Question:
Find the lateral area of the cylinder.

Round your answer to the nearest tenth.

 $\Box$  square meters

Answer: 402.1 square meters

1. Given that the height of the cylinder is 8 meters and the radius of the base is 8 meters.

2. The formula for the lateral area of the cylinder is: L.A. = 2𝜋rh, where r represents the radius of the base and h represents the height.

3. Substitute the given radius r = 8 meters and height h = 8 meters into the formula to get: L.A. = 2𝜋(8)(8).

4. Calculate to get L.A. = 128𝜋.

5. Use π ≈ 3.1416 to replace π for accuracy calculation, and get: L.A. ≈ 128 × 3.1416 = 402.1248.

6. Round the calculation result to one decimal place to get 402.1.

7. Through the above reasoning, the final lateral area of the cylinder is approximately 402.1 square meters.

Cylinder

<note>Cylinder</note> consists of two parallel and <note>identical circular bases</note> and a lateral surface. The <note>bases</note> are two identical <note>circles</note>, with <note>equal radii and diameters</note>, and their <note>centers lie on the same line</note>. The <note>lateral surface</note> is a <note>rectangle</note>, which, when unfolded, has a <note>height equal to the height of the cylinder</note> and a <note>width equal to the circumference of the circle</note>.

Definition of Radius

<note>In the circle at the bottom of the cylinder, the point is the center of the circle, and the distance from any point on the circumference to the center is 8 meters</note>, thus this distance is the radius of the circle.

Formula for Lateral Area of a Cylinder

Based on the known radius of the base of the cylinder r = 8 meters and height h = 8 meters, substitute into the formula: L.A. = 2𝜋(8)(8). Finally, obtain the lateral area of the cylinder and calculate the value.


Question:
Look at this figure:What is the shape of its bases?

- rectangle
- triangle
- circle
- decagon

Answer: - rectangle

1. According to the figure in the problem, identify it as a prism.

2. The properties of a prism indicate that its top and bottom faces are the same polygon and are parallel.

3. From the structure of the figure, confirm that the top and bottom faces of the prism are quadrilaterals.

4. Further observation shows that the opposite sides of the quadrilateral are parallel and equal in length, which indicates that the shape of the top and bottom faces is a rectangle.

5. Therefore, according to the definition of a prism, the top and bottom faces of this cube are rectangles.

6. Through the above reasoning, the final answer is a rectangle.

Definition of Prism

A prism is a polyhedron with two parallel and congruent polygonal bases, and all other faces are parallelograms.

<note>In the figure of this problem, the given shape is identified as a prism</note>. According to <note>the definition of prism</note>, it can be determined that <note>the top and bottom faces of the cube are polygons that are identical in shape and parallel</note>.

Definition of Rectangle

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

The quadrilateral on the base has opposite sides that are parallel and equal in length <note>底面的四边形的对边平行且等长</note>, and <note>all four angles are right angles</note>, indicating that <note>the shape of the base is a rectangle</note>.


Question:
Find the lateral area of the cone.

Round your answer to the nearest tenth.

 $\Box$  square centimeters

Answer: 188.5 square centimeters

1. According to the problem description, the slant height of the cone is 12 cm, and the radius of the base is 5 cm.

2. The formula for calculating the lateral surface area of the cone is: lateral surface area = π × r × 𝓁, where r is the radius of the base of the cone, and 𝓁 is the slant height of the cone.

3. Substitute the given conditions r = 5 cm, 𝓁 = 12 cm, to obtain lateral surface area = π × 5 × 12.

4. Calculate π × 5 × 12 = 60π.

5. Since the approximate value of π is 3.141592653589793, further calculation yields lateral surface area ≈ 60 × 3.141592653589793 ≈ 188.49555921538757.

6. According to the problem requirements, round the answer to one decimal place to obtain lateral surface area ≈ 188.5 square cm.

7. Through the above reasoning, the final answer is 188.5 square cm.

Definition of Cone

A cone in three-dimensional space is a surface generated by rotating a line segment (generatrix) around a fixed straight line (axis) belonging to a plane of a circle, which is not collinear with the endpoint (vertex). The surface consists of the conical surface and the base (circle).

The radius of the base circle of the cone is <note>5 centimeters</note>, and the slant height from the vertex to the boundary is <note>12 centimeters</note>. The lateral surface is formed by connecting the perimeter of the base circle with the <note>vertex</note>.

Generatrix

The generatrix of a cone is the line segment that joins a point on the circumference of the base to the apex.

A point on the <note>base circumference</note> is any point on the circumference, and the <note>vertex</note> is the vertex of the cone. The line segment connecting a point on the <note>base circumference</note> and the <note>vertex</note> is the generatrix. The generatrix is the line segment in the cone from a point on the <note>base circumference</note> to the <note>vertex</note>.

Lateral Surface Area of a Cone

The lateral surface area \(A\) of a cone is calculated using the formula \(A = \pi r l\), where \(r\) is the radius of the base circle and \(l\) is the slant height of the cone.

The formula for calculating the lateral surface area of a cone is <note>π × 5 × 12</note>. Substituting <note>base radius r = 5 cm</note> and <note>slant height 𝓁 = 12 cm</note>, we get <note>lateral surface area = π × 5 × 12</note>.


Question:
Find the lateral area of the cone.

Round your answer to the nearest tenth.

 $\Box$  square feet

Answer: 1,061.9 square feet

1. First, in the given geometry problem, the slant height of the cone is 26 ft, and the diameter of the base is 26 ft, so the radius of the base is 13 ft.

2. With the base radius determined, use the formula for the lateral area of the cone to solve for the lateral area of the cone, which is L.A. = πrl, where r is the base radius of the cone and l is the slant height.

3. Substitute the known base radius and slant height into the formula: L.A. = π * 13 * 26.

4. Calculate the numerical value of the lateral area of the cone: L.A. ≈ π * 338 ≈ 1061.8583 square feet.

5. Round the result to the first decimal place, and the final lateral area of the cone is approximately 1061.9 square feet.

6. After the above reasoning, the final answer is 1061.9 square feet.

Definition of Cone

<note>The diameter of the base circle of the cone is 26 feet</note>, <note>the radius is 13 feet</note>, <note>the length of the line segment from the vertex to any point on the base is the slant height, which is 26 feet</note>.

Lateral Surface Area of a Cone

Original: <note>圆锥底面半径13英尺</note>和<note>斜高26英尺</note>代入公式，得到 L.A. = π * 13 * 26。

Translation: Substituting <note>the cone's base radius of 13 feet</note> and <note>the slant height of 26 feet</note> into the formula, we get L.A. = π * 13 * 26.


Question:
Look at this figure:What is the shape of its base?

- triangle
- nonagon
- heptagon
- pentagon

Answer: - triangle

1. From the figure, we can observe that this is a triangular pyramid structure, consisting of the main body of a tetrahedron and its projection surface.

2. According to the definition of a tetrahedron, it is composed of four triangular faces, with one of these triangular faces being the base of the tetrahedron.

3. The figure shows that each dashed line is a lateral edge connecting the vertices of the base to the remaining vertices, emphasizing the shape of the base.

4. Since the base of the tetrahedron is composed of three vertices, according to the property of the number of vertices in polygons (an n-sided polygon is composed of n edges), the base should be a triangle.

5. Through the above analysis, we can confirm that the base is a triangle.

6. Therefore, the base of this tetrahedron is a triangle.

Definition of Triangle

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

<note>The base of the tetrahedron is a triangle</note>, <note>formed by three vertices and three edges</note>. In the figure, <note>the dashed part supports the configuration of the base in the triangle</note>, confirming <note>the property of the base as a triangle</note>. <note>The three sides of the base are respectively connected to its three vertices</note>, in accordance with <note>the basic definition of a triangle</note>.


Question:
Look at this figure:What is the shape of its base?

- triangle
- hexagon
- pentagon
- square

Answer: - square

1. Observing the figure, it can be seen that this is a cross-sectional structure diagram of a cube, and its upper part is a plane from a vertex to the center of the cross-section, which can be combined into a square pyramid.

2. A cube is a hexahedron, and its base shape is a square, which is a basic known property in solid geometry.

3. Verify the base shape: If other structures of the solid are symmetrical and equal, such as the equal length properties of each face that can assist linear measurement, these properties hold true in the square structure.

4. As shown in the figure, it is part of a cube, so the shape of the base is a square, rather than other polygons such as a triangle, pentagon, or hexagon.

5. Based on the above reasoning, it is confirmed that the shape of the base corresponding to this solid is a square.

Definition of Cube

A cube is a three-dimensional geometric figure with six faces, all of which are squares.

<note>In the figure of this problem, the given solid shape can be regarded as a sectional view of a cube</note>, that is, a part of a cube, <note>all its faces are squares</note>, and it has been verified that <note>its base is a square</note>.

Definition of Square

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

<note>In the figure of this problem, the base is a square</note>. Each side of the square <note>is of equal length in the base</note>, and <note>each of its angles is 90 degrees</note>, thus it meets the definition of a square.

Definition of Cube

In the diagram of this problem, the base of the cube is a square. According to the geometric properties of the cube, it can be verified by observing the <note>relationship between the faces</note>, where <note>all sides of the base are equal in length</note>, and <note>the sides are perpendicular to each other</note>, which conforms to the geometric properties of a square.


Question:
Look at this figure:What is the shape of its bases?

- circle
- rectangle
- nonagon
- pentagon

Answer: - rectangle

1. Observing the given figure, we notice that its shape is a prism. According to the definition in geometry, the top and bottom faces of a prism are identical and parallel polygons.

2. From the figure, it can be seen that the sides of the prism are rectangles, which is characteristic of a cuboid (rectangular prism).

3. A cuboid is a special type of prism, characterized by having identical rectangular top and bottom faces. Therefore, we infer that the base shape is a rectangle.

4. To confirm the base shape is a rectangle, note that according to the definition of a prism, all side faces are perpendicular to the base, which explains why the side faces are rectangles and the base is also a rectangle.

5. Through the above analysis, it can be confirmed that the base of the cuboid is a rectangle.

6. Therefore, it can be concluded that the base shape of the cuboid is a rectangle.

Definition of Prism

<note>This geometric body is a prism</note>, with two <note>identical and parallel rectangular bases</note> and <note>four lateral faces</note>, all of which are <note>rectangles</note>.

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.

In the figure of this problem, <note>the given geometric body is a rectangular prism</note>, <note>its six faces are all rectangles</note>.

<note>A rectangular prism has two identical and parallel rectangular bases and top surfaces</note>.


Question:
In the rectangular prism shown below, which lines are intersecting? Select all that apply.

-

| $\overleftrightarrow{KO}$ | and | $\overleftrightarrow{LP}$ |
-

| $\overleftrightarrow{IJ}$ | and | $\overleftrightarrow{IL}$ |
-

| $\overleftrightarrow{KL}$ | and | $\overleftrightarrow{LP}$ |
-

| $\overleftrightarrow{IJ}$ | and | $\overleftrightarrow{LP}$ |

Answer:
-

| \$\overleftrightarrow{IJ}\$ | and | \$\overleftrightarrow{IL}\$ |
-

| \$\overleftrightarrow{KL}\$ | and | \$\overleftrightarrow{LP}\$ |

1. First, consider the lines \\(ackslash overleftrightarrow{KO}\\) and \\(ackslash overleftrightarrow{LP}\\). Both lines lie in the same plane \\(KLOP\\) and do not intersect. According to the definition of parallel lines, two lines in the same plane that do not intersect are called parallel lines. Therefore, \\(ackslash overleftrightarrow{KO}\\) and \\(ackslash overleftrightarrow{LP}\\) are not intersecting lines.\n\n2. Next, consider the lines \\(ackslash overleftrightarrow{IJ}\\) and \\(ackslash overleftrightarrow{IL}\\). Both lines share an endpoint \\(I\\) and intersect at point \\(I\\). Therefore, \\(ackslash overleftrightarrow{IJ}\\) and \\(ackslash overleftrightarrow{IL}\\) are intersecting lines.\n\n3. Consider the lines \\(ackslash overleftrightarrow{KL}\\) and \\(ackslash overleftrightarrow{LP}\\). Both lines lie in the same plane \\(KLOP\\) and intersect at point \\(L\\). Therefore, \\(ackslash overleftrightarrow{KL}\\) and \\(ackslash overleftrightarrow{LP}\\) are intersecting lines.\n\n4. Finally, consider the lines \\(ackslash overleftrightarrow{IJ}\\) and \\(ackslash overleftrightarrow{LP}\\). \\(ackslash overleftrightarrow{IJ}\\) lies in plane \\(IJNM\\), while \\(ackslash overleftrightarrow{LP}\\) lies in plane \\(KLOP\\), and the two lines lie in different planes and do not have a common point. According to the definition of skew lines, they are skew lines. According to the properties of skew lines, skew lines do not intersect. Therefore, \\(ackslash overleftrightarrow{IJ}\\) and \\(ackslash overleftrightarrow{LP}\\) are not intersecting lines.\n\n5. Through the above analysis, it is finally determined that among the above options, the intersecting lines are \\(ackslash overleftrightarrow{IJ}\\) and \\(ackslash overleftrightarrow{IL}\\) as well as \\(ackslash overleftrightarrow{KL}\\) and \\(ackslash overleftrightarrow{LP}\\).

Definition of Parallel Lines

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

<note>\(\overleftrightarrow{KO}\)</note> and <note>\(\overleftrightarrow{LP}\)</note> lie in the same plane <note>\(KLOP\)</note> and do not intersect, so according to the definition of parallel lines, <note>\(\overleftrightarrow{KO}\)</note> and <note>\(\overleftrightarrow{LP}\)</note> are parallel lines.

Definition of Skew Lines

Two lines are called skew lines if and only if they are not parallel and do not lie in the same plane. Skew lines are also known as non-coplanar lines.

In the diagram of this problem, <note>line IJ and line LP are not parallel and not in the same plane</note>, they are <note>skew lines, also known as oblique lines</note>.

Properties of Skew Lines

Two lines are called skew lines if and only if they are not parallel and do not lie in the same plane. Skew lines do not intersect.

The original text: <note>Line IJ and line LP are not parallel and not in the same plane</note>, they are skew lines, so <note>IJ and LP do not intersect</note>.


Question: The rectangular prism below is labeled with its measured dimensions. Taking measurement error into account, what is the percent error in its calculated volume?Round your answer to the nearest tenth of a percent and include a percent sign (%). $\Box$

Answer: 27.9%

1. According to the rectangular prism volume formula, the measured dimensions of the rectangular prism are length 15 yd, width 4 yd, and height 5 yd. Its measured volume is length × width × height = 15 × 4 × 5 = 300 cubic yd.

2. Meanwhile, the measurement error can reach up to half of the nearest integer yd, which means the maximum possible error in each direction is 0.5 yd.

3. Calculate the maximum possible volume: adding the error to each dimension, the maximum possible volume is (15 + 0.5) × (4 + 0.5) × (5 + 0.5) = 15.5 × 4.5 × 5.5 = 383.625 cubic yd.

4. Calculate the minimum possible volume: subtracting the error from each dimension, the minimum possible volume is (15 - 0.5) × (4 - 0.5) × (5 - 0.5) = 14.5 × 3.5 × 4.5 = 228.375 cubic yd.

5. Calculate the difference between the measured volume and the maximum and minimum volumes to find the largest possible error. The difference between the measured volume and the maximum volume is 383.625 - 300 = 83.625 cubic yd, and the difference between the measured volume and the minimum volume is 300 - 228.375 = 71.625 cubic yd.

6. The maximum volume error is 83.625 cubic yd, so this is the value we need to use.

7. Use the percentage error formula to calculate the percentage error of the measured volume: percentage error = (maximum possible error / measured volume) × 100% = (83.625 / 300) × 100%.

8. The calculation yields: percentage error = 27.875%.

9. According to the problem requirements, round the percentage error to the nearest tenth, resulting in 27.9%.

10. Through the above reasoning, the final answer is 27.9%.

Definition of Rectangular Prism

The length of the rectangular prism is <note>15 yards</note>, the width is <note>4 yards</note>, and the height is <note>5 yards</note>. These three dimensions are used as the actual size of the rectangular prism to calculate the volume.

Volume Formula of Rectangular Prism

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

In the diagram of this problem, in the rectangular prism, <note>the side 15yd represents the length</note>, <note>the side 4yd represents the width</note>, <note>the side 5yd represents the height</note>, so the volume of the rectangular prism is equal to the product of the length, width, and height, i.e., volume = 15 * 4 * 5.


Question:
Look at this figure:What is the shape of its bases?

- circle
- triangle
- octagon
- pentagon

Answer: - circle

1. Observe the given figure, this is a cylinder, which is a geometric body enclosed by the lateral surface of the cylinder and the two bases at the top and bottom.

2. A cylinder is a three-dimensional figure, and its two bases are identical and parallel plane shapes.

3. In the figure, the shape of the base is a circle, so each base of the cylinder is a circle.

4. Since each base is a circle, among the given options, 'circle' should be chosen as the shape of the base.

5. After the above analysis, the final answer is circle.

Cylinder

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

<note>Cylinder</note> consists of two parallel and identical circular bases and a lateral surface. <note>The bases are two identical circles, with equal radii and diameters, and their centers lie on the same line</note>. The lateral surface is a rectangle, and when unfolded, <note>its height is equal to the height of the cylinder, and its width is equal to the circumference of the circle</note>.

Definition of Circle

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

<note>The base of the cylinder is a circle</note>. The definition of a circle is the set of all points in a plane that are at a fixed distance from a fixed point. <note>The center of the base in the figure is the center point of the base</note>, <note>every point on the base is equidistant from the center point</note>, and this distance is the radius. Therefore, the base is a circle.


Question: The rectangular prism below is labeled with its measured dimensions. Taking measurement error into account, what is the percent error in its calculated volume?Round your answer to the nearest tenth of a percent and include a percent sign (%). $\Box$

Answer: 11.9%

1. Given a right rectangular prism with length 10 mm, width 13 mm, and height 19 mm. Calculate its measured volume as: Volume = Length × Width × Height = 10 mm × 13 mm × 19 mm.

2. Calculate the measured volume: Volume = 10 × 13 × 19 = 2470 cubic mm.

3. Consider measurement error: The maximum possible error in each measurement is half of 1 mm, which is 0.5 mm. When calculating the maximum and minimum possible volumes, this error needs to be considered.

4. Calculate the maximum possible volume: Add 0.5 mm to each measurement value, then the product gives the maximum possible volume, i.e., (10 + 0.5) mm × (13 + 0.5) mm × (19 + 0.5) mm = 10.5 × 13.5 × 19.5.

5. The maximum possible volume is: 10.5 × 13.5 × 19.5 = 2764.125 cubic mm.

6. Calculate the minimum possible volume: Subtract 0.5 mm from each measurement value, then the product gives the minimum possible volume, i.e., (10 - 0.5) mm × (13 - 0.5) mm × (19 - 0.5) mm = 9.5 × 12.5 × 18.5.

7. The minimum possible volume is: 9.5 × 12.5 × 18.5 = 2190 cubic mm.

8. Calculate the volume error: Volume error = max(|Measured Volume - Maximum Volume|, |Measured Volume - Minimum Volume|).

9. Calculate the error: |Measured Volume - Maximum Volume| = |2470 - 2764.125| = 294.125.

10. Calculate the error: |Measured Volume - Minimum Volume| = |2470 - 2190| = 280.

11. The maximum error is 294.125 cubic mm.

12. Calculate the percentage error of the volume: Percentage error = (Maximum error / Measured Volume) × 100%.

14. Through the above reasoning, the final answer is 11.9%.

Definition of Rectangular Prism

<note>This geometric figure is a rectangular prism</note>, with a <note>length of 10 millimeters</note>, <note>width of 13 millimeters</note>, and <note>height of 19 millimeters</note>. Each <note>adjacent face intersects perpendicularly</note>.

Volume Formula of Rectangular Prism

<note>Side 13mm represents length</note>, <note>Side 10mm represents width</note>, <note>Side 19mm represents height</note>, so the volume of the rectangular prism is equal to the product of length, width, and height, i.e., Volume = 13 * 10 * 19.


Question: The rectangular prism below is labeled with its measured dimensions. Taking measurement error into account, what is the percent error in its calculated volume?Round your answer to the nearest tenth of a percent and include a percent sign (%). $\Box$

Answer: 8.8%

1. Observe the cuboid, its measured dimensions are 19 m, 15 m, and 19 m. Thus, the measured volume is 19×15×19 = 5,415 cubic meters.

2. The maximum error for each measurement is half of 1 meter, which is 0.5 meters. Therefore, the smallest possible length is each measurement minus 0.5 meters, and the largest possible length is each measurement plus 0.5 meters.

3. Calculate the maximum possible volume by adding the error to each measurement:

4. Maximum possible volume: (19 + 0.5) × (15 + 0.5) × (19 + 0.5) = 19.5 × 15.5 × 19.5 = 5,893.875 cubic meters.

5. Calculate the minimum possible volume by subtracting the error from each measurement:

6. Minimum possible volume: (19 - 0.5) × (15 - 0.5) × (19 - 0.5) = 18.5 × 14.5 × 18.5 = 4,936.125 cubic meters.

7. Calculate the difference between the measured volume and the maximum and minimum volumes:

8. Difference with the minimum volume: 5,415 - 4,936.125 = 478.875 cubic meters.

9. Difference with the maximum volume: 5,893.875 - 5,415 = 478.875 cubic meters.

10. Since the two differences are equal, either can be used. You can choose the difference between the maximum volume and the measured volume, which is 478.875 cubic meters.

11. Calculate the percentage of the maximum possible volume error: 478.875 ÷ 5,415 ≈ 0.088431.

12. Convert this decimal to a percentage, which is approximately 8.8431%.

13. Round to the first decimal place to get 8.8%.

14. After the above reasoning, the final answer is that the error percentage is 8.8%.

Definition of Rectangular Prism

The original text: <note>The side lengths of the rectangular prism are 19 meters, 15 meters, and 19 meters</note>, and <note>all angles are right angles</note>, conforming to the <note>definition of a rectangular parallelepiped</note>.

Volume Formula of Rectangular Prism

In the figure of this problem, <note>the volume formula of a rectangular prism is V = 19 meters × 15 meters × 19 meters</note> = 5,415 cubic meters.


Question: The rectangular prism below is labeled with its measured dimensions. Taking measurement error into account, what is the percent error in its calculated volume?Round your answer to the nearest tenth of a percent and include a percent sign (%). $\Box$

Answer: 12.9%

1. Given the length of the rectangular prism is 20 inches, the width is 18 inches, and the height is 7 inches. According to the volume formula for a rectangular prism V = length × width × height, calculate its measured volume V = 20 × 18 × 7 = 2520 cubic inches.

2. Since the measurements are rounded to the nearest whole inch, the maximum possible error for each measurement is half an inch, i.e., 0.5 inches.

3. Calculate the maximum possible volume. The maximum possible length, width, and height are 20 + 0.5 = 20.5 inches, 18 + 0.5 = 18.5 inches, and 7 + 0.5 = 7.5 inches respectively. Thus, the maximum possible volume = 20.5 × 18.5 × 7.5 = 2844.375 cubic inches.

4. Calculate the minimum possible volume. The minimum possible length, width, and height are 20 - 0.5 = 19.5 inches, 18 - 0.5 = 17.5 inches, and 7 - 0.5 = 6.5 inches respectively. Thus, the minimum possible volume = 19.5 × 17.5 × 6.5 = 2216.625 cubic inches.

5. Calculate the volume difference between the measured volume and the maximum possible volume and the minimum possible volume to determine the largest possible volume error. The maximum error value is max(2844.375 - 2520, 2520 - 2216.625) = max(324.375, 303.375) = 324.375 cubic inches.

6. Percentage error = (maximum volume error / measured volume) × 100%. So the percentage error = (324.375 / 2520) × 100% ≈ 12.8571428571%.

7. Round to the nearest tenth, percentage error ≈ 12.9%.

8. Through the above reasoning, the final answer is 12.9%.

In the figure of this problem, <note>the length of the rectangular prism is 20 inches</note>, <note>the width is 18 inches</note>, and <note>the height is 7 inches</note>. Therefore, its volume is measured as <note>V = 20 × 18 × 7 = 2520 cubic inches</note>.


Question:
Find the lateral area of the cone.

Round your answer to the nearest tenth.

 $\Box$  square feet

Answer: 1,061.9 square feet

1. Given that the slant height l of this cone is 26 feet, and the diameter of the base is 26 feet.

2. Since the base of the cone is a circle, the radius r is half of its diameter, so r = 26/2 = 13 feet.

3. Calculate the lateral surface area of the cone using the formula for the lateral surface area of a cone: A_side = 𝜋 * r * l.

4. Substitute the known values of r and l into the formula: A_side = 𝜋 * 13 * 26.

5. Perform the calculation step by step: 𝜋 * 13 * 26 ≈ 1061.8583 square feet.

6. Round the result to one decimal place, which is ≈ 1061.9 square feet.

7. Through the above reasoning, the final answer is 1061.9 square feet.

Definition of Cone

<note>The base of the cone is a circle with a diameter of 26 feet</note>, <note>the vertex is the tip of the cone</note>, any point from <note>the vertex to the circumference ray</note> forms its side. <note>The slant height 𝓁 of the cone is the distance from the vertex to one side of the circumference, given as 26 feet</note>.

Definition of Diameter

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

In the diagram of this problem, the base of the cone is a circle, <note>the diameter of the circle is 26 feet</note>, connecting <note>the center O of the circle and two points on the circumference</note>, <note>the length is twice the radius, i.e., diameter = 26 feet</note>.

Definition of Radius

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

In the figure of this problem, <note>the radius of the circle r</note> is equal to <note>half of the diameter of the circle</note>, <note>the diameter of the circle is 26 feet</note>, so <note>r=26/2=13 feet</note>.

Lateral Surface Area of a Cone

<note>The radius r of the cone is 13 feet</note>, <note>the slant height l is 26 feet</note>. Substitute these values into the formula <note>A_side = 𝜋 * 13 * 26</note> for calculation.


Question: The rectangular prism below is labeled with its measured dimensions. Taking measurement error into account, what is the percent error in its calculated volume?Round your answer to the nearest tenth of a percent and include a percent sign (%). $\Box$

Answer: 14.7%

1. Given that the three measured edge lengths of a rectangular prism are 19 meters, 16 meters, and 6 meters respectively. According to the volume formula for a rectangular prism, i.e., V = length * width * height, the measured volume is V = 19 * 16 * 6 = 1824 cubic meters.

2. Each measurement is in meters, so the maximum possible error is 0.5 meters. This is because measuring to the nearest integer meter means the error could be within ±0.5 meters.

3. Based on the given maximum possible error of 0.5 meters, calculate the possible maximum volume. The maximum possible dimensions are 19.5 meters, 16.5 meters, and 6.5 meters, so the maximum volume is 19.5 * 16.5 * 6.5 = 2091.375 cubic meters.

4. Similarly, based on the possible maximum error, calculate the possible minimum volume. The minimum possible dimensions are 18.5 meters, 15.5 meters, and 5.5 meters, so the minimum volume is 18.5 * 15.5 * 5.5 = 1570.375 cubic meters.

5. Compare the measured volume of 1824 cubic meters with the calculated maximum and minimum volumes to find the volume error. The error with the maximum volume is 2091.375 - 1824 = 267.375 cubic meters, and the error with the minimum volume is 1824 - 1570.375 = 253.625 cubic meters.

6. Since the calculation error is greater with the maximum volume compared to the measured volume, the maximum possible error is 267.375 cubic meters.

7. According to the percentage error formula: (maximum possible error / measured volume) * 100%, the percentage error is (267.375 / 1824) * 100% ≈ 14.7%.

8. Through the above reasoning, the final answer is 14.7%.

Definition of Rectangular Prism

<note>The lengths of the three edges of the rectangular prism are 19 meters, 16 meters, and 6 meters</note>, <note>these edges are connected by faces to form a rectangular prism</note>.

Volume Formula of Rectangular Prism

In this problem, <note>The volume formula of the rectangular prism is V = 19 meters * 16 meters * 6 meters = 1824 cubic meters</note>.


Question:
What is m $\angle $ K?

m $\angle $ K= $\Box$ °

Answer: m \$\angle \$ K=120°

1. The figure given in the problem is a circle, where points I, H, J are points on the circumference, and K is the center of the circle.

2. The problem states that ∠IHJ = 60°.

3. According to the inscribed angle theorem, an inscribed angle is equal to half of the central angle that subtends the same arc. Therefore, the inscribed angle theorem applies in this case: the chord IJ subtends the inscribed angle ∠IHJ and the central angle ∠IKJ, both sharing the same arc IJ.

4. According to the inscribed angle theorem: m ∠IKJ = 2 × m ∠IHJ.

5. Substituting the given m ∠IHJ = 60° into the formula: m ∠IKJ = 2 × 60°.

6. Calculating, we get m ∠IKJ = 120°.

7. Therefore, through step-by-step reasoning, the final answer is m ∠IKJ = 120°.

Definition of Circle

In the figure of this problem, <note>in circle O</note>, <note>point K is the center</note>, <note>the radii are KI, KH, KJ</note>. All points <note>at a distance from point K equal to KI, KH, KJ</note> are on <note>circle O</note>.

Definition of Inscribed Angle

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

<note>In circle O, the vertex H of angle ∠IHJ is on the circumference</note>, <note>the two sides of angle ∠IHJ intersect circle O at points I and J respectively</note>. Therefore, <note>angle ∠IHJ is an inscribed angle</note>.

Definition of Central Angle

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

In the diagram of this problem, in the circle, <note>points I and J</note> are two points on the circle, <note>the center of the circle is point K</note>. <note>The angle ∠IKJ formed by the lines KI and KJ</note> is called the central angle.

Inscribed Angle Theorem

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

<note>In circle K, points I, H, and J are on the circle</note>, the central angle corresponding to arc IJ is <note>∠IKJ</note>, and the inscribed angle is <note>∠IHJ</note>. According to the Inscribed Angle Theorem, <note>∠IHJ is equal to half of the central angle ∠IKJ corresponding to arc IJ</note>, that is, <note>∠IHJ = 1/2 ∠IKJ</note>.


Question:
What is m $\angle $ R?

m $\angle $ R= $\Box$ °

Answer: m \$\angle \$ R=88°

1. The problem states that the measure of angle TSU is 44°, and we need to calculate the measure of angle R.

2. According to the diagram, angle TSU is the inscribed angle subtended by arc TU, and angle R is the central angle subtended by the same arc TU.

3. Applying the Inscribed Angle Theorem: In the same circle or congruent circles, an inscribed angle is equal to half of its corresponding central angle. Therefore, angle TSU = 1/2 × angle R.

4. Substituting the given condition from the problem, we get: 44° = 1/2 × angle R.

5. To find the measure of angle R, we solve the equation: angle R = 44° × 2.

6. The calculation yields: angle R = 88°.

7. Through the above reasoning, we finally obtain the answer as 88°.

Definition of Inscribed Angle

In the figure of this problem, in the circle, <note>the vertex S of angle TSU is on the circumference</note>, <note>the two sides of angle TSU intersect the circle at points T and U respectively</note>. Therefore, <note>angle TSU is an inscribed angle</note>.

Definition of Central Angle

<note>In a circle, points T and U are two points on the circle</note>, <note>the center of the circle is point R</note>. <note>The angle ∠TRU formed by the lines RT and RU</note> is called the central angle.

Definition of Arc

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

<note>In the figure of this problem, there are two points T and U on the circle</note>, arc TU is the curve segment connecting these two points. According to the definition of arc, <note>arc TU is the curve segment between two points T and U on the circle</note>.

Inscribed Angle Theorem

<note>In the circle, points T, S, U are on the circle</note>, the arc TU corresponds to the <note>central angle ∠TRU, the inscribed angle ∠TSU</note>. According to the Inscribed Angle Theorem, <note>∠TSU is equal to half of the central angle ∠TRU corresponding to the arc TU</note>, that is, <note>∠TSU = 1/2 ∠TRU</note>.


Question:
Find the lateral area of the cone.

Round your answer to the nearest tenth.

 $\Box$  square inches

Answer: 31.4 square inches

1. Given the problem conditions, the slant height of the cone is 5 inches, and the diameter of the base is 4 inches.

2. First, based on the base diameter of 4 inches, we can find the base radius as r = 2 inches.

3. The formula for the lateral surface area of the cone is A_side = πrl, where r is the base radius and l is the slant height.

4. Substitute the given data into the lateral surface area formula: A_side = π * 2 * 5.

5. Calculate to get A_side = 10π.

6. Since the problem requires the result to be rounded to one decimal place, we need to approximate 10π.

7. Approximating 10π ≈ 31.4159 and rounding to one decimal place, we get 31.4.

8. Through the above reasoning, the final answer is 31.4.

Definition of Cone

A cone in three-dimensional space is a surface generated by rotating a line segment (generatrix) around a fixed straight line (axis) belonging to a plane of a circle, which is not collinear with the endpoint (vertex). The surface consists of the conical surface and the base (circle).

<note>A cone is formed by a circle with a radius of r=2 inches at the base and a slant height of l=5 inches</note>.

Definition of Radius

The <note>radius of the base of the cone</note> refers to the length of the <note>line segment from the center of the circle to any point on the circumference</note>. Given that the <note>diameter of the base</note> is <note>4 inches</note>, therefore the <note>radius of the base r=2 inches</note>.

Definition of Diameter

<note>The diameter of the base of the cone is 4 inches</note>, connecting <note>the center O and two points on the circumference</note>, with a length of <note>2 times the radius</note>, that is, <note>diameter = 4 inches</note>, therefore <note>the base radius r = 2 inches</note>.

Lateral Surface Area of a Cone

The lateral surface area \(A\) of a cone is calculated using the formula \(A = \pi r l\), where \(r\) is the radius of the base circle and \(l\) is the slant height of the cone.

<note>Base radius r = 2 inches</note>, <note>Slant height ℓ = 5 inches</note>, thus L.A. = π × 2 × 5 = 10π. After approximation, the result is 31.4 square inches.

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>The diameter of the base of the cone d = 4 inches</note>, therefore <note>the radius of the base r = 2 inches</note>. According to the circumference formula of the circle, <note>the circumference of the circle C equals 2π multiplied by the radius r</note>, that is <note>C = 2π × 2 = 4π</note>.


Question:
What is ST?

ST= $\Box$

Answer: ST=10

1. Given ∆PQR is a triangle, and the circle is its incircle, given QR = 3 and PR = 6.

2. According to the tangent length theorem, the lengths of two tangents drawn from a point to a circle are equal, so QR = RS.

3. Therefore, RS = QR = 3.

4. From PR = PQ + QR, QR = 3, PR = 6, we get PQ = 6 - 3 = 3.

5. According to the tangent length theorem, PQ = PU, thus PU = PQ = 3.

6. From PT = PU + TU, given PT = 13, PU = 3, we get TU = PT - PU = 13 - 3 = 10.

7. Similarly, according to the tangent length theorem, TU = ST, thus ST = TU = 10.

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

Tangent-Segment Theorem

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

<note>From a point P outside the circle, two tangents PQ and PU are drawn to the circle, and their tangent lengths are equal, i.e., PQ = PU</note>; <note>Similarly, from a point R outside the circle, two tangents QR and RS are drawn to the circle, and their tangent lengths are equal, i.e., QR = RS</note>; <note>From a point T outside the circle, two tangents TU and TS are drawn to the circle, and their tangent lengths are equal, i.e., TU = ST</note>.


Question:
Find the lateral area of the cone.

Round your answer to the nearest tenth.

 $\Box$  square yards

Answer: 628.3 square yards

1. Given that the slant height (𝓁) of the cone is 20 yards, and the diameter of the base circle is 20 yards. By definition, the radius (r) of the circle is half of the diameter, so the radius r = 20/2 = 10 yards.

2. The formula for the lateral area (L.A.) of the cone is: L.A. = πrl, where r is the radius of the base circle of the cone, and 𝓁 is the slant height of the cone.

3. Substitute the given data into the lateral area formula: L.A. = π × 10 × 20.

4. Calculate L.A., obtaining L.A. = 200π.

5. Use π ≈ 3.14159 for approximate calculation: L.A. = 200 × 3.14159 = 628.318.

6. According to the problem's requirement, round the result to one decimal place, obtaining 628.3.

7. Through the above reasoning, the final answer is 628.3 square yards.

Definition of Radius

In the figure of this problem, the diameter of the circle at the bottom of the cone is 20 yards, therefore the radius r is half of it, i.e., r = 10 yards.

Generatrix

The generatrix of a cone is the line segment that joins a point on the circumference of the base to the apex.

In the figure of this problem, <note>a line segment from a point on the circumference of the base of the cone to the apex is the generatrix</note>. Given that the slant height of the cone is 20 yards, determine <note>𝓁 = 20 yards</note>.

Lateral Surface Area of a Cone

In the diagram of this problem, since <note>r = 10 yards</note>, <note>𝓁 = 20 yards</note>, therefore the lateral surface area <note>L.A. = π × 10 × 20</note>.


Question: The rectangle below is labeled with its measured dimensions. Taking measurement error into account, what is the percent error in its calculated area?Round your answer to the nearest tenth of a percent and include a percent sign (%). $\Box$

Answer: 11.7%

1. Given a rectangle with a length of 16 mm and a width of 6 mm. We need to consider measurement errors and calculate the percentage error of its area.

2. First, calculate the measured area. The formula for calculating the area of a rectangle is length multiplied by width. Therefore, the measured area is 16 mm × 6 mm = 96 mm².

3. Since each measurement is made to the nearest integer millimeter, the maximum possible error is 0.5 mm.

4. Calculate the maximum possible area by adding the maximum possible error to each measurement value and then multiplying: length is 16 mm + 0.5 mm = 16.5 mm, width is 6 mm + 0.5 mm = 6.5 mm. The maximum possible area is 16.5 mm × 6.5 mm = 107.25 mm².

5. Calculate the minimum possible area by subtracting the maximum possible error from each measurement value and then multiplying: length is 16 mm - 0.5 mm = 15.5 mm, width is 6 mm - 0.5 mm = 5.5 mm. The minimum possible area is 15.5 mm × 5.5 mm = 85.25 mm².

6. Calculate the difference between the maximum area and the measured area, as well as the difference between the minimum area and the measured area. Determine the larger of the two, which is the maximum possible error of the area.

7. The maximum possible error is 107.25 mm² - 96 mm² = 11.25 mm², the minimum possible error is 96 mm² - 85.25 mm² = 10.75 mm². Therefore, the maximum possible error of the area is 11.25 mm².

8. Calculate the percentage error of the area. The percentage error is the maximum possible error divided by the measured area, then multiplied by 100%. Therefore, the percentage error is (11.25 mm² / 96 mm²) × 100% = 11.71875%.

9. Round the result to the nearest hundredth. The percentage error of the area is approximately 11.7%.

10. After the above reasoning, the final answer is 11.7%.

Definition of Rectangle

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

<note>A quadrilateral is a rectangle</note>, its <note>interior angles are all right angles (90 degrees)</note>, and <note>two sides with a length of 16 millimeters are parallel and equal in length</note>, <note>two sides with a width of 6 millimeters are parallel and equal in length</note>.

Formula for the Area of a Rectangle

The area of a rectangle is equal to its length multiplied by its width.

In the figure of this problem, <note>the length of the rectangle is 16 millimeters</note>, <note>the width is 6 millimeters</note>, so <note>the area of the rectangle</note> = <note>16 millimeters * 6 millimeters</note> = <note>96 square millimeters</note>.


Question: The rectangle below is labeled with its measured dimensions. Taking measurement error into account, what is the percent error in its calculated area?Round your answer to the nearest tenth of a percent and include a percent sign (%). $\Box$

Answer: 22.5%

1. Given that the length and width of the rectangle are 10 cm and 3 cm respectively, the measured area is 10 cm × 3 cm = 30 cm².

2. For each measurement, the maximum possible measurement error is 0.5 cm (this is because the measurement is to the nearest whole centimeter, so the maximum possible error is half of that, i.e., 0.5 cm).

3. Calculate the possible maximum area: add the maximum error to each measurement value, i.e., (10 cm + 0.5 cm) × (3 cm + 0.5 cm) = 10.5 cm × 3.5 cm = 36.75 cm².

4. Calculate the possible minimum area: subtract the maximum error from each measurement value, i.e., (10 cm - 0.5 cm) × (3 cm - 0.5 cm) = 9.5 cm × 2.5 cm = 23.75 cm².

5. Find the maximum error in the area: calculate the difference between the minimum area and the measured area, and the difference between the maximum area and the measured area, and find the larger value. The difference between the minimum area and the measured area is |23.75 cm² - 30 cm²| = 6.25 cm², the difference between the maximum area and the measured area is |36.75 cm² - 30 cm²| = 6.75 cm². Therefore, the maximum possible error is 6.75 cm².

6. Calculate the percentage error in the area: percentage error = (maximum possible error) ÷ (measured area) = 6.75 cm² ÷ 30 cm² = 0.225.

7. Convert the decimal to a percentage: 0.225 = 22.5%.

8. After the above reasoning, the final answer is 22.5%.

Definition of Rectangle

In the figure of this problem, <note>the length of the rectangle is 10 cm, and the width is 3 cm</note>. <note>Each interior angle of the rectangle is a right angle (90 degrees)</note>, and <note>the opposite sides are parallel and equal in length</note>.

Formula for the Area of a Rectangle

<note>The length of the rectangle is 10 cm, and the width is 3 cm</note>, according to the formula A = length * width, the area of the rectangle is <note>A = 10 cm * 3 cm = 30 cm²</note>.


Question: The rectangular prism below is labeled with its measured dimensions. Taking measurement error into account, what is the percent error in its calculated volume?Round your answer to the nearest tenth of a percent and include a percent sign (%). $\Box$

Answer: 21.4%

1. Given that the length of the rectangular prism is 10 meters, the width is 4 meters, and the height is 18 meters, we first calculate its measured volume as: Volume = Length × Width × Height = 10 meters × 4 meters × 18 meters = 720 cubic meters.

2. Each measurement is taken to the nearest whole meter, so the maximum error for each measurement is 0.5 meters.

3. Calculate the maximum possible volume: Add the maximum error to each dimension. Maximum possible length = 10 meters + 0.5 meters = 10.5 meters, maximum possible width = 4 meters + 0.5 meters = 4.5 meters, maximum possible height = 18 meters + 0.5 meters = 18.5 meters. Therefore, the maximum possible volume is: Maximum possible volume = 10.5 meters × 4.5 meters × 18.5 meters = 874.125 cubic meters.

4. Calculate the minimum possible volume: Subtract the maximum error from each dimension. Minimum possible length = 10 meters - 0.5 meters = 9.5 meters, minimum possible width = 4 meters - 0.5 meters = 3.5 meters, minimum possible height = 18 meters - 0.5 meters = 17.5 meters. Therefore, the minimum possible volume is: Minimum possible volume = 9.5 meters × 3.5 meters × 17.5 meters = 584.625 cubic meters.

5. Calculate the difference to determine the maximum possible error. Compare the possible volume differences: the difference between 720 cubic meters (measured volume) and 874.125 cubic meters (maximum volume) is 154.125 cubic meters, and the difference between 720 cubic meters (measured volume) and 584.625 cubic meters (minimum volume) is 135.375 cubic meters. The maximum difference is 154.125 cubic meters, so this is the maximum possible error.

6. Calculate the percentage error of the volume: Percentage error = (Maximum possible error / Measured volume) × 100% = (154.125 cubic meters / 720 cubic meters) × 100% ≈ 21.4%.

7. From the above analysis, the final answer is that the maximum possible percentage error is 21.4%.

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.

The length of the rectangular prism is <note>10 meters</note>, the width is <note>4 meters</note>, and the height is <note>18 meters</note>. Each face of the rectangular prism is <note>rectangular</note>, and the <note>opposite faces are parallel to each other</note>.

Volume Formula of Rectangular Prism

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

<note>The length of the rectangular prism is 10 meters</note>, <note>the width is 4 meters</note>, <note>the height is 18 meters</note>. Therefore, the volume of the rectangular prism = length × width × height = <note>10 meters × 4 meters × 18 meters</note> = <note>720 cubic meters</note>.


Question:
What is IJ?

IJ= $\Box$

Answer: IJ=5

1. Given that the incircle of △IMK is tangent to the sides IH, IJ, JK, KL, LM, MH of △IMK at points H, J, L, respectively, according to the tangent length theorem, we get: HI = IJ, JK = LK, LM = MH.

2. Given LM = 10, according to the tangent length theorem, we get MH = 10.

3. Given MH = 10 and IM = 15, we get HI + HM = IM.

4. Substituting the given conditions, we get HI + 10 = 15.

5. Solving the equation HI + 10 = 15, we get HI = 5.

6. Since HI and IJ are tangent segments from I to the incircle, according to the tangent length theorem, we get IJ = HI.

7. Given HI = 5, we get IJ = 5.

8. Through the above reasoning, we finally get the answer IJ = 5.

Tangent-Segment Theorem

<note>In the figure of this problem, two tangents HI and IJ are drawn from an external point I to the circle, and their tangent lengths are equal, i.e., HI = IJ.</note> Similarly, <note>two tangents MH and ML are drawn from an external point M to the circle, and their tangent lengths are equal, i.e., MH = ML.</note>


Question:
Is this a polygon?

- yes
- no

Answer: - no

1. First, define a polygon: A polygon is a closed plane figure composed of a finite number of straight line segments, each segment connecting to exactly two other segments at its endpoints.

2. By observing the figure given in the problem, it can be seen that the figure is a continuous closed curve, not composed of straight line segments.

3. Based on the above observation, the boundary of the figure is not composed of straight line segments, thus it does not satisfy the definition of a polygon.

4. Through the above reasoning, it can be confirmed that the figure is not a polygon because although it is closed, it does not have edges composed of straight line segments.

Polygon

A polygon is a closed figure in a plane formed by a finite number of line segments joined sequentially where each segment intersects exactly two other segments at its endpoints.

<note>The figure is a closed shape on a plane</note>. It consists of a continuous closed curve without using straight line segments, so it <note>is not a polygon</note>.


Question:
Is this a polygon?

- yes
- no

Answer: - no

1. Polygon: A polygon is a closed figure on a plane, formed by connecting multiple line segments in sequence, with each pair of line segments intersecting only at endpoints.

2. Check the properties of each segment in the given figure. The boundary of the figure is clearly formed by a curve, not multiple straight line segments.

3. Observe whether the boundary of the figure is closed. The figure shows a curve that fails to close, indicating that the figure is not closed.

4. Combine the analysis results from steps 2 and 3: The figure does not meet the definition of a polygon because it is neither composed of straight line segments nor completely closed.

5. Based on the above reasoning, the final answer is no.

<note>The figure shows an unclosed curve</note>, which is neither composed of multiple line segments, <note>nor a closed figure</note>, so it is not a polygon.


Question:
Is this shape a regular polygon?

- yes
- no

Answer: - yes

1. Given that all sides have a length of 78, this means that in this polygon, each side is equal.

2. Given that all interior angles are 144°, this means that in this polygon, each interior angle is equal.

3. According to the definition of a regular polygon, a regular polygon is a polygon with all sides and all interior angles equal.

4. Since all sides of this polygon are equal in length and each interior angle is also equal, it meets the definition of a regular polygon.

5. Based on the above analysis, the conclusion is: this polygon is a regular polygon.

Definition of Regular Polygon

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

In the figure of this problem, in a regular polygon, all sides are 78 in length, and all interior angles are 144°. Therefore, this polygon is a regular polygon.


Question:
What is TU?

TU= $\Box$

Answer: TU=1

1. Given that △RVT is inscribed in a circle, with points of tangency at Q, S, and U.

2. According to the tangent-segment theorem, the lengths of two tangent segments from a point to a circle are equal. Therefore, \\(\overline{RS} = \overline{RQ}\\) and \\(\overline{TU} = \overline{TS}\\).

3. In the figure, \\(\overline{RS} = \overline{RQ} = 1\\), \\(\overline{RT} = 2\\).

4. Thus, it can be known that: \\(\overline{RS} + \overline{ST} = \overline{RT}\\).

5. Substituting the known values: 1 + \\(\

Tangent-Segment Theorem

<note>From point R to the inscribed circle, the two tangent segments are RS and RQ</note>. According to the Tangent-Segment Theorem, <note>\(\overline{RS} = \overline{RQ}\)</note>. Similarly, <note>the tangent segments from point T are TS and TU</note>. According to the Tangent-Segment Theorem, <note>\(\overline{TU} = \overline{TS}\)</note>.


Question: The rectangle below is labeled with its measured dimensions. Taking measurement error into account, what is the percent error in its calculated area?Round your answer to the nearest tenth of a percent and include a percent sign (%). $\Box$

Answer: 8.2%

1. Given that the length of the rectangle is 13 yd and the width is 12 yd. Therefore, the measured area A_m = 13 × 12 = 156 square yd.

2. Each dimension is measured in units of 1 yd, so the maximum possible error for each measurement is 0.5 yd.

3. To find the maximum possible area A_Max, add the maximum possible error to each measurement: length is 13 + 0.5 = 13.5 yd, width is 12 + 0.5 = 12.5 yd. Calculating the maximum possible area gives A_Max = 13.5 × 12.5 = 168.75 square yd.

4. To find the minimum possible area A_Min, subtract the maximum possible error from each measurement: length is 13 - 0.5 = 12.5 yd, width is 12 - 0.5 = 11.5 yd. Calculating the minimum possible area gives A_Min = 12.5 × 11.5 = 143.75 square yd.

5. Calculate the difference between the minimum area and the measured area: D_Min = |A_m - A_Min| = |156 - 143.75| = 12.25 square yd.

6. Calculate the difference between the maximum area and the measured area: D_Max = |A_Max - A_m| = |168.75 - 156| = 12.75 square yd.

7. The larger of the two differences is the maximum possible error in the area: E = max(D_Min, D_Max) = 12.75.

8. Calculate the percentage error in the area: Percentage error = (E / A_m) × 100% = (12.75 / 156) × 100% ≈ 8.173%.

9. Round the percentage error to the nearest tenth: 8.173% ≈ 8.2%.

10. After the above reasoning, the final answer is 8.2%.

Definition of Rectangle

<note>A quadrilateral is a rectangle</note>, <note>its interior angles are all right angles (90 degrees)</note>, and <note>two sides with a length of 13 yards are parallel and equal in length</note>, <note>two sides with a length of 12 yards are parallel and equal in length</note>.

Formula for the Area of a Rectangle

In the figure of this problem, in the rectangle, <note>side 13yd is the length of the rectangle</note>, <note>side 12yd is the width of the rectangle</note>, so the area of the rectangle is equal to its <note>length 13 multiplied by its width 12</note>, that is, area = 13 * 12.


Question:
Is this a polygon?

- yes
- no

Answer: - no

1. First, we need to clarify the definition of a Polygon: a polygon is a simple closed figure formed by connecting several line segments end to end, with each endpoint of a line segment being an endpoint of another line segment.

2. Analyze the given figure in the problem and observe its boundary. It can be seen that this figure has a closed nature, but its boundary is not composed of line segments, but of curves.

3. Therefore, based on the definition of a polygon and the conditions of the problem, we conclude that this figure is not a polygon.

Polygon

<note>In the figure of this problem</note>, a polygon is a closed figure on a plane. <note>It is formed by connecting multiple line segments sequentially</note>, and each pair of line segments intersects only at endpoints. The figure is not composed of line segments, so it is not a polygon.


Question:
What is UV?

UV= $\Box$

Answer: UV=4

1. Given X is the tangent point, and WX is the tangent from W to the circle, WV is the tangent from W to the circle. According to the theorem that two tangents drawn from the same external point are equal, WX is equal to WV, so WX = VW = 5.

2. Given SW = 7, and WX is the segment from W to S, SX + WX = SW, so SX + 5 = 7, solving gives SX = 2.

3. S is the intersection point of the two tangents. According to the theorem that two tangents drawn from the same external point are equal, SX is equal to ST, so ST = SX = 2.

4. Given SU = 6, and ST is the segment from S to U, SU = ST + TU, so TU + 2 = 6, solving gives TU = 4.

5. U is the intersection point of the two tangents. According to the theorem that two tangents drawn from the same external point are equal, TU is equal to UV, so UV = TU = 4.

6. Through the above reasoning, the final answer is UV = 4.

Definition of Tangent to a Circle

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

<note>The circle and the line WX have exactly one common point X, which is called the point of tangency</note>. Therefore, the line WX is a tangent to the circle. Similarly, <note>WV</note>, <note>SX</note>, <note>ST</note>, <note>TU</note>, and <note>UV</note> are also tangents to the circle, each having exactly one common point with the circle, namely <note>V</note>, <note>X</note>, <note>T</note>, <note>U</note>, and <note>V</note> respectively.

Tangent-Segment Theorem

<note>From an external point W, two tangents WX and WV are drawn to the circle</note>, their tangent lengths are equal, i.e., <note>WX=WV=5</note>; <note>From an external point S, two tangents SX and ST are drawn to the circle</note>, their tangent lengths are equal, i.e., <note>SX=ST=2</note>; <note>From an external point U, two tangents TU and UV are drawn to the circle</note>, their tangent lengths are equal, i.e., <note>TU=UV=4</note>.


Question:
Is this shape a regular polygon?

- yes
- no

Answer: - no

1. The given figure is shown as a polygon with ten sides. According to the definition, a regular polygon is a convex polygon with all sides equal and all angles equal.

2. Based on observation, it is mentioned in the problem that some angles are greater than 90° while others are less than 90°. According to the definition, all angles of a regular polygon should be equal, so the interior angles of this polygon are not equal.

3. According to the definition of a regular polygon, all sides and angles must be equal, but here the angles are not equal. Therefore, this polygon is not a regular polygon.

4. Based on the above reasoning, the final answer is no.

Definition of Regular Polygon

<note>This figure is a polygon with ten sides</note>. According to the definition of a regular polygon, a regular polygon requires <note>all sides to be equal and all interior angles to be equal</note>. In the problem, <note>some angles are known to be greater than 90°, while other angles are less than 90°</note>, indicating that <note>not all angles are equal</note>. Therefore, <note>this polygon does not meet the definition of a regular polygon</note>.


Question:
What is EF?

EF= $\Box$

Answer: EF=4

1. From the given figure, it is known that △JHF is inscribed in a circle, indicating that JH, HF, FJ are tangents to the circle.

2. According to the tangent length theorem, we know EJ = IJ, GH = HI, FG = EF.

3. From the given data in the figure, we know IJ = 8, HJ = 11, FH = 7.

4. In △HJF, HJ = HI + IJ. That is, HI = HJ - IJ = 11 - 8.

5. Calculating, we get HI = 3.

6. According to the theorem in step 2, GH = HI, therefore GH = 3.

7. On the line GF, through GH + FG = FH, substituting the known values GH = 3, FH = 7, we get: FG = FH - GH = 7 - 3.

8. Calculating, we get FG = 4.

9. According to the theorem in step 2, EF = FG, therefore EF = 4.

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

Definition of Tangent to a Circle

<note>The circle and the lines JF, FH, HJ have exactly one common point E, G, I</note>, these common points are called points of tangency. Therefore, <note>the lines JF, FH, HJ are tangents to the circle</note>.

Tangent-Segment Theorem

<note>From a point J outside the circle, two tangents JE and JI are drawn</note>, their lengths are equal, i.e., <note>JE=JI</note>. Similarly, <note>from a point F outside the circle, two tangents FE and FG are drawn</note>, their lengths are equal, i.e., <note>FE=FG</note>. Similarly, <note>from a point H outside the circle, two tangents HG and HI are drawn</note>, their lengths are equal, i.e., <note>HG=HI</note>.


Question:
Look at this figure:What is the shape of its bases?

- rectangle
- circle
- heptagon
- octagon

Answer: - circle

1. Carefully observe the figure, the figure shows a green three-dimensional shape, which can be simply identified as a type of cylindrical shape.

2. Further analysis, the three-dimensional shape is not a prism, because its surface curve is smooth and continuous, without showing any polygonal sides. Therefore, it is more like a cylinder.

3. Confirming the shape as a cylinder involves using geometric knowledge: the two bases of a cylinder are completely identical and parallel plane figures.

4. In a cylinder, the base must be a closed circle, because the consistent cross-section of the cylinder is also circular, which matches the description of the shape in the figure without any corners or straight lines.

5. Therefore, based on the above analysis and the definition of a cylinder, it is determined that the base shape of the cylinder in the figure is circular.

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

Definition of Prism

A prism is a polyhedron with two parallel and congruent polygonal bases, and all other faces are parallelograms.

The two bases of the <note>green solid shape</note> are parallel and completely identical, and its sides are smooth and continuous without showing any polygonal sides, so it can be confirmed that it is a <note>prism</note>.

Cylinder

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

<note>In the figure of this problem, the cylinder consists of two parallel and identical circular bases and a lateral surface</note>.<note>The bases are two identical circles</note>, their radii and diameters are equal, and their centers are on the same line.<note>The lateral surface is a rectangle</note>, when unfolded, its height is equal to the height of the cylinder, and its width is equal to the circumference of the circle.

Definition of Circle

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

<note>The upper and lower bases of the cylinder are both circular</note>. The definition of a circle is the set of all points in a plane that are at a fixed distance from a fixed point. <note>The distance from all points on the base to the center of the circle is the radius</note>, thus it conforms to the definition of a circle.

Cylinder

<note>A cylinder consists of two parallel and identical circular bases and a lateral surface</note>.<note>The bases are two identical circles</note>,<note>their radii and diameters are equal</note>, and their centers lie on the same line. The lateral surface is a rectangle,<note>when unfolded, its height is equal to the height of the cylinder, and its width is equal to the circumference of the circle</note>.


Question:
What is QV?

QV= $\Box$

Answer: QV=10

1. Given △RTV, the incircle touches sides RT, RV, TV at points S, Q, U respectively. By the tangent-segment theorem, the tangents from point T to the circle, TS and TU, are equal, TS = TU = 4.

2. According to the tangent-segment theorem, the tangents from point V to the circle, QV and UV, are equal, i.e., QV = UV.

3. From the figure, TV = 14, with TU = 4, thus UV = TV - TU = 14 - 4 = 10.

4. Through the above calculation, we get UV = 10. According to the tangent-segment theorem, QV = UV, hence QV = 10.

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

Definition of Tangent to a Circle

<note>The circle and the line TS have exactly one common point S</note>, this common point is called the point of tangency. Therefore, the line TS is the tangent to the circle. Similarly, <note>The circle and the line TU have exactly one common point U</note>, this common point is called the point of tangency. Therefore, the line TU is the tangent to the circle.

Tangent-Segment Theorem

<note>From a point T outside the circle, two tangents TS and TU are drawn</note>, their tangent lengths are equal, i.e., <note>TS=TU=4</note>. <note>The line connecting the center of the circle and point T bisects the angle between the two tangents TS and TU</note>. Similarly, <note>from a point V outside the circle, two tangents QV and UV are drawn</note>, their tangent lengths are equal, i.e., <note>QV=UV</note>. Through <note>UV = TV - TU = 10 (this step uses 14 − 4 = 10)</note>, and from <note>QV = UV, it follows that QV = 10</note>.


Question:
What is VW?

VW= $\Box$

Answer: VW=8

1. Given that point R is the intersection of two tangents RS and RW, according to the tangent length theorem, we get the segment RS = RW, thus RS = RW = 10.

2. According to the figure, the length of RT is 15, we can derive ST + RS = RT.

3. Substituting the known values RS = 10 and RT = 15, we get ST + 10 = 15.

4. Subtracting 10 from both sides, we get ST = 5.

5. Given that point T is the intersection of two tangents TU and ST, according to the tangent length theorem, we get the segment TU = ST, thus TU = ST = 5.

6. According to the figure, the length of TV is 13, we can derive UV + TU = TV.

7. Substituting the known values TU = 5 and TV = 13, we get UV + 5 = 13.

8. Subtracting 5 from both sides, we get UV = 8.

9. Given that point V is the intersection of two tangents VW and UV, according to the tangent length theorem, we get the segment VW = UV, thus VW = UV = 8.

10. Through the above reasoning, we finally get the answer VW = 8.

Definition of Tangent to a Circle

In the figure of this problem, <note>the circle and the line segments RW, RT, TU, UV, and VW have only one common point</note>, these <note>common points are points R, T, U, and V, etc.</note>. Therefore, <note>the line segments RW, RT, TU, UV, and VW are tangents to the circle</note>.

Tangent-Segment Theorem

In the figure of this problem, from an external point R, two tangents to the circle <note>RS</note> and <note>RW</note> are drawn, and their tangent lengths are equal, i.e., <note>RS = RW = 10</note>; from an external point T, two tangents to the circle <note>ST</note> and <note>TU</note> are drawn, and their tangent lengths are equal, i.e., <note>ST = TU = 5</note>; from an external point V, two tangents to the circle <note>UV</note> and <note>VW</note> are drawn, and their tangent lengths are equal, i.e., <note>UV = VW = 8</note>.


Question:
The diagram shows a convex polygon.

What is the sum of the exterior angle measures, one at each vertex, of this polygon?
 $\Box$ °

Answer: 360°

1. Given that the polygon is a convex polygon, according to the definition of the exterior angle of a polygon, the measure of each exterior angle is positive, and the sum of any two exterior angles in the same direction (clockwise or counterclockwise) will not exceed 360°.

2. According to the exterior angle sum theorem of polygons, the sum of the exterior angles (one exterior angle at each vertex) of any polygon is equal to 360°, regardless of the number of sides of the polygon.

3. By reasoning from the convex polygon without exterior angle labels in the figure, it is known that it has the above properties, therefore the sum of all exterior angles of this polygon is 360°.

4. Through the above reasoning, the final answer is 360°.

Definition of Exterior Angle of a Polygon

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

<note>In the figure of this problem, one interior angle of the polygon is ∠ABC</note>, extending the adjacent sides of this interior angle <note>AB and BC to form the angle ∠DBE</note> is called the exterior angle of the interior angle <note>∠ABC</note>.

Exterior Angle Sum Theorem of Polygon

For any polygon, the sum of its exterior angles is equal to 360°.

In the figure of this problem, the given polygon is a <note>convex polygon</note>. According to the <note>Exterior Angle Sum Theorem of Polygon</note>, the sum of the exterior angles at each vertex of this <note>convex polygon</note> is equal to <note>360°</note>.


Question:
Is this shape a regular polygon?

- yes
- no

Answer: - yes

1. Definition of regular polygon: A regular polygon is a polygon with all interior angles equal and all sides equal.

2. Given that all sides of the polygon are 73, therefore all sides of the polygon are equal.

3. Given that all interior angles of the polygon are 140°, it indicates that all interior angles of the polygon are equal.

4. According to the definition of regular polygon, this polygon meets the conditions of all interior angles being equal and all sides being equal.

5. Based on the above reasoning, the final answer is yes.

Definition of Regular Polygon

<note>All sides of the polygon are 73</note>, therefore <note>all sides of the polygon are equal</note>; at the same time, <note>all interior angles of the polygon are 140°</note>, indicating that <note>all interior angles of the polygon are equal</note>.


Question:
Find the value of x in rectangle PQRS.

x= $\Box$

Answer: x=11

1. In rectangle PQRS, it is known that one pair of opposite sides has lengths \\overline{PQ}=4x and \\overline{RS}=x+33. According to the definition of a rectangle, the opposite sides are equal, thus \\overline{PQ}=\\overline{RS}.

2. From the conclusion in step 1 \\overline{PQ}=\\overline{RS}, we get the equation 4x=x+33.

3. By transposing terms to solve the equation 4x=x+33: move x to the left side of the equation, obtaining 4x-x=33.

4. Simplify the equation 4x-x=33 to 3x=33.

5. By dividing both sides by the coefficient 3, we solve for x=33/3.

6. We obtain x=11.

7. Therefore, through the above reasoning steps, the final answer is x=11.

Definition of Rectangle

In the diagram of this problem, quadrilateral PQRS is a rectangle, with its interior angles <note>∠PQR, ∠QRS, ∠RSP, ∠SPQ all being right angles (90 degrees)</note>, and <note>side PQ is parallel and equal in length to side RS</note>, <note>side QR is parallel and equal in length to side SP</note>. Therefore, according to the definition, the lengths of these two pairs of opposite sides are equal, i.e., <note>\overline{PQ} = \overline{RS}</note>.


Question:
Is this shape a regular polygon?

- yes
- no

Answer: - yes

1. Given that the shape in the figure is a triangle, the three interior angles are 60°, and all side lengths are 216.

2. According to the definition of a regular polygon: a regular polygon is a polygon with all sides of equal length and all interior angles equal.

3. In this problem, each interior angle of the triangle is 60°, which means all interior angles are equal.

4. At the same time, the three sides of this triangle are all 216 in length, indicating all sides are of equal length.

5. Therefore, it can be concluded that this triangle meets the definition of a regular polygon (more specifically, an equilateral triangle).

6. Based on the above reasoning, the final answer is yes.

Definition of Regular Polygon

In the figure of this problem, the shape is a triangle, <note>with three interior angles of 60° each</note>, <note>all sides are 216</note>, which indicates that the shape satisfies the definition of a regular polygon. Specifically, <note>the three interior angles of the triangle are 60°, 60°, and 60°</note>; <note>the three sides are 216, 216, and 216</note>.


Question:
Find the value of s in parallelogram FGHI.

s= $\Box$

Answer: s=13

1. Given FGHI is a parallelogram, according to the definition of a parallelogram, the opposite sides of a parallelogram are equal.

2. From step 1, we know that segment FG is equal to segment HI, i.e., FG = HI.

3. Given the length of segment FG is s+26 and the length of segment HI is 3s, according to step 2, we have the equation s+26 = 3s.

4. Solve the equation s+26 = 3s:

5. Move s to the other side to get 26 = 3s - s.

6. Combine like terms to get 26 = 2s.

7. Divide both sides by 2 to get s = 13.

8. Through the above reasoning, the final answer is s = 13.

Definition of Parallelogram

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

<note>Quadrilateral FGHI is a parallelogram</note>, <note>side FG is parallel and equal to side HI</note>, <note>side FI is parallel and equal to side GH</note>.


Question:
Is this shape a regular polygon?

- yes
- no

Answer: - no

1. It is known that the figure is a pentagon. According to the definition of a regular polygon, a regular polygon is a polygon with all interior angles equal and all sides of equal length.

2. From the figure's labels, three of the angles are 82°, 134°, and 121° respectively.

3. Since the interior angles of the pentagon are not all the same, this indicates that it is not a regular pentagon. A regular pentagon should have equal interior angles. The interior angle of a regular pentagon should be 108° (according to the formula for the central angle and interior angle of a regular polygon: the formula for the interior angle is (n-2) * 180° / n, where n is the number of sides, thus for a pentagon it is (5-2) * 180° / 5 = 108°).

4. Similarly, checking the lengths of each side of the pentagon, some sides are labeled as 88, while another side is labeled as 72.

5. Based on the above results of unequal angle degrees and side lengths, the sides of this polygon are also not all equal.

6. The above analysis shows that this pentagon does not meet the conditions of a regular polygon. Therefore, this figure is not a regular polygon.

7. Through the above reasoning, the final answer is no, this figure is not a regular polygon.

Definition of Regular Polygon

<note>In the figure of this problem, the interior angles of the pentagon are 82°, 134°, 121°, 121°, 82°.</note> Since the interior angles are not equal, this pentagon does not meet the definition conditions of a regular polygon. Additionally, <note>the lengths of the sides are 88, 88, 88, 72, 88.</note> Since the side lengths are not all equal, this pentagon also does not meet the definition conditions of a regular polygon. Therefore, this figure is not a regular polygon.

Formulas for the Central Angle and Interior Angle of a Regular Polygon

For a regular polygon with \( n \) sides, the measure of each central angle is given by \( \frac{360^\circ}{n} \). The measure of each interior angle is given by \( \frac{(n - 2) \cdot 180^\circ}{n} \).

In this problem, <note>n is the number of sides of the pentagon, which is 5</note>. According to the formula, <note>the interior angle of a regular pentagon is (5-2) * 180° / 5 = 108°</note>. The problem gives <note>the interior angles of the pentagon as 82°, 134°, 121°, 121°, 82°</note>, which do not match <note>the calculated interior angle of a regular pentagon, 108°</note>. Therefore, this shape is not a regular pentagon.


Question:
Find the value of c in parallelogram HIJK.

c= $\Box$

Answer: c=23

1. Given that in parallelogram HIJK, the length of side HK is 2c and the length of side IJ is c+23.

2. According to the definition of a parallelogram, opposite sides are equal, i.e., HI = JK and HK = IJ.

3. Therefore, based on HK = IJ, we can establish the equation: 2c = c + 23.

4. Solving the equation: subtracting c from both sides of the equation, we get c = 23.

5. Through the above reasoning, the final answer is c = 23.

Definition of Parallelogram

<note>Quadrilateral HIJK is a parallelogram</note>, <note>side HI is parallel and equal to side JK</note>, <note>side HK is parallel and equal to side IJ</note>.


Question:
Find the value of y in rhombus CDEF.

y= $\Box$ °

Answer: y=31°

1. Given that CDEF is a rhombus, according to the property that adjacent angles of a rhombus are supplementary, we can conclude that the adjacent angles of the rhombus are supplementary. Therefore, ∠C and ∠D are supplementary. In other words, ∠C + ∠D = 180°.

2. According to the problem statement, ∠C = y + 25°, ∠D = 4y.

3. Since ∠C and ∠D are adjacent angles, we have the equation: (y + 25°) + 4y = 180°.

4. Simplify the equation: (y + 25°) + 4y = 180°

5. Combine like terms: 5y + 25° = 180°

6. Subtract 25° from both sides of the equation, we get: 5y = 155°

7. Divide both sides of the equation by 5, we get: y = 31°

8. Through the above reasoning, we finally obtain the answer y = 31°.

Definition of Rhombus

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

<note>In quadrilateral CDEF, all sides CD, DE, EF, and FC are equal</note>, therefore quadrilateral CDEF is a rhombus. Additionally, <note>the diagonals CE and DF of quadrilateral CDEF bisect each other perpendicularly</note>.

Adjacent Angles of Rhombus are Supplementary

Any pair of adjacent angles in a rhombus are supplementary.

In the diagram of this problem, in rhombus CDEF, <note>angle C and angle D are a pair of adjacent angles</note>, according to the properties of a rhombus, <note>any pair of adjacent angles of a rhombus are supplementary</note>, that is, <note>angle C + angle D = 180 degrees</note>. Similarly, <note>angle E and angle F are also a pair of adjacent angles</note>, satisfying <note>angle E + angle F = 180 degrees</note>.


Question:
What is KL?

KL= $\Box$

Answer: KL=4

1. Observe the figure, it is known that OP and NO are tangents to the circle, therefore according to the tangent length theorem, it can be concluded that OP = NO. According to the given condition ON = 9.

2. According to the problem, KO = 13, now it is known that OP = ON = 9, therefore we can calculate the length of KP. Thus, we get: KP + OP = KO, KP + 9 = 13.

3. Through equation operations, KP = 13 - 9 = 4.

4. Since K is the point of tangency of the two tangents KP and KL, according to the tangent length theorem, it follows that KP = KL.

5. Based on the result in step 3 and the tangent length theorem, we get KP = 4, according to the above conditions KL = 4.

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

Definition of Tangent to a Circle

In the figure of this problem, <note>circle C and line OP have only one common point P, which is called the point of tangency</note>. Therefore, <note>line OP is the tangent to circle C</note>. Similarly, <note>circle C and line NO have only one common point N, which is called the point of tangency</note>. Therefore, <note>line NO is the tangent to circle C</note>.

Tangent-Segment Theorem

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

<note>From an external point K, two tangents KP and KL are drawn to the circle</note>, and their tangent lengths are equal, i.e., <note>KP=KL</note>. <note>The line connecting the center of the circle C and this point K</note> bisects the angle between the two tangents KP and KL, i.e., <note>angle PKC = angle LKC</note>. Similarly, from <note>an external point O, two tangents OP and ON are drawn to the circle</note>, and their tangent lengths are equal, i.e., <note>OP=ON</note>. <note>The line connecting the center of the circle C and this point O</note> bisects the angle between the two tangents OP and ON, i.e., <note>angle POC = angle NOC</note>.


Question:
Find the lateral area of the triangular prism.

 $\Box$  square yards

Answer: 96 square yards

1. First, identify that the base of the triangular prism is a right triangle, with the three sides of this right triangle being 3 yards, 4 yards, and 5 yards respectively.

2. Calculate the perimeter of the base triangle using the triangle perimeter formula Perimeter = a + b + c, where a, b, and c are the sides of the triangle, resulting in Perimeter = 3 + 4 + 5 = 12 yards.

3. Using the lateral area formula for prisms, multiply the perimeter of the base triangle by the height of the prism, where the height of the prism is 8 yards.

4. Use the prism lateral area formula Lateral Area = Perimeter × Height, substituting the known perimeter and height into the formula to get Lateral Area = 12 × 8 = 96.

5. Therefore, the lateral area of the prism is 96 square yards.

Definition of Right Triangle

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

In the figure of this problem, in the base triangle, <note>the angle is a right angle (90 degrees)</note>, so the base triangle is a right triangle. <note>The 3-yard side and the 4-yard side are the legs</note>, <note>the 5-yard side is the hypotenuse</note>.

Definition of Prism

<note>In the figure of this problem</note>, given is <note>a triangular prism</note>, <note>the base is a right triangle</note>, and the three lateral faces are <note>rectangles</note>.

Lateral Surface Area Formula of a Prism

The lateral surface area of a prism is equal to the perimeter of the base multiplied by the height.

<note>Lateral Surface Area Formula LA = P × H</note>, substituting <note>perimeter 12 yards</note> and <note>height 8 yards</note>, we get <note>Lateral Surface Area LA = 12 × 8 = 96 square yards</note>.

Formula for the Perimeter of a Triangle

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

The original text: <note>The three sides are 3 yards, 4 yards, and 5 yards</note>, according to the formula for the perimeter of a triangle, i.e., Perimeter L = <note>3 yards + 4 yards + 5 yards</note>.


Question:
Find the value of c in parallelogram STUV.

c= $\Box$

Answer: c=2

1. Given: Parallelogram STUV.

2. According to the definition of a parallelogram, sides ST and UV are equal, i.e., \\(\overline{ST} = \overline{UV}\\).

3. The problem states \\(\overline{ST} = c + 16\\) and \\(\overline{UV} = 9c\\).

4. Substitute \\(\overline{ST}\\) and \\(\overline{UV}\\) into the equation: \\(\c + 16 = 9c\\).

5. Solve the equation \\(\c + 16 = 9c\\), first subtract c from both sides to get: \\(\16 = 8c\\).

6. Divide both sides by 8 to get: \\(\c = 2\\).

7. Through the above reasoning, the final answer is \\(\c = 2\\).

Definition of Parallelogram

In the figure of this problem, the quadrilateral <note>STUV is a parallelogram</note>, <note>side ST is parallel and equal to side UV</note>, <note>side SU is parallel and equal to side TV</note>. According to the information given in the problem, <note>\( \overline{ST} = c + 16 \)</note>, and <note>\( \overline{UV} = 9c \)</note>, the two are equal, i.e., <note>\( c + 16 = 9c \)</note>.


Question:
Find the value of x in parallelogram TUVW.

x= $\Box$

Answer: x=28

1. In quadrilateral TUVW, according to the problem statement, the length of side TW is x+28, and the length of side UV is 2x.

2. Given that TUVW is a parallelogram, according to the definition of a parallelogram, which states that opposite sides are equal, we have TW = UV.

3. Substitute the given conditions into the equation TW = UV, resulting in the equation: x + 28 = 2x.

4. Solve the equation x + 28 = 2x, first subtract x from both sides to get 28 = x.

5. Through the above reasoning, the final answer is x = 28.

<note>Quadrilateral TUVW is a parallelogram</note>, side <note>TW is parallel and equal to side UV</note>, side <note>TU is parallel and equal to side VW</note>. In the figure, side <note>TW has a length of x+28</note>, side <note>UV has a length of 2x</note>, according to the definition, <note>TW=UV</note>.


Question:
Is this shape a regular polygon?

- yes
- no

Answer: - no

1. Observing the given polygon, we find that it is a pentagon, with its interior angles and side lengths as follows: angles—111°, 152°, 111°, 83°, 83°; side lengths—57, 130, 57, 83, 83.

2. According to the definition of a regular polygon, all interior angles should be equal, and all side lengths should be equal.

3. By checking each interior angle, we find that the interior angles of this polygon are 111°, 152°, 111°, 83°, 83°.

4. Since at least one angle among angle A, angle B, angle C, angle D, angle E (152°) is different from the other angles (83° and 111°), the interior angles of this polygon are not equal.

5. Next, checking the side lengths of the polygon, we find that the side lengths are 57, 130, 57, 83, 83.

6. Since at least one side length (130) is different from the other values (57 and 83), the side lengths of this polygon are not equal.

7. Based on the above analysis, the interior angles and side lengths of this pentagon are not equal.

8. Since the definition of a regular polygon requires all angles and side lengths to be equal, and this polygon does not meet these conditions, it is not a regular polygon.

9. Through the above reasoning, the final answer is no.

Definition of Regular Polygon

<note>The definition of a regular polygon requires that all interior angles and side lengths of a pentagon must be equal</note>. However, the interior angles of the pentagon in this problem are <note>111°, 152°, 111°, 83°, 83°</note>, and the side lengths are <note>57, 130, 57, 83, 83</note>. Therefore, it does not meet the definition of a regular polygon.


Question:
Is this shape a regular polygon?

- yes
- no

Answer: - yes

1. Given that all sides of the hexagon are 103, according to the definition of a polygon, if all sides of a polygon are equal, the polygon is called an equilateral polygon. Therefore, this hexagon is an equilateral hexagon.

2. Given that each interior angle is 120°, according to the definition of a polygon, if all interior angles of a polygon are equal, the polygon is called an equiangular polygon. Therefore, this hexagon is an equiangular hexagon.

3. According to the definition of a regular polygon, a regular polygon must satisfy both equilateral and equiangular conditions simultaneously.

4. From steps 1 and 2, it can be concluded that this hexagon satisfies the dual conditions of a regular polygon, i.e., all sides are equal and all angles are equal.

5. Through the above reasoning, the final answer is: Yes, this figure is a regular polygon.

<note>In the figure of this problem, in the regular hexagon</note>, all sides are <note>103</note>, and all interior angles are <note>120°</note>. Therefore, this hexagon is a regular polygon.


Question:
$\overline{CD}$ is shown on the graph below. $\overline{CD}$ is dilated by a scale factor of 1/3 centered at (3,–5) to create $\overline{C'D'}$ .

What is the length of $\overline{C'D'}$ ?
Write your answer as a whole number or as a decimal rounded to the nearest tenth.

 $\Box$  units

Answer: 4.1 units

1. According to the problem statement, given the line segment \overline{CD}, its endpoints are C(-4, 3) and D(8, 1).

2. To calculate the length of the line segment \overline{CD}, we can use the distance formula d = \sqrt{(x2 - x1)^2 + (y2 - y1)^2}.

3. Substitute the coordinates of the endpoints C(-4, 3) and D(8, 1) into the formula: d = \sqrt{(8 - (-4))^2 + (1 - 3)^2}.

6. Substitute the results: d = \sqrt{144 + 4} = \sqrt{148}.

7. \sqrt{148} is approximately equal to 12.166, so the length of \overline{CD} is approximately 12.166 units.

8. According to the problem statement, \overline{CD} is scaled through the center located at (3, -5) with a scaling factor of \frac{1}{3}.

9. According to the definition of the scaling factor in the Cartesian coordinate system, the scaling factor is \frac{1}{3}, and the length of \overline{C'D'} is \frac{1}{3} \times \text{length} \overline{CD}.

11. Round the result to the nearest tenth, obtaining the length of \overline{C'D'} as 4.1.

12. Based on the above reasoning, the final answer is 4.1.

Distance Formula Between Two Points

In a Cartesian coordinate plane, the distance \( d \) between two points \((x_1, y_1)\) and \((x_2, y_2)\) can be calculated using the formula \( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \).

The length of the line segment <note>\overline{CD}</note> can be calculated using the coordinates of its two endpoints <note>C(-4, 3)</note> and <note>D(8, 1)</note>.

Definition of Scale Factor in Rectangular Coordinate Systems

In a rectangular coordinate system, if a polygon xxx is transformed to xxx by scaling, then the scale factor is defined as the ratio of the length of any side of the scaled polygon to the length of the corresponding side of the original polygon.

<note>After scaling, the line segment CD becomes the line segment C'D'</note>, scale factor = <note>length of C'D' / length of CD</note>, i.e., scale factor = <note>1/3</note>.


Question:
Find the value of p in rhombus EFGH.

p= $\Box$ °

Answer: p=62°

1. Given that quadrilateral EFGH is a rhombus, according to the property that adjacent angles of a rhombus are supplementary, any two pairs of adjacent angles in EFGH are supplementary angles. Therefore, ∠E and ∠H are supplementary angles.

2. According to the property of supplementary angles in a rhombus, the sum of the measures of supplementary angles is 180°. Therefore, ∠E + ∠H = 180°.

3. According to the given conditions, ∠E = p and ∠H = p + 56°.

4. Substitute the known angle measures into the equation for supplementary angles, we get: p + (p + 56°) = 180°.

5. Simplify the equation to get 2p + 56° = 180°.

6. Subtract 56° from both sides of the equation to get: 2p = 124°.

7. Divide both sides of the equation by 2 to get: p = 62°.

8. Through the above reasoning, the final answer is 62°.

Definition of Rhombus

In the figure of this problem, <note>quadrilateral EFGH is a rhombus</note>, so <note>all sides EF, FG, GH, HE are equal</note>. Additionally, <note>the diagonals EG and FH of quadrilateral EFGH bisect each other perpendicularly</note>.

In the figure of this problem, in the rhombus <note>EFGH</note>, <note>angle E and angle H are a pair of adjacent angles</note>. According to the properties of the rhombus, any pair of adjacent angles of a rhombus are supplementary, that is, <note>angle E + angle H = 180 degrees</note>. Similarly, <note>angle G and angle F are also a pair of adjacent angles</note>, satisfying <note>angle G + angle F = 180 degrees</note>.


Question:
Look at this figure:What is the shape of its bases?

- triangle
- rectangle
- heptagon
- decagon

Answer: - rectangle

1. Observe the figure and determine that this is a cuboid. A cuboid is a special type of trapezoidal prism with two parallel and congruent bases.

2. According to the definition of a cuboid, its base is a type of parallelogram, and in a cuboid, the base usually appears as a rectangle.

3. By observing all the faces of the cuboid, each pair of opposite faces are congruent rectangles, so its base is also a rectangle.

4. Based on the above analysis, it is clear that the shape of the base of the cuboid is a rectangle.

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

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>In the figure of this problem, the shape is identified as a rectangular prism</note>, having six rectangular faces, with opposite faces being congruent and parallel. Among these rectangular faces, by observing <note>any two opposite faces, such as the top and bottom faces</note>, it can be determined that their shape is <note>rectangular</note>.

Definition of Rectangle

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

<note>Each face of a cuboid is a rectangle</note>, including its base. <note>The base meets the definition of a rectangle</note>, having <note>four right angles</note>, <note>opposite sides parallel and equal</note>. The base of the cuboid is <note>a quadrilateral</note>, in which <note>each interior angle is a right angle (90 degrees)</note>, and <note>opposite sides are parallel and equal in length</note>.


Question:
Find the value of w in rectangle ABCD.

w= $\Box$

Answer: w=11

1. Given ABCD is a rectangle, according to the definition of a rectangle, the opposite sides are parallel and equal in length. Therefore, \\overline{AD} = \\overline{BC} and \\overline{AB} = \\overline{CD}.

2. According to the problem statement, \\overline{AD} = w + 11, \\overline{BC} = 2w.

3. According to the property in step 1, \\overline{AD} = \\overline{BC}, substitute the values of \\overline{AD} and \\overline{BC}, we get the equation: w + 11 = 2w.

4. Solve the equation from step 3: first subtract w from both sides of the equation, we get 11 = 2w - w.

5. Simplify the result from step 4, we get 11 = w.

6. Through the above reasoning, the final answer is w = 11.

<note>Quadrilateral ABCD is a rectangle</note>, with its interior angles <note>∠DAB, ∠ABC, ∠BCD, ∠CDA all being right angles (90 degrees)</note>, and <note>side AD is parallel and equal in length to side BC</note>, <note>side AB is parallel and equal in length to side CD</note>. The specific side length information given in the problem is <note>AD = w + 11 and BC = 2w</note>.


Question:
Is parallelogram STUV a rhombus?

- yes
- no

Answer: - no

1. Given that quadrilateral STUV is a parallelogram, and according to the definition of a parallelogram, the corresponding sides are parallel and equal.

2. The diagonals SU and TV of quadrilateral STUV intersect at point R.

3. According to the properties of the diagonals of a rhombus, if STUV is a rhombus, then the diagonals should be perpendicular to each other.

4. It is now known that ∠TRU = 95°.

5. To determine whether SU and TV are perpendicular, we need ∠TRU = 90°, but the actual situation is ∠TRU = 95°.

6. Therefore, it can be concluded that SU and TV are not perpendicular.

7. Since SU and TV are not perpendicular, based on the definition of a rhombus, which requires the diagonals to be perpendicular to each other, it can be concluded that STUV is not a rhombus.

8. Through the above reasoning, the final conclusion is that STUV is not a rhombus.

Definition of Parallelogram

<note>Quadrilateral STUV</note> is a parallelogram, <note>side ST is parallel and equal to side UV</note>, <note>side SV is parallel and equal to side TU</note>.

Definition of Rhombus

<note>STUV is a rhombus</note>, then <note>sides ST, TU, UV, and VS are equal</note>, and <note>diagonals SU and TV intersect perpendicularly at point R</note>, and <note>∠TRU should be a right angle (90 degrees)</note>.

Properties of the Diagonals of a Rhombus

In a rhombus, the diagonals bisect each other and are perpendicular to each other.

In the figure of this problem, in rhombus STUV, <note>the diagonals SU and TV bisect each other</note> and <note>are perpendicular to each other</note>. Specifically, <note>point R is the intersection point of diagonals SU and TV</note>, and <note>SR = RU and TR = RV</note>. Additionally, <note>angles ∠TRU and ∠SRV are both right angles (90 degrees)</note>, so <note>the diagonals SU and TV are perpendicular to each other</note>.


Question:
Is parallelogram UVWX a rhombus?

- yes
- no

Answer: - no

1. Given quadrilateral UVWX is a parallelogram. In a parallelogram, the diagonals bisect each other.

2. Let ∠UTX = 94°. In a rhombus, the diagonals are perpendicular to each other. Therefore, if the parallelogram UVWX is a rhombus, then ∠UTX should be equal to 90°.

3. According to the problem, ∠UTX = 94°, not 90°, so the diagonals UW and VX are not perpendicular to each other.

4. Since the diagonals are not perpendicular, according to the properties of the diagonals in a rhombus, the parallelogram UVWX cannot be a rhombus.

5. Based on the above reasoning, the final answer is no.

Definition of Parallelogram

In the figure of this problem, the quadrilateral <note>UVWX</note> is a parallelogram, with side <note>UV</note> parallel and equal to side <note>WX</note>, and side <note>VW</note> parallel and equal to side <note>UX</note>.

Definition of Rhombus

<note>Quadrilateral UVWX is a rhombus</note>, then all sides <note>UV, VW, WX, XU should be equal</note>. Additionally, the diagonals <note>UW and VX</note> of quadrilateral UVWX should <note>perpendicularly bisect each other</note>, i.e., <note>diagonals UW and VX intersect at point T</note>, and <note>angle ∠UTX is a right angle (90 degrees)</note>, and <note>UT = TW and VT = TX</note>.

In the rhombus <note>UVWX</note>, the diagonals <note>UW and VX</note> bisect each other and are perpendicular to each other. Specifically, <note>point T is the intersection point of diagonals UW and VX, and UT = TW and VT = TX</note>. Additionally, <note>angle ∠UTX and angle ∠WTX are both right angles (90 degrees)</note>, so the diagonals <note>UW and VX are perpendicular to each other</note>.