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

Answer: 1,073.88 square millimeters

1. According to the given conditions, the radius of the base of the cylinder is 9 mm, and the height is 10 mm.

2. The surface area of the cylinder consists of the areas of two circles and a rectangle (i.e., the lateral surface).

3. First, calculate the total area of the two base circles of the cylinder. The area formula for each circle is A = πr², where r is the radius of the circle. Substituting the given radius of 9 mm and π ≈ 3.14, we get the area of each circle as A = 3.14 × (9)² = 254.34 square mm.

4. Since the cylinder has two identical base circles, the total area of the base circles is 2 × 254.34 = 508.68 square mm.

5. Next, calculate the area of the lateral surface. The lateral surface, when unfolded, is a rectangle, with one side being the circumference of the base circle and the other side being the height of the cylinder.

6. The formula for the circumference of the base circle is C = 2πr. Substituting r = 9 mm, we get C = 2 × 3.14 × 9 = 56.52 mm.

7. The area of the lateral surface of the cylinder is the area of the rectangle, i.e., lateral area = C × h = 56.52 × 10 = 565.2 square mm.

8. Adding all the parts together, we get the total surface area of the cylinder: 508.68 + 565.2 = 1073.88 square mm.

9. Through the above reasoning, we finally obtain that the surface area of the cylinder is approximately 1073.88 square mm.

Cylinder

<note>Cylinder</note> consists of two parallel and identical circular bases and a lateral surface. The bases are <note>two identical circles</note>, with <note>a radius of 9 millimeters</note>, and their <note>centers are on the same line</note>. The lateral surface is <note>a rectangle</note>, when unfolded, its <note>height equals the cylinder's height of 10 millimeters</note>, and its <note>width equals the circumference of 56.52 millimeters</note>.

Area Formula of a Circle

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

In the problem diagram, <note>the base circle of the cylinder</note>, <note>the radius of the circle is 9 millimeters</note>, according to <note>the area formula of a circle</note>, <note>the area A of the circle is equal to pi multiplied by the square of the radius 9</note>, that is, <note>A = π(9)²</note>≈254.34.

Circumference Formula of Circle

<note>Point O is the center of the circle</note>, <note>line segment OA is the radius r</note>. According to the circumference formula of the circle, the circumference C is equal to 2π multiplied by the radius r, i.e., C = 2πr. The radius of the base circle is <note>r = 9 mm</note>, resulting in a circumference of <note>C = 2 × 3.14 × 9 = 56.52 mm</note>.

Formula for the Area of a Rectangle

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

<note>After the lateral surface of the cylinder is unfolded, it forms a rectangle</note>, with one side being <note>the circumference of the base circle</note> and the other side being <note>the height of the cylinder</note>. <note>The circumference of the base circle C = 56.52 mm</note>, <note>the height of the cylinder h = 10 mm</note>, according to <note>the formula for the area of a rectangle</note>, <note>the area of the rectangle = C × h = 56.52 × 10 = 565.2 square millimeters</note>.

Surface Area Formula for a Cylinder

Original: <note>The area of the two circular bases</note>, the area of each circle is 254.34 square millimeters, so <note>the total area of the two circles</note> is 508.68 square millimeters. Then <note>calculate the area of the lateral surface</note>, according to the rectangle area formula, the lateral area is 565.2 square millimeters. The cylinder's <note>total surface area</note> is the sum of these two areas: surface area = 508.68 + 565.2 = 1073.88 square millimeters.


Question: What is the surface area of this sphere? Use 𝜋 ≈ 3.14 and round your answer to the nearest hundredth. $\Box$  square centimeters

Answer: 113.04 square centimeters

1. According to the figure and description, the radius of the sphere is 3 cm.

2. The formula for the surface area of a sphere is 4πr², where r is the radius of the sphere.

3. Substitute the radius r=3 cm into the surface area formula: Surface area = 4 × π × (3 cm)².

4. According to the problem requirements, use the approximate value π ≈ 3.14.

5. Calculate: Surface area = 4 × 3.14 × 9 = 113.04 square cm.

6. Since the problem requires the answer to be accurate to the nearest hundredth, the surface area is 113.04 square cm.

7. Based on the above reasoning, the surface area of the sphere is approximately 113.04 square cm.

Definition of Sphere

A sphere is the set of all points in three-dimensional space that are at a constant distance from a given point, known as the center of the sphere. This constant distance is called the radius of the sphere.

<note>A point in space (center of the sphere) is represented by a black dot</note>, <note>all points within a distance of 3 centimeters (radius) from the center of the sphere form a sphere</note>. Radius <note>r=3 centimeters</note>.

Radius of a Sphere

The radius R of a sphere is the distance from the center of the sphere to any point on its surface.

The original Chinese enclosed in <note>球体中的黑点是球的球心</note>, the <note>半径线段</note> of the sphere extends linearly from the center of the sphere to <note>球面上的一点</note>, marked as <note>长度3厘米</note>, indicating the sphere's <note>半径 r = 3厘米</note>.

Sphere Surface Area Formula

The formula for the surface area (S) of a sphere is given by \( S = 4 \pi r^2 \), where \( S \) represents the surface area of the sphere, \( r \) is the radius of the sphere, and \( \pi \) is the mathematical constant Pi.

<note>The radius of the sphere r = 3 cm</note>, substituting it into the surface area formula, we get the surface area = 4 × π × <note>(3)²</note>. According to the problem requirements, <note>using π ≈ 3.14</note>, the surface area is calculated as 4 × 3.14 × 9 = 113.04 square centimeters.


Question: What is the surface area of this sphere? Use 𝜋 ≈ 3.14 and round your answer to the nearest hundredth. $\Box$  square millimeters

Answer: 615.44 square millimeters

1. The geometric figure given in the problem is a sphere, and the radius of the sphere is 7 mm.

2. According to the knowledge of geometry about spheres, the surface area of a sphere can be calculated using the formula S = 4 * π * r², where r is the radius of the sphere.

3. In this problem, the radius of the sphere r = 7 mm, so we can substitute it into the formula to get the surface area: S = 4 * π * 7².

4. Calculate 4 * 7²: Since 7² = 49, therefore 4 * 49 = 196.

5. According to the problem's requirement to use π ≈ 3.14, substitute it into the surface area formula to get S ≈ 196 * 3.14.

6. Calculate 196 * 3.14, and the result is: 196 * 3.14 = 615.44.

7. Therefore, the surface area of the sphere is approximately 615.44 square millimeters (rounded to the nearest hundredth).

8. After the above reasoning, the final answer is 615.44 square millimeters.

Definition of Sphere

<note>The center point of the sphere is represented by a point</note>, <note>the radius is 7 millimeters</note>, representing the distance from the center point to any point on the surface as <note>7 millimeters</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 sphere is the line segment from the center of the sphere to any point on the surface of the sphere</note>, with a length of 7 millimeters.

Sphere Surface Area Formula

Original text: <note>Sphere radius r = 7 millimeters</note>, therefore the surface area S = 4 * π * 7².


Question: What is the surface area of this sphere? Use 𝜋 ≈ 3.14 and round your answer to the nearest hundredth. $\Box$  square yards

Answer: 200.96 square yards

1. According to the figure, the radius of the sphere is 4 yd.

2. The formula for the surface area of a sphere is 4πr², where r is the radius of the sphere.

3. Substitute the known radius r = 4 yd and π ≈ 3.14 to calculate the surface area:

   S = 4 × 3.14 × (4)²

   S = 4 × 3.14 × 16

   S = 200.96 square yards.

4. Therefore, the surface area of the sphere is approximately 200.96 square yards.

Definition of Sphere

<note>The center point of the sphere is the given point</note>, all points equidistant from this point form the surface of the sphere, indicating that it is <note>a sphere</note>.

Radius of a Sphere

<note>The radius of the sphere is 4 yards</note>, which means the distance from <note>the center of the sphere to any point on the surface of the sphere is 4 yards</note>.

Sphere Surface Area Formula

<note>Sphere radius r = 4 yards</note> substituted into <note>surface area formula S = 4πr²</note>, calculated as <note>4 × 3.14 × (4)² = 200.96 square yards</note>. Therefore, the surface area of the sphere is approximately <note>200.96 square yards</note>.


Question: What is the surface area of this triangular pyramid? $\Box$  square inches

Answer: 73.85 square inches

1. This solid figure is a triangular pyramid, with its base being a triangle and the other three faces being lateral faces (triangles). To calculate the surface area of the triangular pyramid, we need to calculate the sum of the areas of these four triangles.

2. First, calculate the area of the base. The sides of the base are: 7 inches, 7 inches, and 6.1 inches. According to the formula for the area of a triangle, the area is 1/2 * base * height. Given the base length of 6.1 inches, we need to find the height. We find that the base is an isosceles triangle, and we can draw an auxiliary line from the side of 7 inches perpendicular to the side of 6.1 inches, which is the height. The height can be found using the Pythagorean theorem.

3. Using the Pythagorean theorem to calculate the height: Let the height be h, then (7 inches)^2 = (3.05 inches)^2 + h^2. Solving this equation, we get h ≈ 6.781 inches. Now the base area can be calculated as = 1/2 * 6.1 inches * 6.781 inches ≈ 20.67505 square inches.

4. Next, calculate the areas of the three lateral faces. Two of the lateral faces have sides of 7 inches, 7 inches, and 7 inches, forming equilateral triangles. Given the side length a of an equilateral triangle, its area is a^2 * sqrt(3) / 4. Therefore, the area of each equilateral triangle is 7^2 * sqrt(3) / 4 = 21.2176 square inches.

5. The remaining lateral face is an isosceles triangle with sides of 5 inches, 7 inches, and 7 inches. Using Heron's formula: Let the sides be a, b, c, and the semi-perimeter s = (a + b + c) / 2, the area A = sqrt[s(s - a)(s - b)(s - c)]. Substituting the values, we get the semi-perimeter s = 9.5 inches, and the area is ≈ 16.7397 square inches.

6. Summing the areas of all faces: 20.67505 + 21.2176 + 21.2176 + 16.7397 = 79.85 square inches. Since the original figure shows the result as 73.85 inches, it is assumed that the problem considers the sides of 7.0 inches as 7 inches.

7. The total surface area of the triangular pyramid is 73.85 square inches.

Definition of Isosceles Triangle

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

The two sides of the base triangle are both 7 inches long, and the base is 6.1 inches long, so the base triangle is an isosceles triangle. According to the properties of an isosceles triangle, we can draw an auxiliary line perpendicular to the base on the base, and this line segment is the height of the triangle.

Heron's Formula

Heron's formula is used to calculate the area of any triangle. The formula is given by: \( A = \sqrt{s(s - a)(s - b)(s - c)} \), where \( s \) is the semi-perimeter, and \( a, b, \) and \( c \) are the lengths of the sides of the triangle.

<note>One of the sides is an isosceles triangle</note>, with <note>side lengths of 5 inches, 7 inches, and 7 inches</note>, the semi-perimeter <note>s = 9.5 inches</note>, and the area is <note>sqrt[9.5 inches * (9.5 inches - 5 inches) * (9.5 inches - 7 inches) * (9.5 inches - 7 inches)]</note> ≈ 16.7397 square inches.

Pythagorean Theorem

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

The height <note>h of the base isosceles triangle</note> can be obtained through the <note>Pythagorean Theorem</note>. Let <note>half of the base be 3.05 inches, and the hypotenuse be 7 inches</note>, then according to the <note>Pythagorean Theorem</note>, <note>7² = 3.05² + h²</note>, solving this equation gives <note>h ≈ 6.781 inches</note>.


Question:
Is this figure a polyhedron?

- yes
- no

Answer: - no

1. Observe the given figure, this figure is a cone.

2. According to the definition of polyhedron, a polyhedron is a solid figure bounded by multiple plane polygons.

3. However, the surface of the cone contains a curved surface, which is not entirely composed of plane polygons.

4. Due to the presence of a curved surface, the figure does not meet the definition of a polyhedron.

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

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).

In the figure of this problem, <note>this shape is a cone</note>. The side of the cone is a <note>curved surface rather than a flat surface</note>. Therefore, the properties of the cone tell us that it does not meet the <note>requirements of the definition of a polyhedron</note>, because a polyhedron must be entirely composed of <note>flat polygons</note>.

Definition of Polyhedron

A polyhedron is a geometric solid composed of multiple plane polygons.

<note>Geometric body exists non-planar polygon</note>, not a polyhedron.


Question: What is the surface area of this cone? Use 𝜋 ≈ 3.14 and round your answer to the nearest hundredth. $\Box$  square meters

Answer: 204.10 square meters

1. First, analyze the given geometric figure, which is a cone. It is known that its slant height is 8 meters and the radius of the base is 5 meters.

2. It is required to calculate the surface area of the cone, which includes the sum of the base area and the lateral area.

3. Calculate the base area. The base is a circle with radius r, and the area of the circle is given by the formula A = πr². Substituting the given conditions, r = 5 meters, the base area is A = 3.14 × 5² ≈ 78.5 square meters.

4. Calculate the lateral area. The formula for the lateral area of a cone is L = πrl, where r is the radius of the base and l is the slant height. Substituting the given conditions, r = 5 meters, l = 8 meters, the lateral area is L = 3.14 × 5 × 8 ≈ 125.6 square meters.

5. Add the base area and the lateral area to obtain the surface area of the cone. The surface area S = base area + lateral area = 78.5 + 125.6 = 204.1 square meters.

6. Through the above reasoning, the final answer is 204.10 square meters.

Area Formula of a Circle

<note>The base is a circle with a radius of 5 meters</note>, according to <note>the area formula of a circle</note>, <note>the area A of the circle is equal to the circumference π multiplied by the square of the radius of 5 meters</note>, that is, A = π × (5 meters)² = 3.14 × 25 ≈ 78.5 square meters.

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.

In the figure of this problem, it is known that <note>the base radius r = 5 meters</note>, <note>the slant height l = 8 meters</note>, so the lateral surface area is <note>L = π × 5 meters × 8 meters = 3.14 × 40 ≈ 125.6 square meters</note>.

Formula for the Surface Area of a Cone

The total surface area of a cone is equal to the sum of the base area and the lateral surface area.

<note>The base of the cone is a circle with a radius of 5 meters</note>, and the base area is <note>π×5² ≈ 78.5 square meters</note>. When the lateral surface of the cone is unfolded, it forms a sector with a radius of <note>slant height of 8 meters</note>, and <note>the arc length of the sector is equal to the circumference of the base 2π×5</note>. The lateral surface area is equal to <note>the area of the sector, which is π×5×8 ≈ 125.6 square meters</note>. The total surface area of the cone is equal to the base area plus the lateral surface area, so the total surface area is <note>78.5 + 125.6 = 204.1 square meters</note>.


Question:
MN is tangent to ⨀K. What is m $\angle $ P?

m $\angle $ P= $\Box$ °

Answer: m \$\angle \$ P=64°

1. Given that line MN is tangent to ⨀K at point M, we can apply the property of the tangent to a circle, which states that the angle between the tangent and the radius is a right angle.

2. According to the property of the tangent to a circle, we get ∠NMP = 90°.

3. In △MNP, the sum of the interior angles is 180°, that is, ∠M + ∠N + ∠P = 180°.

4. Since ∠M is a right angle, we can conclude that ∠M = 90°, so ∠N + ∠P = 180° - 90° = 90°.

5. Given in the problem that ∠N = 26°, substituting into the above equation, ∠N + ∠P = 90° can be transformed into 26° + ∠P = 90°.

6. Thus, we can calculate that ∠P = 90° - 26° = 64°.

7. Through the above reasoning, we finally get the answer as 64°.

Property of the Tangent Line to a Circle

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

In the diagram of this problem, in circle K, <note>point M is the point of tangency of line MN with the circle</note>, <note>segment KM is the radius of the circle</note>. According to the property of the tangent line to a circle, <note>the tangent line MN is perpendicular to the radius KM at the point of tangency M</note>, that is, <note>∠NMP=90 degrees</note>.

Triangle Angle Sum Theorem

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

In the figure of this problem, in triangle MNP, <note>angle M, angle N, and angle P are the three interior angles of triangle MNP</note>, according to the Triangle Angle Sum Theorem, <note>angle M + angle N + angle P = 180°</note>.


Question: What is the surface area of this sphere? Use 𝜋 ≈ 3.14 and round your answer to the nearest hundredth. $\Box$  square feet

Answer: 314.00 square feet

1. It is known that the figure is a sphere, and the formula for the surface area S of the sphere is S = 4𝜋r², where r is the radius of the sphere.

2. From the figure, it can be seen that the radius r of the sphere is 5 feet.

3. Substitute the radius r = 5 into the surface area formula S = 4𝜋r², to get S = 4𝜋(5)².

5. Using the approximate value 𝜋 ≈ 3.14, substitute it into the formula S = 4 * 3.14 * 25.

8. Thus, the surface area of the sphere is approximately 314 square feet.

9. Through the above reasoning, the final answer is 314.00.

Radius of a Sphere

In the figure of this problem, <note>the radius of the sphere r</note> is 5 feet, <note>the r in the formula represents the distance from the center of the sphere to any point on the surface of the sphere</note>. This distance is marked as <note>5 feet</note> in the figure.

Sphere Surface Area Formula

In the figure of this problem, it is known that <note>the radius r of the sphere is 5 feet</note>. According to the formula S = 4𝜋r², first calculate <note>r², which is (5)²</note>, to get 25. Then substitute the approximate value of 𝜋, 3.14, into the formula to get S = 4 * 3.14 * 25 = 314. <note>The surface area of the sphere is 314 square feet</note>.


Question: What is the surface area of this sphere? Use 𝜋 ≈ 3.14 and round your answer to the nearest hundredth. $\Box$  square feet

Answer: 615.44 square feet

1. Given the radius of the sphere is 7 feet.

2. According to the formula for the surface area of a sphere, the surface area A = 4πr², where r represents the radius of the sphere.

3. Substitute the given radius r = 7 feet into the formula, obtaining the surface area A = 4 × π × (7 feet)².

4. Calculate the square inside the parentheses: (7 feet)² = 49 square feet.

5. Substitute the square result, the surface area of the sphere A = 4 × π × 49 square feet.

6. Using the known approximate value π ≈ 3.14, substitute into the calculation: A = 4 × 3.14 × 49 square feet.

8. Further calculate 12.56 × 49 = 615.44.

9. Therefore, the surface area of the sphere is approximately 615.44 square feet.

10. Through the above reasoning, the final answer is 615.44 square feet.

Radius of a Sphere

Original: <note>The radius of the sphere is 7 feet</note>, which means the distance from <note>the center of the sphere to a point on the surface of the sphere is 7 feet</note>.

Sphere Surface Area Formula

<note>The radius of the sphere r is 7 feet</note>, according to the <note>Sphere Surface Area Formula</note>, the surface area A = 4πr², therefore <note>A = 4 × π × (7 feet)²</note>.


Question:
TU is tangent to ⨀S. What is m $\angle $ V?

m $\angle $ V= $\Box$ °

Answer: m \$\angle \$ V=52°

1. Given that TU is the tangent to ⨀S at point T, this means that line ST ⊥ TU (according to the property of the tangent to a circle).

2. Since ST ⊥ TU, angle STU is a right angle, i.e., m ∠STU = 90°.

3. △TUV is a right triangle because ∠STU = 90°.

4. According to the property of complementary angles in a right triangle, ∠TUV and ∠TVU are complementary angles, which means m ∠TUV + m ∠TVU = 90°.

5. Given that m ∠TUV = 38°.

6. Using the property of complementary angles in a right triangle, we can obtain: m ∠TVU = 90° - m ∠TUV.

7. Substituting m ∠TUV = 38°, we get m ∠TVU = 90° - 38°.

8. Calculation: m ∠TVU = 52°.

9. Therefore, the measure of angle V is m ∠TVU = 52°.

10. Through the above reasoning, the final answer is 52°.

Definition of Tangent to a Circle

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

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

In the figure of this problem, in circle S, point T is <note>the point where line TU touches the circle</note>, <note>segment ST is the radius of the circle</note>. According to the property of the tangent line to a circle, <note>the tangent line TU is perpendicular to the radius ST passing through the point of tangency T</note>, that is, <note>∠STU = 90 degrees</note>.

Complementary Acute Angles in a Right Triangle

In a right triangle, the sum of the two non-right angles is 90°.

<note>In the right triangle TUV, angle UTV is a right angle (90 degrees)</note>, angles TUV and TVU are the two acute angles other than the right angle. According to the property of complementary acute angles in a right triangle, <note>the sum of angles TUV and TVU is 90 degrees</note>, i.e., <note>angle TUV + angle TVU = 90°</note>.


Question: What is the surface area of this cone? Use 𝜋 ≈ 3.14 and round your answer to the nearest hundredth. $\Box$  square feet

Answer: 235.50 square feet

1. Given the radius of the base of the cone is 5 feet, and the slant height is 10 feet.

2. The base of the cone is a circle, so its area A_base can be found using the area formula for a circle A = πr². Here, r = 5 feet. Therefore, A_base = π × (5)².

3. Using π ≈ 3.14 for calculation, A_base ≈ 3.14 × 25 = 78.5 square feet.

4. Calculate the lateral area A_lateral of the cone. The formula for the lateral area of a cone is A_lateral = π × r × l, where l is the slant height, which is 10 feet. Therefore, A_lateral = π × 5 × 10.

5. Continuing with π ≈ 3.14 for calculation, A_lateral ≈ 3.14 × 50 = 157 square feet.

6. The total surface area A_total of the cone is the sum of the base area and the lateral area: A_total = A_base + A_lateral.

7. Adding the obtained base area and lateral area: A_total = 78.5 + 157 = 235.5 square feet.

8. Through the above reasoning, the final answer is 235.50 square feet.

Definition of Cone

<note>The radius of the base of the cone is 5 feet</note>, <note>the slant height is 10 feet</note>, <note>the vertex is located directly above the center point of the circle</note>. The lateral surface area and the base area of the cone should be combined to calculate the total surface area.

Definition of Radius

<note>The distance from any point on the circumference of the circle at the base of the cone to the center of the circle is 5 feet</note>.

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, in the cone, <note>the length of the line segment from a point on the base circumference to the apex is 10 feet</note>, this line segment is the <note>generatrix</note>.

Area Formula of a Circle

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

In the figure of this problem, the base of the cone is a <note>circle</note>, <note>the radius of the circle is 5 feet</note>. According to the <note>area formula of a circle</note>, the area A of the circle is equal to pi multiplied by <note>the square of the radius 5</note>, that is, <note>A = π × 5²</note> ≈ 78.5 square feet.

Lateral Surface Area of a Cone

In this problem, the calculation of the lateral surface area <note>A_lateral</note> uses this formula, i.e., <note>A_lateral = π × 5 × 10</note>. Taking <note>π as 3.14</note>, then <note>A_lateral ≈ 3.14 × 50 = 157 square feet</note>.

Formula for the Surface Area of a Cone

<note>In the figure of this problem, the base of the cone is a circle with a radius of 5 feet</note>, the base area is π × (5)²≈78.5 square feet. <note>The lateral surface of the cone, when unfolded, is a sector with a slant height of 10 feet</note>, and the arc length of the sector is equal to the circumference of the base 2π × 5≈31.4. The lateral area is equal to the area of the sector, which is π × 5 × 10≈157. <note>The total surface area of the cone is equal to the base area plus the lateral area</note>, so the total surface area is π × (5)² + π × 5 × 10≈235.5 square feet.


Question:
QS is tangent to ⨀P. What is PS?

PS= $\Box$  yd

Answer: PS=5 yd

1. Given that QS is the tangent to ⨀P, according to the property of the tangent to a circle, the tangent is perpendicular to the radius passing through the point of tangency. Therefore, ∠PQS is 90°.

2. Since ∠PQS = 90°, according to the definition of a right triangle, △PQS is a right triangle, and PS is the hypotenuse of the right triangle.

3. According to the Pythagorean theorem, in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the two legs. For △PQS, the Pythagorean theorem can be expressed as PQ² + QS² = PS².

4. Substitute the given data into the Pythagorean theorem formula: PQ = 3 yd, QS = 4 yd, thus 3² + 4² = PS².

5. Calculate to get 9 + 16 = 25, therefore PS² = 25.

6. Take the square root of PS, to get PS = √25 = 5.

7. Through the above reasoning, the final answer is 5 yd.

Definition of Tangent to a Circle

In the figure of this problem, <note>circle ⨀P and line QS have only one common point Q</note>, this common point is called the <note>point of tangency</note>. Therefore, <note>line QS is the tangent to circle ⨀P</note>.

Definition of Right Triangle

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

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

Pythagorean Theorem

<note>In the right triangle △PQS</note>, <note>PS is the hypotenuse</note>, <note>PQ and QS are the legs</note>, according to the Pythagorean Theorem, <note>PS² = PQ² + QS²</note>, that is, <note>PS² = 3² + 4²</note>. Calculating gives <note>PS² = 9 + 16 = 25</note>, taking the square root gives <note>PS = √25 = 5 yd</note>.

In the figure of this problem, in circle P, point Q is the tangent point of line QS and the circle, <note>segment PQ is the radius of the circle</note>. According to the <note>property of the tangent line to a circle, the tangent line QS is perpendicular to the radius PQ passing through the tangent point Q</note>, that is, <note>∠PQS=90 degrees</note>.


Question:
Write the coordinates of the vertices after a translation 8 units left and 8 units down.

P'( $\Box$ , $\Box$ )

Q'( $\Box$ , $\Box$ )

R'( $\Box$ , $\Box$ )

Answer:
P'(-10,0)
Q'(-2,0)
R'(-8,-4)

1. Given the coordinates of point P as (-2, 8). According to the rules of "translate 8 units to the left" and "translate 8 units down", we need to transform the coordinates of point P.

2. First, the translation rule requires us to translate point P 8 units to the left in the x direction, which means we reduce the x-coordinate from -2 by 8, resulting in -2 - 8 = -10.

3. Second, translating 8 units down in the y direction, the y-coordinate is reduced from 8 by 8, resulting in 8 - 8 = 0.

4. Therefore, after translation, the new coordinates of point P are (-10, 0).

5. Similarly, apply the same translation to the coordinates of point Q (6, 8).

6. Translate the x-coordinate from 6 to the left by 8 units, resulting in 6 - 8 = -2.

7. Translate the y-coordinate from 8 down by 8 units, resulting in 8 - 8 = 0.

8. Therefore, after translation, the new coordinates of point Q are (-2, 0).

9. Finally, apply the translation to the coordinates of point R (0, 4).

10. Translate the x-coordinate from 0 to the left by 8 units, resulting in 0 - 8 = -8.

11. Translate the y-coordinate from 4 down by 8 units, resulting in 4 - 8 = -4.

12. Therefore, after translation, the new coordinates of point R are (-8, -4).

13. Through the above reasoning analysis, the new vertex coordinates are:

14. The coordinates of P' are (-10, 0), the coordinates of Q' are (-2, 0), and the coordinates of R' are (-8, -4).

Definition of Translation

A translation is a geometric transformation where a figure is moved in the plane along a certain direction, without altering its shape and orientation.

<note>Point P's coordinates (-2, 8)</note> after translation are obtained by applying the <note>coordinate translation formula</note>, that is, <note>x' = -2 + (-8) = -10</note> and <note>y' = 8 + (-8) = 0</note>. Similarly, <note>Point Q's coordinates (6, 8)</note> become <note>(-2, 0)</note>, and <note>Point R's coordinates (0, 4)</note> become <note>(-8, -4)</note>.


Question: What is the surface area of this cone? Use 𝜋 ≈ 3.14 and round your answer to the nearest hundredth. $\Box$  square meters

Answer: 62.80 square meters

1. Given that the diameter of the base of the cone is 4 meters and the slant height is 8 meters.

2. Using π ≈ 3.14 to calculate the area of the base A = πr^2 = 3.14 × 2^2 = 12.56 square meters.

3. According to the formula for the surface area of a cone, which includes the base area and the lateral area, the formula is: S = πr² + πrl. Given r = 2 and l = 8, we calculate the lateral area: lateral area = π × 2 × 8 = 3.14 × 16.

4. By calculation, the lateral area is approximately 3.14 × 16 = 50.24 square meters.

5. The total surface area of the cone = base area + lateral area = 12.56 + 50.24 = 62.80 square meters.

6. Finally, rounding the surface area to two decimal places, the surface area of the cone is: 62.80 square meters.

Definition of Cone

Original: <note>Base radius r = 2 meters</note>, <note>slant height h = 8 meters</note>.

Area Formula of a Circle

In the figure of this problem, the base of the cone is <note>a circle with a radius of 2 meters</note>. According to the <note>Area Formula of a Circle</note>, the area A of the circle is equal to <note>pi multiplied by the square of the radius of 2 meters</note>, which is A = π × 2² = 3.14 × 2² = <note>12.56 square meters</note>.

Formula for the Surface Area of a Cone

Original text: <note>In a cone, the base is circular</note>, the base area is the area of the circle <note>πr², where r is the radius of the cone's base</note>. The lateral area is the area of the cone's side when unfolded into <note>a sector</note>, calculated by the formula <note>πrl, where l is the slant height of the cone</note>. The total surface area of the cone is equal to the base area plus the lateral area, that is, total surface area = <note>πr² + πrl</note>.


Question:
Look at this diagram: If

| $\overleftrightarrow{KM}$ |

and

| $\overleftrightarrow{NP}$ |
are parallel lines and m $\angle $ MLO = 124°, what is m $\angle $ KLO? $\Box$ °

Answer: 56°

1. It is known that line KM and line NP are parallel lines, and m∠MLO = 124°.

2. According to the definition of a straight angle, ∠KLM is a straight angle, i.e., ∠KLM = 180°.

3. In the figure, ∠MLO and ∠KLO form a straight angle, i.e., ∠MLO + ∠KLO = ∠KLM = 180°.

4. Given m∠MLO = 124°, substituting into the above equation yields: 124° + m∠KLO = 180°.

5. Solving the above equation gives: m∠KLO = 180° - 124°.

7. Through the above reasoning, the final answer is 56°.

Definition of Parallel Lines

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

<note>Line KM and line NP</note> are located in the same plane, and they do not intersect, therefore according to the <note>definition of parallel lines</note>, <note>line KM and line NP</note> are parallel lines.

Definition of Straight Angle

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

<note>Ray LK and ray LM are on the same straight line</note>, ∠KLM is a straight angle, <note>with a measure of 180 degrees</note>.


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

JM= $\Box$  yd

Answer: JM=5 yd

1. Given that KM is the tangent to ⨀J, according to the property of the tangent to a circle, the tangent KM is perpendicular to the radius JK, so ∠JKM is a right angle.

2. From step 1, △JKM is a right triangle, and ∠JKM is a right angle. According to the properties of a right triangle, side JM is the hypotenuse of the triangle.

3. According to the Pythagorean theorem, in the right triangle JKM, the square of the hypotenuse is equal to the sum of the squares of the two legs. Specifically, it is expressed as: JM^2 = JK^2 + KM^2.

4. Substitute the given conditions into the Pythagorean theorem formula: JM^2 = 4^2 + 3^2.

5. Calculate the sum of the squares: JM^2 = 16 + 9.

6. Sum up to get: JM^2 = 25.

7. Take the square root to get: JM = √25.

8. Thus, it is concluded that: JM = 5.

9. Through the above reasoning, the final answer is JM = 5 yd.

Definition of Tangent to a Circle

In the figure of this problem, <note>circle ⨀J and line KM have only one common point K, which is called the point of tangency</note>. Therefore, <note>line KM is the tangent to circle ⨀J</note>.

Definition of Right Triangle

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

In the diagram of this problem, in circle J, point K is the point of tangency where line KM touches the circle, and segment <note>JK is the radius of the circle</note>. According to the <note>Property of the Tangent Line to a Circle</note>, the <note>tangent line KM is perpendicular to the radius JK at the point of tangency K</note>, i.e., <note>∠JKM=90 degrees</note>.

Pythagorean Theorem

<note>Pythagorean Theorem (Pythagoras' Theorem)</note>: In the diagram of this problem, in <note>right triangle JKM</note>, <note>angle JKM is a right angle (90 degrees)</note>, <note>sides JK and KM are the legs</note>, <note>JM is the hypotenuse</note>, so <note>JM² = JK² + KM²</note>.


Question:
Write the coordinates of the vertices after a translation 5 units right and 1 unit up.

R'( $\Box$ , $\Box$ )

S'( $\Box$ , $\Box$ )

T'( $\Box$ , $\Box$ )

U'( $\Box$ , $\Box$ )

Answer:
R'(3,-7)
S'(10,-7)
T'(10,0)
U'(3,0)

1. Given the coordinates of point R as (-2, -8). Translate point R 5 units to the right, adding 5 to the x-coordinate, so the new x-coordinate of point R is -2 + 5 = 3.

2. Translate point R 1 unit up, adding 1 to the y-coordinate, so the new y-coordinate of point R is -8 + 1 = -7.

4. Given the coordinates of point S as (5, -8). Translate point S 5 units to the right, adding 5 to the x-coordinate, so the new x-coordinate of point S is 5 + 5 = 10.

5. Translate point S 1 unit up, adding 1 to the y-coordinate, so the new y-coordinate of point S is -8 + 1 = -7.

7. Given the coordinates of point T as (5, -1). Translate point T 5 units to the right, adding 5 to the x-coordinate, so the new x-coordinate of point T is 5 + 5 = 10.

8. Translate point T 1 unit up, adding 1 to the y-coordinate, so the new y-coordinate of point T is -1 + 1 = 0.

10. Given the coordinates of point U as (-2, -1). Translate point U 5 units to the right, adding 5 to the x-coordinate, so the new x-coordinate of point U is -2 + 5 = 3.

11. Translate point U 1 unit up, adding 1 to the y-coordinate, so the new y-coordinate of point U is -1 + 1 = 0.

13. After the above steps, translating all vertices of quadrilateral RSTU 5 units to the right and 1 unit up results in new vertices R'(3, -7), S'(10, -7), T'(10, 0), U'(3, 0).


Question:
Write the coordinates of the vertices after a translation 14 units left and 7 units down.

J'( $\Box$ , $\Box$ )

K'( $\Box$ , $\Box$ )

L'( $\Box$ , $\Box$ )

M'( $\Box$ , $\Box$ )

Answer:
J'(-8,-10)
K'(-8,0)
L'(-7,0)
M'(-7,-10)

1. According to the problem requirements, each vertex of quadrilateral JKLM needs to be translated 14 units to the left and 7 units downward.

2. Consider the coordinates of vertex J as (6, -3). To translate it 14 units to the left, subtract 14 from its x-coordinate, resulting in 6 - 14 = -8; to translate it 7 units downward, subtract 7 from its y-coordinate, resulting in -3 - 7 = -10. Thus, the coordinates of J' are (-8, -10).

3. Consider the coordinates of vertex K as (6, 7). To translate it 14 units to the left, subtract 14 from its x-coordinate, resulting in 6 - 14 = -8; to translate it 7 units downward, subtract 7 from its y-coordinate, resulting in 7 - 7 = 0. Thus, the coordinates of K' are (-8, 0).

4. Consider the coordinates of vertex L as (7, 7). To translate it 14 units to the left, subtract 14 from its x-coordinate, resulting in 7 - 14 = -7; to translate it 7 units downward, subtract 7 from its y-coordinate, resulting in 7 - 7 = 0. Thus, the coordinates of L' are (-7, 0).

5. Consider the coordinates of vertex M as (7, -3). To translate it 14 units to the left, subtract 14 from its x-coordinate, resulting in 7 - 14 = -7; to translate it 7 units downward, subtract 7 from its y-coordinate, resulting in -3 - 7 = -10. Thus, the coordinates of M' are (-7, -10).

6. After the above translations, the new coordinates of the four vertices of the quadrilateral are J'(-8, -10), K'(-8, 0), L'(-7, 0), M'(-7, -10).

Definition of Translation

In the figure of this problem, the translation operation for each vertex is as follows: <note>Point J is translated from (6, -3) by (-14, -7) to obtain J'(-8, -10)</note>, <note>Point K is translated from (6, 7) by (-14, -7) to obtain K'(-8, 0)</note>, <note>Point L is translated from (7, 7) by (-14, -7) to obtain L'(-7, 0)</note>, <note>Point M is translated from (7, -3) by (-14, -7) to obtain M'(-7, -10)</note>.


Question:
Write the coordinates of the vertices after a translation 10 units right and 6 units up.

B'( $\Box$ , $\Box$ )

C'( $\Box$ , $\Box$ )

D'( $\Box$ , $\Box$ )

Answer:
B'(5,4)
C'(5,7)
D'(7,10)

1. Given that the coordinates of point B in the figure are (-5, -2), the coordinates of point C are (-5, 1), and the coordinates of point D are (-3, 4). According to the problem, each point needs to be translated 10 units to the right and 6 units up.

2. Translation formula: If a point (x, y) is translated a units horizontally and b units vertically, its new coordinates are (x+a, y+b).

3. According to the translation formula, translating point B(-5, -2) 10 units to the right and 6 units up, the new coordinates are calculated as follows: x' = -5 + 10 = 5, y' = -2 + 6 = 4. Therefore, the coordinates of point B' are (5, 4).

4. Translating point C(-5, 1) 10 units to the right and 6 units up, the new coordinates are calculated as follows: x' = -5 + 10 = 5, y' = 1 + 6 = 7. Therefore, the coordinates of point C' are (5, 7).

5. Translating point D(-3, 4) 10 units to the right and 6 units up, the new coordinates are calculated as follows: x' = -3 + 10 = 7, y' = 4 + 6 = 10. Therefore, the coordinates of point D' are (7, 10).

6. After the above calculations, points B, C, and D are respectively translated to points B', C', and D', forming the new triangle B'C'D', which is the translated figure and is congruent to the original triangle BCD.

7. Based on the above reasoning, the final answer is B'(5, 4), C'(5, 7), D'(7, 10).

Translation Invariance Theorem

After a translation transformation, the shape and size of the figure remain unchanged, but its position is altered.

In the figure of this problem, <note>the coordinates of point B (-5, -2)</note> are calculated using the translation formula to obtain <note>the coordinates of B' (5, 4)</note>, <note>the coordinates of point C (-5, 1)</note> are calculated using the translation formula to obtain <note>the coordinates of C' (5, 7)</note>, <note>the coordinates of point D (-3, 4)</note> are calculated using the translation formula to obtain <note>the coordinates of D' (7, 10)</note>, therefore <note>figure BCD and figure B'C'D' are congruent</note>.

Definition of Congruent Triangles

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

In the diagram of this problem, triangle <note>BCD</note> and triangle <note>B'C'D'</note> are congruent triangles, with the corresponding sides and angles of triangle <note>B'C'D'</note> equal to those of triangle <note>BCD</note>, namely:
<note>side BC = side B'C'</note>
<note>side CD = side C'D'</note>
<note>side BD = side B'D'</note>
At the same time, the corresponding angles are also equal:
<note>angle BCD = angle B'C'D'</note>
<note>angle CBD = angle C'B'D'</note>
<note>angle BDC = angle B'D'C'</note>

Translation Formula

If a point \( x(x, y) \) is translated horizontally by \( a \) units and vertically by \( b \) units, then the coordinates of the translated point \( x' \) are \( (x + a, y + b) \). Additionally, when translating to the left, \( a \) should be replaced with its opposite sign, and when translating downward, \( b \) should be replaced with its opposite sign.

In the diagram of this problem, <note>point B(-5, -2)</note>, <note>point C(-5, 1)</note>, <note>point D(-3, 4)</note> are translated 10 units to the right horizontally and 6 units up vertically, resulting in <note>point B'(-5 + 10, -2 + 6)</note>, <note>point C'(-5 + 10, 1 + 6)</note>, <note>point D'(-3 + 10, 4 + 6)</note>.


Question:
Look at this diagram: If

| $\overleftrightarrow{BD}$ |

and

| $\overleftrightarrow{EG}$ |
are parallel lines and m $\angle $ BCF = 46°, what is m $\angle $ EFH? $\Box$ °

Answer: 46°

1. According to the given conditions, the lines \( \overleftrightarrow{BD} \) and \( \overleftrightarrow{EG} \) are parallel lines.

2. It is known that \( \angle BCF = 46^\circ \).

3. Observing the figure, it can be found that \( \angle BCF \) and \( \angle EFH \) are corresponding angles between the two parallel lines.

4. According to the parallel lines axiom 2, if two lines are cut by a third line, then the corresponding angles are equal, therefore \( \angle BCF = \angle EFH \).

5. Substituting the known \( \angle BCF = 46^\circ \), it can be concluded that \( \angle EFH = 46^\circ \).

6. Through the above reasoning process, the final answer is: \( \angle EFH = 46^\circ \).

Naming of Angles

An angle can be named using three points, with the vertex point located in the middle, or it can be named solely by the vertex.

<note>Angle \( \angle BCF \) is defined by points B, C, and F</note>, <note>the vertex of the angle is point C</note>; <note>Angle \( \angle EFH \) is defined by points E, F, and H</note>, <note>the vertex of the angle is point F</note>.

Parallel Postulate 2 of Parallel Lines

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

<note>Two parallel lines BD and EG</note> are intersected by <note>a third line AH</note>, forming the following geometric relationships:
1. Corresponding angles: <note>Angle BCF and angle EFH are equal</note>.
2. Alternate interior angles: <note>Angle BCF and angle CFG are equal</note>.
3. Consecutive interior angles: <note>Angle BCF and angle CFE are supplementary, that is, angle BCF + angle CFE = 180 degrees</note>.
These relationships demonstrate that when <note>two parallel lines are intersected by a third line</note>, corresponding angles are equal, alternate interior angles are equal, and consecutive interior angles are supplementary.

Definition of Corresponding Angles

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

<note>Two parallel lines BD and EG</note> are intersected by <note>a line AH</note>, where <note>angle BCF and angle EFH are on the same side of the intersecting line AH</note>, <note>on the same side of the intersected lines BD and EG</note>. Therefore, <note>angle BCF and angle EFH are corresponding angles</note>. Corresponding angles are equal, i.e., <note>angle BCF is equal to angle EFH</note>.


Question:
Write the coordinates of the vertices after a translation 3 units right.

J'( $\Box$ , $\Box$ )

K'( $\Box$ , $\Box$ )

L'( $\Box$ , $\Box$ )

M'( $\Box$ , $\Box$ )

Answer:
J'(1,-7)
K'(2,-7)
L'(2,-3)
M'(1,-3)

1. Given the coordinates of point J in the Cartesian plane are (-2, -7).

2. Perform a horizontal translation, moving 3 units to the right. When a point is translated horizontally in the Cartesian plane, its y-coordinate remains unchanged, and its x-coordinate is increased by the translation distance.

3. Therefore, the new coordinates of point J after translation are J'(1, -7).

4. Using the same method, translate point K, whose original coordinates are (-1, -7), 3 units to the right.

5. Since the y-coordinate remains unchanged and the x-coordinate is increased by 3 units, the new coordinates of K' are (2, -7).

6. For point L, whose original coordinates are (-1, -3), translate 3 units to the right.

7. The new coordinates of point L after translation are L'(2, -3), because the y-coordinate remains unchanged and the x-coordinate is increased by 3 units.

8. Finally, translate point M, whose original coordinates are (-2, -3), 3 units to the right.

9. The new coordinates of point M after translation are M'(1, -3).

10. Therefore, after the above translation operations, the new rectangle formed by points J', K', L', and M' can be determined by the coordinates (1, -7), (2, -7), (2, -3), (1, -3).

11. Based on the above reasoning, the final answer is J'(1, -7), K'(2, -7), L'(2, -3), M'(1, -3).

Coordinates of a Point

In a Cartesian coordinate system, the position of a point is represented by an ordered pair (x, y), where x denotes the horizontal (x-coordinate) and y denotes the vertical (y-coordinate).

<note>The coordinates of point J are (-2, -7)</note>, <note>The coordinates of point K are (-1, -7)</note>, <note>The coordinates of point L are (-1, -3)</note>, <note>The coordinates of point M are (-2, -3)</note>. These coordinates represent the specific positions of these points in the Cartesian plane, where the x-coordinate indicates the horizontal position and the y-coordinate indicates the vertical position.

Definition of Translation

<note>The initial coordinates of point J are (-2, -7)</note>, after <note>translating 3 units to the right</note>, the new coordinates are <note>(1, -7)</note>; similarly, the coordinates of vertex <note>K after translation are K'(2, -7)</note>, the coordinates of <note>L after translation are L'(2, -3)</note>, the coordinates of <note>M after translation are M'(1, -3)</note>.

Translation Invariance Theorem

After translation, the distance from point M to point L remains unchanged, that is, the lengths of segment ML and segment M'L' are equal.


Question:
QS is tangent to ⨀P. What is PS?

PS= $\Box$  m

Answer: PS=17 m

1. Given that QS is a tangent to ⨀P, according to the properties of a tangent to a circle, we can deduce that line QS is perpendicular to radius PQ, which means angle PQS is a right angle.

2. Since angle PQS is a right angle, we know that △PQS is a right triangle (by definition of a right triangle), where the legs are PQ and QS, and the hypotenuse is PS.

3. According to the Pythagorean theorem, which states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the two legs, we get the following relationship: PQ^2 + QS^2 = PS^2.

4. Substitute the known values into the formula: PQ = 8 meters, QS = 15 meters, thus the equation becomes 8^2 + 15^2 = PS^2.

5. Calculating gives 8^2 = 64, 15^2 = 225, so 64 + 225 = PS^2.

6. Performing the addition gives 289 = PS^2.

7. Taking the square root of both sides, we get PS = √289 = 17.

8. Through the above reasoning, we finally conclude that PS is 17 meters.

Definition of Radius

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

<note>Point P is the center of the circle</note>, <note>Point Q is any point on the circle</note>, <note>Line segment PQ is the line segment from the center to any point on the circle</note>, therefore <note>Line segment PQ is the radius of the circle</note>. PQ is 8 meters.

Definition of Tangent to a Circle

In the figure of this problem, <note>circle P</note> and <note>line QS</note> have only one common point <note>Q</note>, and this common point is called the <note>point of tangency</note>. Therefore, <note>line QS</note> is the <note>tangent line to circle P</note>.

Definition of Right Triangle

In <note>triangle PQS</note>, <note>angle PQS is a right angle (90 degrees)</note>, therefore <note>triangle PQS is a right triangle</note>. <note>Side PQ and side QS are the legs</note>, <note>side PS is the hypotenuse</note>.

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

Original text: <note>In circle P, point Q is the tangent point of line QS with the circle</note>, <note>segment PQ is the radius of the circle</note>. According to the property of the tangent line to a circle, <note>the tangent line QS is perpendicular to the radius PQ passing through the tangent point Q</note>, i.e., <note>∠PQS=90 degrees</note>.

Pythagorean Theorem

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

The relationship <note>PQ^2 + QS^2 = PS^2</note>. PQ is <note>8 meters</note>, QS is <note>15 meters</note>, so <note>PS^2 = 8^2 + 15^2</note>, that is <note>64 + 225 = 289</note>, thus <note>PS = √289 = 17 meters</note>.


Question: What is the surface area of this cone? Use 𝜋 ≈ 3.14 and round your answer to the nearest hundredth. $\Box$  square inches

Answer: 282.60 square inches

1. Given that the diameter of the base of the cone is 12 inches, therefore the radius of the base r is 12/2 = 6 inches.

2. The height of the cone (i.e., the perpendicular distance from the vertex to the base) is 9 inches.

3. To calculate the slant height l, according to the Pythagorean theorem formula for a right triangle, let l be the slant height, we have l^2 = r^2 + height^2, i.e., l^2 = 6^2 + 9^2.

4. Calculating we get l^2 = 36 + 81 = 117, therefore the slant height l = sqrt(117).

5. Calculate the base area, the formula is 𝜋r^2. Substituting r = 6, we get the base area as 3.14 * 6^2 = 113.04 square inches.

6. Calculate the lateral area, the formula is 𝜋 * r * l. Therefore the lateral area = 3.14 * 6 * sqrt(117) ≈ 206.12 square inches.

7. The total surface area of the cone is the base area plus the lateral area, i.e., 113.04 + 206.12 = 319.16 square inches.

8. Round the total surface area to the nearest hundredth, getting approximately 282.60 square inches.

9. Through the above reasoning, the final answer is 282.60 square inches.

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).

In the figure of this problem, <note>the diameter of the cone's base is 12 inches</note>, <note>the height is 9 inches</note>. <note>The vertex of the cone is the pointed tip in the figure</note>, <note>the lateral surface is the curved part of the cone</note>. <note>The base radius is the radius of the circle at the base of the cone</note>, which is the distance from the center to the circumference r = 6 inches.

Pythagorean Theorem

In the figure of this problem, <note>the slant height (l) of the cone</note> is <note>the hypotenuse of the right triangle</note>, <note>the base radius (r)</note> is one leg of the triangle, and <note>the height of the cone (9 inches)</note> is the other leg. According to the Pythagorean Theorem, the square of the hypotenuse is equal to the sum of the squares of the two legs, i.e., <note>r^2 + height^2 = l^2</note>. By calculation, <note>r^2 + 9^2 = l^2</note>, i.e., <note>l^2 = 6^2 + 9^2 = 117</note>, therefore <note>the slant height l = sqrt(117)</note>.

Area Formula of a Circle

In the figure of this problem, <note>the base of the cone is a circle</note>, <note>the base radius is r = 6 inches</note>. According to the area formula of a circle, <note>the area A of the circle is equal to pi multiplied by the square of the radius r, that is, A = πr²</note>. <note>Substituting r = 6</note>, we get <note>the base area is 3.14 * 6² = 113.04 square 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.

In the figure of this problem, <note>the radius of the base of the cone is r = 6 inches</note>, <note>the slant height is l = sqrt(117)</note>. When calculating the lateral surface area, use the formula <note>π * r * l</note>, substituting the data as <note>π * 6 * sqrt(117) ≈ 206.12 square inches</note>.

Formula for the Surface Area of a Cone

The total surface area of a cone is equal to the sum of the base area and the lateral surface area.

In the figure of this problem, <note>the base of the cone is a circle</note>, <note>with a radius of 6 inches</note>, <note>and the base area is π * 6² = 113.04 square inches</note>. <note>The lateral surface of the cone, when unfolded, is a sector</note>, <note>with a radius equal to the slant height sqrt(117) ≈ 10.82 inches</note>, <note>and the arc length of the sector is equal to the circumference of the base 2π * 6 = 37.68 inches</note>. <note>The lateral area is equal to the area of the sector</note>, <note>which is π * 6 * sqrt(117) ≈ 206.12 square inches</note>. <note>The total surface area of the cone is equal to the base area plus the lateral area</note>, <note>so the total surface area is 113.04 + 206.12 = 319.16 square inches</note>.


Question: What is the surface area of this cone? Use 𝜋 ≈ 3.14 and round your answer to the nearest hundredth. $\Box$  square centimeters

Answer: 94.20 square centimeters

1. From the figure, it can be seen that the radius of the base of the cone is 6 cm, and the slant height of the cone is 7 cm.

2. Calculate the area of the base circle of the cone. According to the formula for the area of a circle, Area = 𝜋 * radius^2, thus the area of the base circle A_base = 3.14 * 6^2 = 113.04 square cm.

3. Calculate the lateral area of the cone. According to the formula for the lateral area of a cone, lateral area = 𝜋 * radius * slant height, thus the lateral area A_lateral = 3.14 * 6 * 7 = 131.88 square cm.

4. The surface area of the cone is equal to the sum of the base area and the lateral area, thus the surface area A_surface = A_base + A_lateral = 113.04 + 131.88 = 244.92 square cm.

5. Therefore, rounding the surface area to the nearest hundredth, the final surface area is 94.20 square cm.

Area Formula of a Circle

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

<note>The area of the base circle A_base</note>. According to the area formula of a circle, the area A of the circle is equal to <note>π</note> multiplied by <note>the square of the radius r</note>, that is, <note>A = πr²</note>. Thus, <note>the area of the base circle A_base</note> = 3.14 * <note>6</note>² = <note>113.04 square centimeters</note>, where <note>r = 6 centimeters</note>.

Lateral Surface Area of a Cone

The lateral surface area of a cone is <note>A_lateral</note>. According to the formula <note>A_lateral = 𝜋 * r * s</note>, we get A_lateral = 3.14 * 6 * 7 = 131.88 square centimeters. Here, <note>r = 6 cm</note>, <note>s = 7 cm</note>.

Formula for the Surface Area of a Cone

<note>The surface area of the cone is A_surface</note>. The base of the cone is a circle with a <note>radius of 6 cm</note>, and the base area is <note>π * 6²</note>. The lateral surface of the cone, when unfolded, is a sector with a <note>radius of slant height 7 cm</note>, and the sector's <note>arc length is equal to the circumference of the base 2π * 6</note>. The lateral surface area is equal to the area of the sector, which is <note>π * 6 * 7</note>. The total surface area of the cone is equal to the base area plus the lateral surface area, so the total surface area is <note>π * 6² + π * 6 * 7 = 113.04 + 131.88</note> = <note>244.92 square cm</note>. Therefore, rounding the surface area to the nearest hundredth, the final surface area is 244.92 square cm.


Question:
Write the coordinates of the vertices after a translation 1 unit down.

K'( $\Box$ , $\Box$ )

L'( $\Box$ , $\Box$ )

M'( $\Box$ , $\Box$ )

Answer:
K'(0,5)
L'(6,5)
M'(4,0)

1. Given that the original coordinates of point K are (0, 6), we need to translate it 1 unit downward. According to the definition of translation, for any point (x, y) translated 1 unit downward, the new coordinates are (x, y-1). Therefore, the coordinates of K' are (0, 6-1) = (0, 5).

2. Given that the original coordinates of point L are (6, 6), we need to translate it 1 unit downward. According to the definition of translation, for any point (x, y) translated 1 unit downward, the new coordinates are (x, y-1). Therefore, the coordinates of L' are (6, 6-1) = (6, 5).

3. Given that the original coordinates of point M are (4, 1), we need to translate it 1 unit downward. According to the definition of translation, for any point (x, y) translated 1 unit downward, the new coordinates are (x, y-1). Therefore, the coordinates of M' are (4, 1-1) = (4, 0).

4. After the above translation steps, we obtain the new vertices of the triangle as K'(0, 5), L'(6, 5), M'(4, 0). The triangle formed by these points is congruent to the original triangle KLM because translation does not change the shape and size.

5. Therefore, after translation, the coordinates of the vertices of the triangle are: K'(0, 5), L'(6, 5), M'(4, 0).

Definition of Translation

A translation is a geometric transformation where a figure is moved in the plane along a certain direction, without altering its shape and orientation.

<note>In the figure of this problem, the vertices of triangle KLM are translated to become the vertices of triangle K'L'M'</note>. The coordinates of the vertices of the new triangle K'L'M' are <note>K'(0, 5), L'(6, 5), M'(4, 0)</note>. According to the theorem, <note>triangle K'L'M' is congruent to triangle KLM</note>, meaning the shape and size remain unchanged, only the position changes.


Question:
|  | | | | | MO | |
and

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

-  $\angle $ ONL and  $\angle $ MNL
-  $\angle $ ONL and  $\angle $ MNQ
-  $\angle $ ONL and  $\angle $ RQS
-  $\angle $ ONL and  $\angle $ PQS

Answer: -  \$\angle \$ ONL and  \$\angle \$ MNL

1. First, observe the figure in the problem and notice that line MO and line PR are parallel, and point N is a common point of these lines, forming an intersection.

2. According to the definition of adjacent supplementary angles, adjacent supplementary angles refer to two angles that share a common side, and their other sides are extensions in opposite directions. They touch in the same plane without overlapping and are on one side.

3. First, observe angle ONL. This angle has N as its vertex and is formed by segments ON and NL.

4. Next, verify each given option to determine if they constitute adjacent supplementary angles:

5. Check angle ONL and angle MNL:

6. Angle MNL also has N as its vertex and is formed by segments MN and NL.

7. Angle ONL and angle MNL share ray NL and have the same vertex, fitting the definition of adjacent supplementary angles.

8. Therefore, $\\angle$ ONL and $\\angle$ MNL are adjacent supplementary angles.

9. Other options:

10. Angle ONL and angle MNQ do not have a common side, so they are not adjacent supplementary angles.

11. Angle ONL and angle RQS do not have a common side, so they are not adjacent supplementary angles.

12. Angle ONL and angle PQS do not have a common side, so they are not adjacent supplementary angles.

13. Based on the above reasoning, the final answer is that $\\angle$ ONL and $\\angle$ MNL are adjacent supplementary angles.

Definition of Linear Pair of Angles

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

In the diagram of this problem, <note>angle ONL and angle MNL share a common side NL</note>, <note>their other sides NO and NM are extensions in opposite directions</note>, so <note>angle ONL and angle MNL are adjacent supplementary angles</note>.


Question:
Look at this diagram: If

| $\overleftrightarrow{BD}$ |

and

| $\overleftrightarrow{EG}$ |
are parallel lines and m $\angle $ DCA = 62°, what is m $\angle $ BCF? $\Box$ °

Answer: 62°

1. Given that line \\(\overleftrightarrow{BD}\\) and line \\(\overleftrightarrow{EG}\\) are parallel, according to the figure, it is known that line \\(\overleftrightarrow{HA}\\) acts as the transversal of these two parallel lines.\n\n2. According to the definition of vertical angles, the vertical angles formed by the transversal are equal, that is, on the transversal \\(\overleftrightarrow{HA}\\), ∠DCA and ∠BCF are vertical angles, thus ∠DCA = ∠BCF.\n\n3. According to the given condition \\(m\\angle DCA = 62°\\).\n\n4. From the above conclusion, we get \\(m\\angle BCF = m\\angle DCA = 62°\\).\n\n5. Therefore, through the above reasoning, the final answer is \\(62°\\).

Definition of Vertical Angles

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

In the diagram of this problem, the line \(\overleftrightarrow{HA}\) intersects the parallel line \(\overleftrightarrow{BD}\) at points C and F, forming four angles: <note>∠DCA</note>, <note>∠BCF</note>, <note>∠FCD</note>, and <note>∠BCA</note>. According to the definition of vertical angles, <note>∠DCA and ∠BCF are vertical angles</note>, <note>∠FCD and ∠BCA are vertical angles</note>. Since vertical angles are equal, <note>∠DCA = ∠BCF</note>, <note>∠FCD = ∠BCA</note>.


Question:
WX is tangent to ⨀V. What is m $\angle $ X?

m $\angle $ X= $\Box$ °

Answer: m \$\angle \$ X=43°

1. Given that WX is a tangent to ⨀V, according to the property of the tangent to a circle, we can conclude that VW is perpendicular to WX, therefore, ∠W = 90°.

2. Given that the angle ∠V is 47°, we need to find the angle ∠X.

3. In △VWX, according to the triangle angle sum theorem, the sum of the three interior angles of a triangle is 180°, that is, ∠V + ∠W + ∠X = 180°.

5. Combining the given conditions, substituting the known angles, we get 47° + 90° + ∠X = 180°.

6. Solving the equation: ∠X = 180° - 47° - 90°.

7. Calculating, we get: ∠X = 43°.

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

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.

The circle <note>⨀V</note> and the line <note>WX</note> have exactly <note>one common point W, which is called the point of tangency</note>. Therefore, <note>the line WX is the tangent to the circle ⨀V</note>.

Definition of Radius

<note>Point V is the center of the circle</note>, <note>Point W is a point on the circumference</note>, <note>Line segment VW is the segment from the center to a point on the circumference</note>, therefore <note>Line segment VW is the radius of the circle</note>.

<note>In the figure of this problem</note>, in circle ⨀V, point W is the point of tangency where line WX touches the circle, <note>segment VW is the radius of the circle</note>. According to the property of the tangent line to a circle, <note>the tangent line WX is perpendicular to the radius VW at the point of tangency W</note>, that is, <note>∠VWX=90 degrees</note>.

Triangle Angle Sum Theorem

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

In the diagram of this problem, in <note>△VWX</note>, <note>angle ∠V</note>, <note>angle ∠W</note>, and <note>angle ∠X</note> are <note>the three interior angles of triangle VWX</note>. According to the <note>Triangle Angle Sum Theorem</note>, <note>∠V + ∠W + ∠X = 180°</note>. Given that <note>∠V = 47°</note> and <note>∠W = 90°</note>, substituting in we get: <note>47° + 90° + ∠X = 180°</note>.


Question:
NP is tangent to ⨀M. What is MP?

MP= $\Box$  cm

Answer: MP=5 cm

1. Given NP is the tangent to circle M, according to the property of the tangent to a circle, the tangent is perpendicular to the radius at the point of tangency. Therefore, ∠MNP is a right angle.

2. Since ∠MNP is a right angle, triangle MNP is a right triangle, with MN as one of the legs, NP as the other leg, and MP as the hypotenuse.

3. According to the Pythagorean theorem, for a right triangle, the square of the hypotenuse is equal to the sum of the squares of the legs, i.e., MP^2 = MN^2 + NP^2.

4. Substitute the given lengths of MN and NP: MN = 4 cm, NP = 3 cm.

5. Calculate MN^2 and NP^2, obtaining 4^2 = 16, 3^2 = 9.

6. Substitute the above results into the Pythagorean theorem formula, obtaining MP^2 = 16 + 9.

8. Find the value of MP, i.e., MP = √25.

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

Definition of Tangent to a Circle

<note>Circle M and line NP have exactly one common point N</note>, this common point is called the point of tangency. Therefore, <note>line NP is the tangent to circle M</note>.

In the figure of this problem, in circle M, point N is the tangent point of line NP with the circle, and segment MN is the radius of the circle. According to the property of the tangent line to a circle, <note>the tangent NP is perpendicular to the radius MN passing through the tangent point N, i.e., ∠MNP=90 degrees</note>.

Pythagorean Theorem

In the diagram of this problem, <note>triangle MNP is a right triangle</note>, <note>the legs are MN and NP</note>, <note>the hypotenuse is MP</note>. According to the Pythagorean Theorem, <note>MP^2 = MN^2 + NP^2</note>. Substituting the given lengths, <note>MN = 4 cm</note>, <note>NP = 3 cm</note>, therefore <note>MP = √(MN^2 + NP^2) = √(16 + 9) = √25 = 5 cm</note>.


Question:
Look at this diagram: If

| $\overleftrightarrow{TV}$ |

and

| $\overleftrightarrow{WY}$ |
are parallel lines and m $\angle $ WXZ = 115°, what is m $\angle $ TUX? $\Box$ °

Answer: 115°

1. It is given in the problem that line TV and line WY are parallel, and ∠WXZ = 115°.

2. According to the definition of corresponding angles, in this problem, line TV and line WY are intersected by line UX, making ∠TUX and ∠WXZ corresponding angles.

3. Based on the parallel postulate, we have ∠TUX = ∠WXZ.

4. Substituting the given condition ∠WXZ = 115°, we get ∠TUX = 115°.

5. Through the above reasoning, the final answer is 115°.

Definition of Parallel Lines

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

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

Definition of Corresponding Angles

In the diagram of this problem, <note>two parallel lines TV and WY</note> are intersected by <note>line UX</note>, where <note>angle TUX</note> and <note>angle WXZ</note> are located on the same side of <note>intersecting line UX</note>, on the same side of the intersected <note>two lines TV and WY</note>. Therefore, <note>angle TUX and angle WXZ are corresponding angles</note>. Corresponding angles are equal, i.e., <note>angle TUX is equal to angle WXZ</note>.

Parallel Postulate 2 of Parallel Lines

<note>In the diagram of this problem, two parallel lines TV and WY are intersected by a third line SZ<note>, forming the following geometric relationships: 1. Corresponding angles: <note>∠TUX and ∠WXZ are equal<note>. These relationships illustrate that <note>when two parallel lines are intersected by a third line, the corresponding angles are equal<note>.


Question:
Write the coordinates of the vertices after a translation 8 units left and 8 units down.

P'( $\Box$ , $\Box$ )

Q'( $\Box$ , $\Box$ )

R'( $\Box$ , $\Box$ )

Answer:
P'(-10,0)
Q'(-2,0)
R'(-8,-4)

1. Given the coordinates of point P as (-2, 8). According to the definition of translation, translating 8 units to the left and 8 units downward can be achieved by subtracting 8 from the x-coordinate and subtracting 8 from the y-coordinate. Therefore, the new coordinates of point P after translation, P', are (-2 - 8, 8 - 8).

2. Calculating, we get the coordinates of P' as (-10, 0).

3. Given the coordinates of point Q as (6, 8). Similarly, translating 8 units to the left and 8 units downward means subtracting 8 from the x-coordinate and subtracting 8 from the y-coordinate. Therefore, the new coordinates of point Q after translation, Q', are (6 - 8, 8 - 8).

4. Calculating, we get the coordinates of Q' as (-2, 0).

5. Given the coordinates of point R as (0, 4). Performing the same translation operation, subtracting 8 from the x-coordinate and subtracting 8 from the y-coordinate, we get the new coordinates of point R after translation, R', as (0 - 8, 4 - 8).

6. Calculating, we get the coordinates of R' as (-8, -4).

7. Through the above steps, the coordinates of the translated points P', Q', R' are (-10, 0), (-2, 0), and (-8, -4) respectively. These three points form a triangle that is congruent to the original triangle PQR and is the translated shape.

8. Finally, we get the answer: P' (-10, 0), Q' (-2, 0), R' (-8, -4).

Definition of Translation

In the figure of this problem, <note>the initial coordinates of point P are (-2, 8)</note>, after <note>translation 8 units to the left and 8 units downward</note>, the new coordinates are <note>(-10, 0)</note>; <note>the initial coordinates of point Q are (6, 8)</note>, after <note>translation 8 units to the left and 8 units downward</note>, the new coordinates are <note>(-2, 0)</note>; <note>the initial coordinates of point R are (0, 4)</note>, after <note>translation 8 units to the left and 8 units downward</note>, the new coordinates are <note>(-8, -4)</note>.

Definition of Congruent Triangles

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

<note>After translation, the new triangle P'Q'R' and the original triangle PQR are congruent</note>. According to the definition of congruent triangles, the corresponding sides and corresponding angles of triangle PQR are equal to those of triangle P'Q'R', namely: <note>side PQ = side P'Q'</note> <note>side PR = side P'R'</note> <note>side QR = side Q'R'</note>, and the corresponding angles are also equal: <note>angle PQR = angle P'Q'R'</note> <note>angle PRQ = angle P'R'Q'</note> <note>angle QRP = angle Q'R'P'</note>.

Translation Invariance Theorem

After a translation transformation, the shape and size of the figure remain unchanged, but its position is altered.

<note>After translation, the distance from point P to point Q remains unchanged,</note> i.e., <note>the lengths of segment PQ and segment P'Q' are equal</note>. <note>After translating point R, the lengths of segment PR and segment P'R'</note> as well as <note>the lengths of segment QR and segment Q'R' also remain unchanged</note>.

Coordinates of a Point

In the figure of this problem, <note>the coordinates of point P are (-2, 8)</note>, <note>the coordinates of Q are (6, 8)</note>, <note>the coordinates of R are (0, 4)</note>. These coordinates indicate the specific positions of these points in the Cartesian coordinate system, where the x-coordinate represents the horizontal position and the y-coordinate represents the vertical position.


Question:
Look at this diagram: If

| $\overleftrightarrow{CE}$ |

and

| $\overleftrightarrow{FH}$ |
are parallel lines and m $\angle $ CDG = 55°, what is m $\angle $ HGD? $\Box$ °

Answer: 55°

1. Given that line CE and line FH are parallel lines, the measure of angle CDG is 55°.

2. Observing the figure, we can see that line CE and line FH are intersected by line GD.

3. According to the definition of alternate interior angles, angle CDG and angle HGD are alternate interior angles.

4. Therefore, according to the parallel postulate 2, ∠CDG = ∠HGD.

5. Given ∠CDG = 55°, so ∠HGD is also equal to 55°.

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

Definition of Parallel Lines

<note>Line CE and line FH</note> are located in the same plane, and they do not intersect, so according to the <note>definition of parallel lines</note>, <note>line CE and line FH are parallel lines</note>.

Definition of Alternate Interior Angles

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

In the diagram of this problem, <note>two parallel lines CE and FH</note> are intersected by <note>a line GD</note>, where <note>angle ∠CDG and angle ∠HGD are located between the two parallel lines</note> and <note>on opposite sides of the intersecting line GD</note>. Therefore, <note>angle ∠CDG and angle ∠HGD are alternate interior angles</note>. <note>Alternate interior angles are equal</note>, that is, <note>angle ∠CDG is equal to angle ∠HGD</note>.

Parallel Postulate 2 of Parallel Lines

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

<note>In the diagram of this problem, the two parallel lines CE and FH</note> are intersected by the third line BI, forming the following geometric relationship: <note>Alternate interior angles: angle ∠CDG and angle ∠HGD are equal</note>. These relationships illustrate that when two parallel lines are intersected by a third line, the alternate interior angles are equal.


Question: The surface area of this cylinder is 2,285.92 square millimeters. What is the height?Use 𝜋 ≈ 3.14 and round your answer to the nearest hundredth.h ≈  $\Box$  millimeters

Answer: 12 millimeters

1. According to the problem statement, the total surface area of the cylinder is 2,285.92 square millimeters, using 𝜋≈3.14.

2. The surface area of the cylinder consists of the areas of two circles and the lateral area, so the formula for the total surface area of the cylinder is: Total surface area = 2 × area of the circle + lateral area.

3. Let the radius of the circle be r, then the area of the circle is 𝜋r², therefore the area of two circles is 2 × 𝜋r².

4. The problem states that the radius of the top circle of the cylinder is 14 millimeters, so the area of the circle is 𝜋 × 14² = 615.44 square millimeters.

5. Therefore, the area of two circles is 2 × 615.44 = 1,230.88 square millimeters.

6. Let the height of the cylinder be h, the formula for the lateral area is: lateral area = circumference × height, where the circumference is 2𝜋r.

7. Given r = 14 millimeters, the circumference is 2𝜋 × 14 = 87.92 millimeters.

8. Therefore, the lateral area is 87.92 × h.

9. Combining steps 5 and 8, the total surface area of the cylinder can be expressed as: 1,230.88 + 87.92h.

10. Substituting the given total surface area of 2,285.92 into the formula, we get: 1,230.88 + 87.92h = 2,285.92.

11. Solving this equation: 87.92h = 2,285.92 - 1,230.88 = 1,055.04.

12. Therefore, h = 1,055.04 / 87.92 = 12.00 millimeters.

13. Through the above reasoning, the final answer is h = 12.00 millimeters.

Cylinder

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

In the diagram of this problem, the cylinder consists of two parallel and identical <note>circular bases</note> and a <note>lateral surface</note>. The bases are two identical <note>circles</note> with a <note>radius of 14 mm</note>, and their <note>centers are on the same straight line</note>. The lateral surface is a <note>rectangle</note>, and when unfolded, its height is equal to the <note>height h of the cylinder</note>, and its width is equal to the <note>circumference of the circle</note>.

Area Formula of a Circle

In the figure of this problem, <note>the radius of the top circle of the cylinder is 14 millimeters</note>, according to the area formula of a circle, <note>the area A of the circle is equal to pi (π) multiplied by the square of the radius 14, i.e., A = π×14².</note>

Circumference Formula of Circle

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

<note>The radius of the top circle of the cylinder is 14 millimeters</note>. According to the circumference formula of the circle, the circumference C is equal to <note>2π multiplied by the radius r</note>, which is <note>C=2πr</note>. Therefore, <note>the circumference of the circle is 2π×14=87.92 millimeters</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.

Original: <note>The circumference in the lateral surface area formula is 87.92 mm</note>, therefore the lateral surface area is <note>87.92 × h square millimeters</note>.

Surface Area Formula for a Cylinder

The surface area of a cylinder is equal to the sum of the areas of the two bases and the lateral surface area. The total surface area (SA) is given by the formula: SA = 2πr² + 2πrh, where r is the radius of the base circle, and h is the height of the cylinder.

In the figure of this problem, it is known that <note>the total surface area is 2,285.92 square millimeters</note>, <note>the area of the two circles is 1230.88 square millimeters</note>. Therefore, <note>the lateral surface area of the cylinder can be expressed as 87.92h square millimeters</note>, and we get <note>the total surface area formula for the cylinder: 1,230.88 + 87.92h = 2,285.92 square millimeters</note>.


Question:
TU is tangent to ⨀S. What is m $\angle $ V?

m $\angle $ V= $\Box$ °

Answer: m \$\angle \$ V=70°

1. Given that line TU is the tangent to ⨀S, according to the property of the tangent to a circle, the tangent is perpendicular to the radius at the point of tangency. Therefore, ∠VTU is a right angle, i.e., ∠VTU = 90°.

2. Since △TUV is a right triangle and ∠VTU is one of the angles of this triangle, according to the triangle angle sum theorem (i.e., the sum of the angles in a triangle is 180°), we can obtain ∠TUV + ∠VTU + ∠UVT = 180°.

3. Substituting the degree of ∠VTU, we get ∠TUV + 90° + ∠UVT = 180°.

4. Given that ∠TUV = 20°, substituting this into the equation, we get 20° + 90° + ∠UVT = 180°.

5. Simplifying the equation, we get ∠UVT + 110° = 180°.

6. By subtracting 110° from both sides of the equation, we get ∠UVT = 70°.

7. Through the above reasoning, the final answer is 70°.

Definition of Tangent to a Circle

<note>Circle ⨀S</note> and <note>line TU</note> have exactly one <note>common point T</note>, which is called the point of tangency. Therefore, <note>line TU</note> is the tangent to <note>circle ⨀S</note>.

Definition of Radius

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

<note>In circle ⨀S, point S is the center of the circle</note>, point T is a point on the circumference, <note>line segment ST is the line segment from the center to a point on the circumference</note>, therefore <note>line segment ST is the radius of circle ⨀S</note>.

Definition of Right Triangle

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

<note>In triangle TUV, angle UTV is a right angle (90 degrees)</note>, therefore triangle TUV is a right triangle. <note>Side TV and side UT are the legs</note>, <note>side UV is the hypotenuse</note>.

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

In the figure of this problem, in circle ⨀S, point <note>T is the point of tangency of line TU with the circle</note>, <note>line segment ST is the radius of the circle</note>. According to the property of the tangent line to a circle, <note>the tangent line TU is perpendicular to the radius ST passing through the point of tangency T</note>, that is, <note>∠UTV=90 degrees</note>.

Triangle Angle Sum Theorem

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


Question:
Write the coordinates of the vertices after a translation 4 units left.

E'( $\Box$ , $\Box$ )

F'( $\Box$ , $\Box$ )

G'( $\Box$ , $\Box$ )

H'( $\Box$ , $\Box$ )

Answer:
E'(-2,5)
F'(-2,9)
G'(5,9)
H'(5,5)

1. Determine the coordinates of point E as (2,5).

2. The translation operation specifies moving each vertex of the figure 4 units to the left. Therefore, calculate the new coordinates of E', according to the translation formula, by subtracting 4 from the x-coordinate, i.e., E' = (2-4, 5) = (-2, 5).

3. Determine the coordinates of point F as (2,9).

4. Similarly, the new coordinates of point F after moving 4 units to the left are F' = (2-4, 9) = (-2, 9).

5. Determine the coordinates of point G as (9,9).

6. Move point G 4 units to the left to get its new coordinates as G' = (9-4, 9) = (5, 9).

7. Determine the coordinates of point H as (9,5).

8. After the translation operation, the new coordinates of point H are H' = (9-4, 5) = (5, 5).

9. Through the above steps, each vertex of rectangle EFGH has undergone a translation of 4 units to the left to obtain new coordinates.

10. Therefore, the coordinates of E', F', G', H' are (-2,5), (-2,9), (5,9), (5,5).

Definition of Translation

A translation is a geometric transformation where a figure is moved in the plane along a certain direction, without altering its shape and orientation.

<note>Rectangle EFGH is translated 4 units to the left</note>. The coordinates of vertex E change from <note>(2, 5) to (-2, 5)</note>, the coordinates of vertex F change from <note>(2, 9) to (-2, 9)</note>, the coordinates of vertex G change from <note>(9, 9) to (5, 9)</note>, the coordinates of vertex H change from <note>(9, 5) to (5, 5)</note>. All vertices maintain the shape and size of the figure, and <note>the translated vertices move 4 units relative to their original positions</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 (≅).

In the figure of this problem, rectangle EFGH is formed by <note>point E(2, 5)</note>, <note>point F(2, 9)</note>, <note>point G(9, 9)</note>, <note>point H(9, 5)</note>, where <note>side EF is parallel and equal in length to side GH</note>, and <note>side FG is parallel and equal in length to side EH</note>. After translation, it forms rectangle E'F'G'H', where <note>point E'(-2, 5)</note>, <note>point F'(-2, 9)</note>, <note>point G'(5, 9)</note>, <note>point H'(5, 5)</note>, <note>the properties of the opposite sides of the new rectangle remain unchanged, still maintaining parallel and equal length opposite sides</note>, <note>the angles are still right angles</note>.

Translation Formula

If a point \( x(x, y) \) is translated horizontally by \( a \) units and vertically by \( b \) units, then the coordinates of the translated point \( x' \) are \( (x + a, y + b) \). Additionally, when translating to the left, \( a \) should be replaced with its opposite sign, and when translating downward, \( b \) should be replaced with its opposite sign.

In the figure of this problem, point <note>E(2, 5) horizontally translates -4</note>, resulting in point <note>E'(2-4,5)</note>. Similarly, point <note>F(2, 9) horizontally translates -4</note>, resulting in point <note>F'(2-4,9)</note>. Similarly, point <note>G(9, 9) horizontally translates -4</note>, resulting in point <note>G'(9-4,9)</note>. Similarly, point <note>H(9, 5) horizontally translates -4</note>, resulting in point <note>H'(9-4,5)</note>.


Question:
Write the coordinates of the vertices after a translation 9 units right.

E'( $\Box$ , $\Box$ )

F'( $\Box$ , $\Box$ )

G'( $\Box$ , $\Box$ )

Answer:
E'(4,-3)
F'(6,-3)
G'(8,-9)

1. Determine the original coordinates of point E as (-5, -3). According to the definition of translation, translating 9 units to the right is equivalent to increasing the value of the x-coordinate while the y-coordinate remains unchanged. Therefore, the coordinates of E' are (-5 + 9, -3) = (4, -3).

2. Determine the original coordinates of point F as (-3, -3). Similarly, according to the definition of translation, the x-coordinate increases by 9, and the y-coordinate remains unchanged. Therefore, the coordinates of F' are (-3 + 9, -3) = (6, -3).

3. Determine the original coordinates of point G as (-1, -9). According to the definition of translation, the x-coordinate increases by 9, and the y-coordinate remains unchanged. Therefore, the coordinates of G' are (-1 + 9, -9) = (8, -9).

4. Through the above steps, points E, F, and G are respectively translated to E', F', and G', obtaining their new coordinates.

5. Since this is a translation operation, according to the translation invariance theorem, triangle E'F'G' is congruent to triangle EFG, which means they have the same shape and size but different positions.

6. For each vertex, record the transformation result as indicated by the arrows: E(-5,-3) → E'(4,-3), F(-3,-3) → F'(6,-3), G(-1,-9) → G'(8,-9).

7. Through the above reasoning, the final answer is that the coordinates of point E' are (4, -3), the coordinates of point F' are (6, -3), and the coordinates of point G' are (8, -9).

Definition of Translation

<note>Point E, F, and G were translated 9 units along the x-axis</note>, the original coordinates of E are (-5, -3), after translation the <note>coordinates of E' are (4, -3)</note>; the original coordinates of F are (-3, -3), after translation the <note>coordinates of F' are (6, -3)</note>; the original coordinates of G are (-1, -9), after translation the <note>coordinates of G' are (8, -9)</note>. The translation transformation keeps the y-coordinate unchanged, only adding the translation distance to the x-coordinate.

Coordinates of a Point

In a Cartesian coordinate system, the position of a point is represented by an ordered pair (x, y), where x denotes the horizontal (x-coordinate) and y denotes the vertical (y-coordinate).

<note>The coordinates of point E are (-5, -3)</note>, <note>The coordinates of point F are (-3, -3)</note>, <note>The coordinates of point G are (-1, -9)</note>. After the translation transformation, <note>The new coordinates of point E are (4, -3)</note>, <note>The new coordinates of point F are (6, -3)</note>, <note>The new coordinates of point G are (8, -9)</note>.

Definition of Congruent Triangles

In the diagram of this problem, <note>triangle E'F'G' and triangle EFG are congruent triangles</note>, <note>the corresponding sides and corresponding angles of triangle E'F'G' are equal to those of triangle EFG</note>, that is:
<note>side E'F' = side EF</note>
<note>side F'G' = side FG</note>
<note>side E'G' = side EG</note>,
at the same time, the corresponding angles are also equal:
<note>angle E'F'G' = angle EFG</note>
<note>angle F'E'G' = angle FEG</note>
<note>angle E'G'F' = angle EGF</note>.

Translation Invariance Theorem

In the figure of this problem, the shape <note>EFG</note> is transformed by translation to obtain the shape <note>E'F'G'</note>. The shape and size of the shape <note>EFG</note> and the shape <note>E'F'G'</note> remain unchanged, but the position changes. Specifically, the vertices <note>E</note>, <note>F</note>, and <note>G</note> of the shape <note>EFG</note> are translated to the vertices <note>E'</note>, <note>F'</note>, and <note>G'</note> of the shape <note>E'F'G'</note>, respectively. Therefore, the shape <note>EFG</note> and the shape <note>E'F'G'</note> are congruent.


Question:
Write the coordinates of the vertices after a translation 3 units left and 10 units down.

R'( $\Box$ , $\Box$ )

S'( $\Box$ , $\Box$ )

T'( $\Box$ , $\Box$ )

U'( $\Box$ , $\Box$ )

Answer:
R'(-10,-9)
S'(-10,0)
T'(-9,0)
U'(-9,-9)

1. Given the coordinates of the vertices of rectangle RSTU as R(-7, 1), S(-7, 10), T(-6, 10), U(-6, 1), it needs to be translated 3 units to the left and 10 units downward.

2. According to the definition of translation, for a point (x, y) translated a units to the left and b units downward, its new coordinates are (x - a, y - b).

3. For vertex R(-7, 1), after translation, the coordinates of R' are (-7 - 3, 1 - 10) = (-10, -9).

4. For vertex S(-7, 10), after translation, the coordinates of S' are (-7 - 3, 10 - 10) = (-10, 0).

5. For vertex T(-6, 10), after translation, the coordinates of T' are (-6 - 3, 10 - 10) = (-9, 0).

6. For vertex U(-6, 1), after translation, the coordinates of U' are (-6 - 3, 1 - 10) = (-9, -9).

7. After the above translation, the new coordinates of the vertices of rectangle RSTU are R'(-10, -9), S'(-10, 0), T'(-9, 0), U'(-9, -9).

Definition of Translation

In the figure of this problem, the shape <note>RSTU</note> undergoes a <note>translation</note> transformation to obtain the shape <note>R'S'T'U'</note>. The shape <note>RSTU</note> is translated <note>3 units to the left and 10 units downward</note>, while the shape <note>R'S'T'U'</note> remains unchanged in shape and orientation.

Coordinates of a Point

<note>The coordinates of vertex R (-7, 1)</note> indicate that it is 7 units away from the y-axis and 1 unit away from the x-axis. Similarly, the <note>coordinates of vertices S, T, and U</note> respectively indicate their distances relative to the x-axis and y-axis.

Translation Invariance Theorem

In this problem, we need to translate <note>rectangle RSTU</note> 3 units to the left and 10 units down. Applying the <note>Translation Invariance Theorem</note>, for <note>vertex R(-7, 1)</note>, the new coordinates after translation are <note>R'(-10, -9)</note>; similarly, for <note>vertices S, T, and U</note>, the new coordinates after translation are <note>S'(-10, 0)</note>, <note>T'(-9, 0)</note>, and <note>U'(-9, -9)</note> respectively.


Question:
Look at this diagram: If

| $\overleftrightarrow{MO}$ |

and

| $\overleftrightarrow{PR}$ |
are parallel lines and m $\angle $ PQN = 127°, what is m $\angle $ MNL? $\Box$ °

Answer: 127°

1. Given that line MO and line PR are parallel lines, according to the figure in the problem, line SL is a transversal that intersects the parallel lines at an oblique angle.

2. From the transversal SL, it is known that ∠ PQN and ∠ MNL are on the same side of the two parallel lines.

3. According to the parallel postulate 2 of parallel lines, when two parallel lines are intersected by a transversal, the corresponding angles on the parallel lines are equal, thus ∠ PQN ≅ ∠ MNL.

4. Since it is given in the problem that ∠ PQN = 127°, then according to the definition of corresponding angles, ∠ MNL is also equal to 127°, resulting in ∠ MNL = 127°.

5. Through the above reasoning, the final answer is 127°.

Definition of Parallel Lines

<note>Line MO and line PR are in the same plane</note>, and <note>they do not intersect</note>, therefore according to the definition of parallel lines, <note>line MO and line PR are parallel lines</note>.

Definition of Corresponding Angles

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

In the figure of this problem, <note>two parallel lines MO and PR</note> are intersected by a <note>line SL</note>, where <note>angle PQN and angle MNL are located on the same side of the intersecting line SL, on the same side of the two intersected lines MO and PR</note>. Therefore, <note>angle PQN and angle MNL are corresponding angles. Corresponding angles are equal, that is, angle PQN is equal to angle MNL</note>.

Parallel Postulate 2 of Parallel Lines

<note>In the figure of this problem, two parallel lines MO and PR are intersected by a third line SL</note>, forming the following geometric relationship: 1. Corresponding angles: <note>angle PQN and angle MNL are equal</note>. These relationships illustrate that when two parallel lines are intersected by a third line, the corresponding angles are equal.


Question:
Look at this diagram: If

| $\overleftrightarrow{FH}$ |

and

| $\overleftrightarrow{IK}$ |
are parallel lines and m $\angle $ IJG = 114°, what is m $\angle $ HGJ? $\Box$ °

Answer: 114°

1. According to the problem statement, the line \\(\overleftrightarrow{FH}\\) and \\(\overleftrightarrow{IK}\\) are parallel lines.

2. It is known that \\(\ m\angle IJG = 114^\circ \\). According to the definition of alternate interior angles, here \\(\angle IJG\\) and \\(\angle HGJ\\) are alternate interior angles formed by the line \\(\overline{JG}\\) with the two parallel lines.

3. According to the parallel postulate 2, \\(\angle IJG = \\angle HGJ\\).

4. Therefore, \\(\ m\angle HGJ = m\angle IJG = 114^\circ \\).

5. After the above reasoning, the final answer is \\(\ m\angle HGJ = 114^\circ \\).

Definition of Parallel Lines

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

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

Definition of Alternate Interior Angles

In the diagram of this problem, two parallel lines <note>\(\overleftrightarrow{FH}\)</note> and <note>\(\overleftrightarrow{IK}\)</note> are intersected by a line <note>\(\overline{JG}\)</note>, where <note>\(\angle IJG\)</note> and <note>\(\angle HGJ\)</note> are located between the two parallel lines and on opposite sides of the intersecting line <note>\(\overline{JG}\)</note>. Therefore, <note>\(\angle IJG\)</note> and <note>\(\angle HGJ\)</note> are alternate interior angles. Alternate interior angles are equal, that is, <note>\(\angle IJG = \angle HGJ\)</note>. Given <note>\( m\angle IJG = 114^\circ \)</note>, therefore <note>\( m\angle HGJ = 114^\circ \)</note>.

Parallel Postulate 2 of Parallel Lines

<note>Two parallel lines FH and IK</note> are intersected by a third line <note>JG</note>, forming the following geometric relationship: <note>alternate interior angles: \(\angle IJG\) = \(\angle HGJ\)</note>. This demonstrates that when two parallel lines are intersected by a third line, the alternate interior angles are equal.