 12 cubic inches.

7. Finally, calculate the total volume of the entire figure. According to the principle of addition, the total volume is equal to the sum of the volumes of each part:

8. Total volume = volume of the upper rectangular prism + volume of the lower rectangular prism = 162 + 12 = 174 cubic inches.

9. Through the above reasoning, the final answer is 174 cubic inches.

Definition of Rectangular Prism

The <note>rectangular prism</note> is defined by length <note>l = 9 inches</note>, width <note>w = 3 inches</note>, and height <note>h = 6 inches</note>. The following <note>rectangular prism</note> is defined by length <note>l = 6 inches</note>, width <note>w = 1 inch</note>, and height <note>h = 2 inches</note>.

Volume Formula of Rectangular Prism

<note>The volume of the rectangular prism is calculated as V = 9 × 3 × 6 = 162 cubic inches</note>.<note>The total volume is the sum of the volumes of two rectangular prisms, which is 162 + 12 = 174 cubic inches</note>.


Question:
What is the volume of this figure?

 $\Box$ cubic inches

Answer: 432cubic inches

1. Observe the given figure, it is a three-dimensional figure composed of two rectangular prisms.

2. The first step is to calculate the volume of the upper rectangular prism. According to the rectangular prism volume formula Volume = length × width × height, the volume of the upper rectangular prism can be obtained.

3. The length of the upper rectangular prism is 15 inches, the width is 4 inches, and the height is 5 inches. Therefore, the volume V = 15 × 4 × 5 cubic inches.

4. The calculated result is V = 300 cubic inches. The volume of the upper rectangular prism is 300 cubic inches.

5. Next, calculate the volume of the lower rectangular prism. Use the same rectangular prism volume formula Volume = length × width × height.

6. The length of the lower rectangular prism is 11 inches, the width is 3 inches, and the height is 4 inches. Therefore, the volume V = 11 × 3 × 4 cubic inches.

7. The calculated result is V = 132 cubic inches. The volume of the lower rectangular prism is 132 cubic inches.

8. To obtain the total volume of the entire combined figure, add the volumes of the two rectangular prisms.

9. Calculate the sum: 300 + 132 = 432 cubic inches.

10. After the above reasoning, the final answer is 432 cubic inches.

Definition of Rectangular Prism

In the figure of this problem, there are two rectangular prisms. <note>The first rectangular prism (top) has a length of 15 inches, a width of 4 inches, and a height of 5 inches</note>. <note>The second rectangular prism (bottom) has a length of 11 inches, a width of 3 inches, and a height of 4 inches</note>.

Volume Formula of Rectangular Prism

<note>Side 15in represents length</note>, <note>Side 4in represents width</note>, <note>Side 5in represents height</note>, so the volume of the rectangular prism is equal to the product of length, width, and height, that is, Volume = 15 * 4 * 5 = 300 cubic inches. <note>Side 11in represents length</note>, <note>Side 3in represents width</note>, <note>Side 4in represents height</note>, so the volume of the rectangular prism is equal to the product of length, width, and height, that is, Volume = 11 * 3 * 4 = 132 cubic inches.


Question:
What is the volume of this figure?

 $\Box$ cubic yards

Answer: 66cubic yards

1. The problem requires finding the volume of the entire solid figure. From the figure, it can be discerned that this solid figure can be divided into two rectangular prisms (i.e., cuboids), which we will call the rear cuboid and the front cuboid.

2. First, calculate the volume of the rear cuboid: its length, width, and height are 5 yards, 3 yards, and 4 yards respectively. According to the cuboid volume formula V = ℓ·w·h, where V is the volume, ℓ is the length, w is the width, and h is the height, we can calculate the volume as V = 5 * 3 * 4 = 60 cubic yards.

3. Next, calculate the volume of the front cuboid: its length, width, and height are 3 yards, 1 yard, and 2 yards respectively. Again, using the cuboid volume formula V = ℓ·w·h, we get the volume as V = 3 * 1 * 2 = 6 cubic yards.

4. The total volume of the entire solid figure is the sum of the volumes of the rear cuboid and the front cuboid, so the total volume is 60 cubic yards + 6 cubic yards = 66 cubic yards.

5. Through the above reasoning, the final answer is 66 cubic yards.

Definition of Rectangular Prism

In the diagram of this problem, <note>the rear cube and the front cube are respectively two rectangular prisms</note>. The rear cube has a length of 5 yards, a width of 3 yards, and a height of 4 yards; the front cube has a length of 3 yards, a width of 1 yard, and a height of 2 yards. They <note>conform to the definition of rectangular prisms</note>, having <note>six rectangular faces</note>, and all their <note>spatial angles are 90 degrees</note>.

Volume Formula of Rectangular Prism

In the diagram of this problem, use the <note>Volume Formula of Rectangular Prism</note> to calculate the volume of two rectangular prisms. The volume of the rear rectangular prism is calculated as follows: <note>V = 5 * 3 * 4 = 60 cubic yards</note>; the volume of the front rectangular prism is calculated as follows: <note>V = 3 * 1 * 2 = 6 cubic yards</note>. Finally, add the volumes of the two rectangular prisms to get the total volume of <note>60 cubic yards + 6 cubic yards = 66 cubic yards</note>.


Question: What is the volume? $\Box$  cubic centimeters

Answer: 54 cubic centimeters

1. As shown in the figure, we have a rectangular prism with a base that is a square with a length of 3 cm and a width of 3 cm, and a height of 6 cm.

2. According to the formula for the volume of a rectangular prism: in a rectangular prism, its volume V is the base area multiplied by the height, i.e., V = base area × height.

3. First, calculate the base area. Since the base is a square, the base area = length × width = 3 cm × 3 cm = 9 square cm.

4. Substitute the base area into the volume formula to get the volume V = 9 square cm × 6 cm = 54 cubic cm.

5. Through the above reasoning, the final answer is 54 cubic cm.

Definition of Rectangular Prism

The geometric figure is a rectangular prism, with a square base of 3 cm in length and 3 cm in width, and a height of 6 cm. <note>符合长方体的定义</note>, with the base being a square, thus we can consider it as a rectangular prism with special properties, that is, part of a cube.

Volume Formula of Rectangular Prism

The original: <note>Base area A</note> = 9 square centimeters, <note>height h</note> = 6 centimeters, so <note>volume V</note> = 9 square centimeters × 6 centimeters = 54 cubic centimeters.


Question:
What is the volume of this figure?

 $\Box$ cubic yards

Answer: 258cubic yards

1. Observing the given geometric figure, it can be divided into two rectangular prisms, namely the upper part and the lower part.

2. Calculate the volume of the upper rectangular prism. The length of the upper part is 14 yd, the width is 3 yd, and the height is 3 yd.

3. According to the formula for the volume of a rectangular prism V=length×width×height, calculate the volume of the upper rectangular prism V_1=14×3×3.

4. The calculated volume of the upper rectangular prism is V_1=126 cubic yd.

5. Calculate the volume of the lower rectangular prism. The length of the lower part is 11 yd, the width is 3 yd, and the height is 4 yd.

6. According to the formula for the volume of a rectangular prism V=length×width×height, calculate the volume of the lower rectangular prism V_2=11×3×4.

7. The calculated volume of the lower rectangular prism is V_2=132 cubic yd.

8. Solve for the volume of the entire geometric figure by adding the volumes of the two rectangular prisms to obtain the total volume.

9. Calculate the total volume V=V_1+V_2=126+132.

10. The calculated total volume of the entire geometric figure is 258 cubic yd.

11. Through the above reasoning, the final answer is 258 cubic yd.

Definition of Rectangular Prism

<note>There are two rectangular prisms in the picture</note>, <note>one rectangular prism has dimensions of length 14 yards, width 3 yards, height 3 yards</note>; <note>the other rectangular prism has dimensions of length 11 yards, width 3 yards, height 4 yards</note>.

Volume Formula of Rectangular Prism

In the figure of this problem, use this formula to calculate <note>the volumes of two rectangular prisms</note>. For <note>the upper rectangular prism, V = 14 yd × 3 yd × 3 yd = 126 cubic yards</note>; for <note>the lower rectangular prism, V = 11 yd × 3 yd × 4 yd = 132 cubic yards</note>.


Question:
In circle P, $\overset{\frown}{QR}$ is highlighted. $\overset{\frown}{QR}$ measures 45°.

What fraction of the circle is highlighted?
Simplify your answer.

 $\Box$
Which expression represents the length of $\overset{\frown}{QR}$ in centimeters?

- 1/8(36𝜋)
- 1/6(36𝜋)
- 1/8(12𝜋)
- 1/6(12𝜋)

Answer:
1/8

 1/8(12𝜋)

1. Given circle P, arc QR is intercepted by the central angle ∠QPR, and it measures 45°.

2. According to the property of central angles, the central angle and its intercepted arc have the same angle measure, so the angle of arc QR is also 45°.

3. The angle corresponding to the circumference of the circle is 360°, so the portion of the highlighted area of arc QR relative to the entire area of the circle can be calculated using the ratio 45/360.

4. Simplifying the fraction 45/360, we get 1/8. This means arc QR occupies 1/8 of the circle.

5. Using the formula for the circumference of a circle C = 2πr, where r is the radius of the circle. In the problem, r is 6 cm, so C = 2π * 6 = 12π cm.

6. Since the highlighted portion of arc QR occupies 1/8 of the circle, its length is 1/8 multiplied by the circumference of the circle, which is (1/8) * 12π.

7. Calculating this, we get the length of arc QR as 1/8 * 12π = 12π/8 = 3π/2 cm.

8. Through the above reasoning, the final answer is that the length of arc QR is 3π/2 cm.

Property of Central Angle

In the diagram of this problem, the central angle corresponding to <note>arc QR</note> is <note>angle QPR</note>, and the degree measure of <note>arc QR</note> is equal to the degree measure of <note>angle QPR</note>.

Definition of Arc

In the figure of this problem, <note>there are two points Q and R on circle P</note>, <note>arc QR is a segment of the curve connecting these two points</note>. According to the definition of arc, <note>arc QR is a segment of the curve between two points Q and R on the circle</note>.

Definition of Central Angle

<note>In the figure of this problem, ∠QPR is the central angle of circle P</note>, <note>with vertex P at the center</note>, <note>sides PQ and PR intersecting circle P respectively</note>, and the measure of ∠QPR is 45°.

Arc Length Formula of a Circle

<note>The length of arc QR L</note> can be calculated as (45/360) * 12π = 1/8 * 12π = 3π/2 cm.

Circumference Formula of Circle

In the figure of this problem, <note>in circle P, point P is the center of the circle</note>, <note>line segments PQ and PR are the radius r</note>. According to the circumference formula of the circle, the circumference of the circle <note>C is equal to 2π multiplied by the radius r, i.e., C=2πr</note>, where <note>r=6 cm</note>, so <note>the circumference of circle P C = 2π * 6 = 12π cm</note>.


Question:
In circle F, $\overset{\frown}{GH}$ is highlighted. $\overset{\frown}{GH}$ measures 135°.

What fraction of the circle is highlighted?
Simplify your answer.

 $\Box$
Which expression represents the length of $\overset{\frown}{GH}$ in inches?

- 1/3(14𝜋)
- 3/8(14𝜋)
- 3/8(49𝜋)
- 1/3(49𝜋)

Answer:
3/8

- 3/8(14𝜋)

1. In circle F, the central angle ∠GFH of arc \\overset{\\frown}{GH} is 135°.

2. According to the property of central angles, the degree measure of the central angle ∠GFH is the same as the degree measure of the arc \\overset{\\frown}{GH} it intercepts, so the degree measure of \\overset{\\frown}{GH} is also 135°.

3. According to the angle properties of a circle, the total degree measure of a circle is 360°, so the highlighted arc \\overset{\\frown}{GH} occupies the proportion of 135°/360° of the entire circle.

5. This means the highlighted part \\overset{\\frown}{GH} occupies 3/8 of the entire circumference of the circle.

6. The formula for the circumference of a circle is C=2\\pi r, where r is the radius. In the problem, r = 7 inches.

7. Substituting the radius into the circumference formula, we get C=2\\pi\\times7=14\\pi.

8. Therefore, the length of the highlighted part \\overset{\\frown}{GH} is 3/8\\times 14\\pi.

9. Calculating, we get the length of \\overset{\\frown}{GH} as \\frac{3}{8}(14\\pi)=\\frac{42\\pi}{8}=\\frac{21\\pi}{4}.

10. Through the above reasoning, the final answer is 3/8, and the length of \\overset{\\frown}{GH} is 3/8(14\\pi) inches.

Definition of Central Angle

<note>In circle F, points G and H are two points on the circle</note>, <note>the center of the circle is point F</note>. <note>The angle ∠GFH formed by the lines FG and FH is called the central angle</note>.

Circumference Formula of Circle

<note>In circle F, point F is the center of the circle, and line segment FG is the radius r</note>. According to the circumference formula of the circle, <note>the circumference C of the circle is equal to 2π multiplied by the radius r</note>, that is, C = 2πr. In the problem, <note>r = 7 inches</note>, therefore <note>the circumference of the circle C = 2π × 7 = 14π</note>.

Property of Central Angle

The central angle corresponding to <note>arc GH</note> is <note>angle GFH</note>, <note>arc GH degrees = angle GFH degrees</note>.

Angle Properties of a Circle

The total measure of the angles in a complete circular arc is 360 degrees, and the sum of the central angles subtended by the entire circumference of a circle is 360 degrees.

<note>G, H所成的两条弧绕圆一周</note>, the sum of the arc measures is <note>360 degrees</note>, and the sum of their corresponding angles at the circumference is also <note>360 degrees</note>.


Question:
What is the volume of this figure?

 $\Box$ cubic meters

Answer: 174cubic meters

1. Decompose the given geometric figure into two rectangular prisms and calculate their volumes separately.

2. First, calculate the volume of the upper rectangular prism. According to the figure and labels given in the problem, the length of the upper rectangular prism is 7 m, the width is 2 m, and the height is 3 m.

3. The formula for calculating the volume of a rectangular prism is: Volume = Length × Width × Height.

4. Substitute the measurements of the upper rectangular prism: Volume = 7 m × 2 m × 3 m = 42 cubic meters.

5. Next, calculate the volume of the lower rectangular prism. According to the labels in the figure, the length of the lower rectangular prism is 11 m, the width is 3 m, and the height is 4 m.

6. Use the volume formula for a rectangular prism again: Volume = Length × Width × Height.

7. Substitute the measurements of the lower rectangular prism: Volume = 11 m × 3 m × 4 m = 132 cubic meters.

8. To calculate the total volume of the entire geometric figure, add the volumes of the upper and lower rectangular prisms: Total Volume = 42 cubic meters + 132 cubic meters = 174 cubic meters.

9. After the above reasoning, the final answer is 174 cubic meters.

Definition of Rectangular Prism

<note>The length of the upper rectangular prism is 7 meters</note>, <note>the width is 2 meters</note>, <note>the height is 3 meters</note>; <note>the length of the lower rectangular prism is 11 meters</note>, <note>the width is 3 meters</note>, <note>the height is 4 meters</note>.

Volume Formula of Rectangular Prism

In the figure of this problem, <note>the length of the upper rectangular prism is 7 meters, the width is 2 meters, and the height is 3 meters</note>, so its volume is calculated as: Volume = <note>7 meters × 2 meters × 3 meters = 42 cubic meters</note>; <note>the length of the lower rectangular prism is 11 meters, the width is 3 meters, and the height is 4 meters</note>, so its volume is calculated as: Volume = <note>11 meters × 3 meters × 4 meters = 132 cubic meters</note>. Then the total volume is the sum of the volumes of the two parts, that is, total volume = <note>42 cubic meters + 132 cubic meters = 174 cubic meters</note>.


Question: What is the volume of this cylinder?Use 𝜋 ≈ 3.14 and round your answer to the nearest hundredth. $\Box$  cubic millimeters

Answer: 502.4 cubic millimeters

1. Given that the diameter of the cylinder is 8 mm.

2. According to the definition of diameter, the radius r is equal to half of the diameter, so the radius r of the cylinder is 8 mm ÷ 2 = 4 mm.

3. Given that the height h of the cylinder is 10 mm.

4. The formula for the volume V of the cylinder is V = πr²h.

5. Substitute the known radius r=4 mm and height h=10 mm into the volume formula, then V = π × (4 mm)² × 10 mm.

7. Substitute the result into the formula to get V = 3.14 × 16 mm² × 10 mm.

9. Then multiply the result by the height 10 mm, to get V = 50.24 mm² × 10 mm = 502.4 mm³.

10. After the above calculations, the final volume of the cylinder is approximately 502.4 mm³.

Cylinder

A cylinder is a geometric solid consisting of two parallel and congruent circular bases and a curved lateral surface connecting the bases.

In the diagram of this problem, the cylinder consists of <note>two parallel and identical circular bases</note> and <note>a lateral surface</note>. <note>The bases are two completely identical circles</note>, with <note>a radius of 4 millimeters</note>, <note>a diameter of 8 millimeters</note>, and <note>their centers are on the same line</note>. The lateral surface is <note>a rectangle</note>, which, when unfolded, has <note>a height equal to the cylinder's height of 10 millimeters</note> and <note>a width equal to the circumference of the circle</note>.

Volume Formula of Cylinder

The volume of a cylinder is equal to the area of its base multiplied by its height. \( V = \pi r^2 h \), where \( r \) is the radius of the base of the cylinder, and \( h \) is the height of the cylinder.

<note>The radius r of the cylinder is 4 millimeters</note>, <note>the height h is 10 millimeters</note>, using the volume formula, it is calculated that <note>V = π × (4 millimeters)² × 10 millimeters</note>.

Definition of Diameter

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

<note>Line segment ab is the diameter of the circle</note>. Line segment ab passes through <note>the center of the circle o</note>, and both endpoints <note>a and b</note> are on the circle. According to the definition of diameter, <note>line segment ab is the longest chord of the circle</note>, with a length equal to twice the radius, i.e., <note>ab = 2 * radius</note>.


Question: What is the volume? $\Box$  cubic inches

Answer: 180 cubic inches

1. Given that the geometric figure is a triangular prism, its base is a triangle with a base length of 5 inches and a height of 8 inches. The height of the triangular prism (i.e., the length of the side) is 9 inches.

2. Calculate the area of the triangular base. According to the formula for the area of a triangle, the area is equal to the base multiplied by the height and then divided by 2, i.e.: Area = (5 × 8) / 2.

3. Perform the calculation: (5 × 8) / 2 = 40 / 2 = 20 square inches.

4. Calculate the volume of the prism. According to the formula for the volume of a prism, the volume is equal to the base area multiplied by the height, so: Volume = 20 × 9.

5. Perform the calculation: 20 × 9 = 180 cubic inches.

6. Through the above reasoning, the final answer is 180 cubic inches.

Definition of Triangular Prism

A triangular prism is a type of hexahedron that is formed by two parallel and congruent triangular bases and three rectangular lateral faces.

In the figure of this problem, the geometric body is a <note>triangular prism</note>, with the <note>base triangle sides being 5 inches and 8 inches</note>, and the <note>height of the triangular prism's lateral side being 9 inches</note>.

Volume Formula of Rectangular Prism

<note>Volume</note> = <note>20 × 9</note>. The calculated <note>volume of the triangular prism is 180 cubic inches</note>.

Area Formula of a Triangle

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

In the figure of this problem, in the triangle, <note>the base is 5 inches</note>, <note>the height on the base is 8 inches</note>, so the area of the triangle is equal to the base multiplied by the height divided by 2, that is, area = (<note>5</note> * <note>8</note>) / 2.


Question:
In circle X, $\overset{\frown}{YZ}$ is highlighted. $\overset{\frown}{YZ}$ measures 90°.

What fraction of the circle is highlighted?
Simplify your answer.

 $\Box$
Which expression represents the length of $\overset{\frown}{YZ}$ in inches?

- 1/4(9𝜋)
- 1/3(9𝜋)
- 1/4(6𝜋)
- 1/3(6𝜋)

Answer:
1/4

 1/4(6𝜋)

1. Given that in circle X, the arc \\overset{\\frown}{YZ} has a radian measure of 90°.

2. According to the properties of the central angle, the degree measure of the arc is equal to the degree measure of the central angle that subtends the arc. Therefore, the angle \\angle YXZ also measures 90°, i.e., \\angle YXZ = 90°.

3. According to the properties of a circle, the complete radian measure of a circle is 360°. Therefore, the proportion of arc \\overset{\\frown}{YZ} in the circle is 90°/360°.

4. Simplifying the fraction above, we get 1/4. Therefore, the arc \\overset{\\frown}{YZ} occupies 1/4 of circle X.

5. The length of arc \\overset{\\frown}{YZ} is 1/4 of the circumference of the circle, i.e., the length is 1/4 * C.

6. Using the formula for the circumference of a circle C = 2πr, where r is the radius of the circle. Given in the problem, r = 3 inches.

7. Substituting the radius into the circumference formula, we have C = 2π * 3 = 6π.

8. Using the arc length formula, the length of arc \\overset{\\frown}{YZ} is 1/4 * (6π) = (6π)/4.

9. Simplifying the fraction (6π)/4, we get 3π/2.

10. Through the above reasoning, we finally obtain the length expression of arc \\overset{\\frown}{YZ} as 1/4(6π), corresponding to the option '1/4(6π)'.

11. The final answer is that arc \\overset{\\frown}{YZ} occupies 1/4 of circle X.

Definition of Arc

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

<note>There are two points Y and Z on circle X</note>, <note>arc \overset{\frown}{YZ} is a segment of the curve connecting these two points</note>. According to the definition of arc, <note>arc \overset{\frown}{YZ} is a segment of the curve between two points Y and Z on the circle</note>.

Circumference Formula of Circle

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

In the figure of this problem, <note>in circle X, point X is the center</note>, <note>line segments XY and XZ are the radius r</note>. According to the <note>circumference formula of circle</note>, <note>the circumference C of the circle is equal to 2π multiplied by the radius r</note>, i.e., C=2πr, where <note>r=3 inches</note>, therefore its <note>circumference C=2π*3=6π</note>.

Arc Length Formula of a Circle

The length of the arc <note>\overset{\frown}{YZ}</note> is \(\frac{1}{4}\) multiplied by the circumference of the circle, so the arc length is \(\frac{1}{4} * 6π = \frac{6π}{4}\), which simplifies to <note>3π/2</note>.

Property of Central Angle

The arc <note>YZ</note> corresponds to the central angle <note>YXZ</note>, and the degree measure of arc <note>YZ</note> is equal to the degree measure of angle <note>YXZ</note>.

Angle Properties of a Circle

Original text: <note>Minor arc YZ and major arc ZY around the circle</note>, the total radians are 360 degrees, and the corresponding <note>sum of the angles at the circumference is also 360 degrees</note>, that is, <note>minor angle YXZ + major angle ZXY = 360 degrees</note>.


Question: What is the volume? $\Box$  cubic millimeters

Answer: 168 cubic millimeters

1. Observe the given figure, which is a rectangular prism with a length of 7 mm, a width of 3 mm, and a height of 8 mm.

2. According to the formula for the volume of a rectangular prism, the volume is equal to the product of the length, width, and height.

3. Expressed in mathematical formula: Volume = length × width × height.

4. Substitute the known length, width, and height to calculate the volume: Volume = 7 mm × 3 mm × 8 mm.

5. Calculate the product: 7 × 3 = 21, then calculate 21 × 8 = 168.

6. Therefore, the volume of the rectangular prism is 168 cubic millimeters.

Definition of Rectangular Prism

In the figure of this problem, <note>the given geometric shape is a rectangular prism</note>, with a length of <note>7 millimeters</note>, a width of <note>3 millimeters</note>, and a height of <note>8 millimeters</note>.

Volume Formula of Rectangular Prism

<note>Volume of Rectangular Prism</note> in the context of this problem diagram, when applying this formula to calculate the volume, the values substituted are <note>7 millimeters, 3 millimeters, 8 millimeters</note>. The calculation steps are as follows: <note>Volume = 7 millimeters × 3 millimeters × 8 millimeters = 168 cubic millimeters</note>.


Question: What is the volume of this cylinder?Use 𝜋 ≈ 3.14 and round your answer to the nearest hundredth. $\Box$  cubic feet

Answer: 12.56 cubic feet

1. Given that the radius of the base of the cylinder is 1 ft and the height is 4 ft.

2. According to the formula for the volume of a cylinder, the volume V = 𝜋r²h, where r represents the radius of the base and h represents the height.

3. Substitute the given radius r = 1 ft and height h = 4 ft into the formula: V = 𝜋 × 1² × 4.

4. Calculate to get V = 3.14 × 1 × 1 × 4.

5. Further calculation gives V = 3.14 × 4.

7. After the above calculations, the final volume of the cylinder is 12.56 cubic feet.

Cylinder

<note>The cylinder consists of two parallel and identical circular bases</note> and <note>a lateral surface</note>. The bases are <note>two identical circles</note>, with <note>equal radius and diameter</note>, and their <note>centers are on the same line</note>. The lateral surface is a <note>rectangle</note>, which, when unfolded, has a <note>height equal to the height of the cylinder (4 feet)</note>, and a <note>width equal to the circumference of the circle (2π × 1 foot)</note>.

Volume Formula of Cylinder

<note>Knowing the radius of the base of the cylinder r = 1 foot, height h = 4 feet</note>, substitute into the formula: <note>V = 𝜋 × 1² × 4</note>.
Calculate to get <note>V = 3.14 × 1 × 1 × 4 = 12.56 cubic feet</note>.


Question:
In circle S, $\overset{\frown}{TVU}$ is highlighted.  $\angle $ TSU measures 90°.

What fraction of the circle is highlighted?
Simplify your answer.

 $\Box$
Which expression represents the length of $\overset{\frown}{TVU}$ in inches?

- 3/4(2𝜋)
- 4/5(4𝜋)
- 4/5(2𝜋)
- 3/4(4𝜋)

Answer:
3/4

 3/4(2𝜋)

1. Given that in circle S, arc \\overset{\\frown}{TVU} is highlighted, and the angle \\angle TSU is 90°.

2. According to the property of central angles, the degree of an arc is equal to the degree of its corresponding central angle. Therefore, the central angle of arc \\overset{\\frown}{TU}, \\angle TSU, is 90°.

3. Since the total angle of the entire circle is 360°, arc \\overset{\\frown}{TU} occupies \\frac{90}{360} = \\frac{1}{4} of the entire circle.

4. Because arc \\overset{\\frown}{TU} occupies \\frac{1}{4} of the circle, the non-highlighted part of the arc occupies \\frac{1}{4}, and the highlighted part of the arc occupies the remaining \\frac{3}{4} of the circle.

5. Therefore, the length of arc \\overset{\\frown}{TVU} is \\frac{3}{4} of the entire circumference.

6. The formula for calculating the circumference of a circle is C = \\pi d, where d is the diameter. Given that the diameter d = 2 in, so C = \\pi \\times 2 = 2\\pi.

7. Therefore, the length of arc \\overset{\\frown}{TVU} = \\frac{3}{4} \\times 2\\pi.

8. After calculation, choose the option that matches \\frac{3}{4}(2\\pi).

9. Through the above reasoning, the final answer is: \\frac{3}{4}(2\\pi).

Definition of Arc

On <note>circle S</note>, there are three points <note>T, V, and U</note>. The <note>arc \overset{\frown}{TVU}</note> is a segment of the curve connecting these three points. According to the definition of an arc, the <note>arc \overset{\frown}{TVU}</note> is a segment of the curve between <note>two points T and U on the circle</note> and passes through point <note>V</note>.

Circumference Formula of Circle

In the figure of this problem, <note>in circle S, point S is the center of the circle</note>, <note>line segment TS is the radius r</note>. According to the circumference formula of the circle, the circumference C is equal to 2π multiplied by the radius r, that is, C = 2πr. <note>Given the diameter d = 2 in, so the radius r = 1 in</note>, <note>circumference C = 2π × 1 = 2π</note>.

Property of Central Angle

The arc <note>TU</note> corresponds to the <note>central angle TSU, and the degree measure of arc TU equals the degree measure of angle TSU</note>.


Question: What is the volume? $\Box$  cubic meters

Answer: 120 cubic meters

1. Observe the triangular prism in the problem, the base is a triangle, with a known base of 8 meters, height of 5 meters, and the prism is formed with a rectangular prism of length 6 meters.

2. According to the area formula of a triangle, the area of any triangle = 1/2 × base × height, therefore the area of the base triangle = 1/2 × 8 meters × 5 meters = 20 square meters.

3. According to the volume formula of a prism, the volume of a prism = base area × height, therefore the volume of this triangular prism = 20 square meters × 6 meters.

4. The calculated volume is 120 cubic meters.

n. Through the above reasoning, the final answer is 120 cubic meters.

Definition of Triangular Prism

<note>A triangular prism consists of two parallel and equal triangular bases</note>, which are connected by <note>three rectangular lateral faces</note>. <note>The two bases are parallel and equal triangles</note>, while <note>the lateral faces are rectangles</note>. Thus, among the <note>six faces of the triangular prism, two are triangles and three are rectangles</note>, forming a <note>hexahedron</note>.

Volume Formula of Prism

The volume of a prism is equal to the base area multiplied by the height.

In the figure of this problem, in the prism, <note>the area of the base is 20 square meters</note>, <note>the height of the prism is 6 meters</note>. Therefore, according to the volume formula of the prism, <note>the volume of the prism is equal to the base area of 20 square meters multiplied by the height of 6 meters</note>.

Area Formula of a Triangle

In the figure of this problem, in the triangle, the side <note>8m is the base</note>, <note>the segment 5m is the height on this base</note>, so the area of the triangle is equal to the base 8m multiplied by the height 5m divided by 2, that is, Area = (8 * 5) / 2 = 20 square meters.