Question:
What is the area of this figure?

 $\Box$  square miles

Answer: 19 square miles

1. Given the figure, based on the hints and the provided figure, we can see that the figure is formed by aligning two adjacent rectangles.

2. Consider the small rectangle A in the upper left corner, its left and right sides are each 2 miles, and its top and bottom sides are each 2 miles. This side length information gives us a clear size of rectangle A.

3. Using the rectangle area formula: Area = Length × Width, calculate the area of rectangle A: 2 miles × 2 miles = 4 square miles.

4. Next, consider the larger rectangle B on the right. This rectangle is 5 miles horizontally and 3 miles vertically, thus confirming the size of this rectangle.

5. Using the same rectangle area formula, calculate the area of rectangle B: 5 miles × 3 miles = 15 square miles.

6. By adding the areas of the two parts, we can obtain the total area of the entire figure, which is: 4 square miles + 15 square miles = 19 square miles.

7. Through the above reasoning, the final answer is 19.

Definition of Rectangle

<note>Small rectangle A and large rectangle B both meet the definition of a rectangle: all four of their angles are right angles (90 degrees), and their opposite sides are equal and parallel</note>. Therefore, A and B are both rectangles.

Formula for the Area of a Rectangle

<note>The side lengths of the small rectangle in the upper left corner are 2 miles and 2 miles</note>, so the area of the small rectangle = 2 miles × 2 miles = 4 square miles. <note>The side lengths of the large rectangle on the right are 5 miles and 3 miles</note>, so the area of the large rectangle = 5 miles × 3 miles = 15 square miles.


Question:
What is the area of this figure?

 $\Box$  square meters

Answer: 42 square meters

1. First, consider the shape of the figure. The figure consists of two parts, an irregular shape like the letter L.

2. Use auxiliary lines to divide the irregular shape into two rectangles, denoted as rectangle A and rectangle B.

3. Rectangle A: length of 8 meters, height of 4 meters. Find the area of rectangle A.

4. According to the rectangle area formula: rectangle area = length × width, calculate the area of rectangle A as: 8 meters × 4 meters = 32 square meters.

5. Rectangle B: width of 2 meters, height of 5 meters. Find the area of rectangle B.

6. Similarly, use the rectangle area formula: rectangle area = length × width, calculate the area of rectangle B as: 2 meters × 5 meters = 10 square meters.

7. The total area is the sum of the areas of the two rectangles, which is: 32 square meters + 10 square meters = 42 square meters.

8. Through the above reasoning, the final answer is 42 square meters.

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

<note>Rectangle A has a length of 8 meters and a width of 4 meters</note>, so its area is <note>8 meters × 4 meters = 32 square meters</note>. <note>Rectangle B has a length of 2 meters and a height of 5 meters</note>, so its area is <note>2 meters × 5 meters = 10 square meters</note>. Adding the areas of the two rectangles gives a total area of <note>42 square meters</note>.

Definition of Rectangle

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

<note>In the figure of this problem, in quadrilateral A, each interior angle is a right angle (90 degrees)</note>, and <note>the side of 8m is parallel and equal in length to the side of 8m</note>, <note>the side of 4m is parallel and equal in length to the side of 4m</note>. Therefore, quadrilateral A is a rectangle. <note>In quadrilateral B, each interior angle is a right angle (90 degrees)</note>, and <note>the side of 5m is parallel and equal in length to the side of 5m</note>, <note>the side of 2m is parallel and equal in length to the side of 2m</note>. Therefore, quadrilateral B is a rectangle.


Question:
What is the area of this square?

 $\Box$  square millimeters

Answer: 625 square millimeters

1. The problem states that a square has a side length of 25 mm.

2. According to the definition of a square, a square is a quadrilateral with all sides equal and each internal angle being 90 degrees. We know that the area of a square can be calculated by: Area = side length × side length.

3. Therefore, the area of this square = 25 mm × 25 mm.

4. Calculated as: Area = 625 square mm.

5. Through the above calculation process, the final area of the square is 625 square mm.

Definition of Square

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

<note>Each of the four sides is 25 millimeters long</note>, and <note>each interior angle is 90 degrees</note>, so the quadrilateral is a square.

Area Formula for Square

The area of a square is equal to the square of its side length, that is, Area = side length × side length.

In the figure of this problem, we know <note>the side length of the square is 25 millimeters</note>, so we can use the formula to calculate the area: <note>Area = 25 millimeters × 25 millimeters = 625 square millimeters</note>.


Question:
What is the area of this figure?

 $\Box$  square inches

Answer: 22 square inches

1. Given the figure is composed of two rectangles, where rectangle A has a base length of 7 inches and a height of 2 inches; rectangle B has a width of 2 inches and a height of 4 inches parallel to the base. The figure's corner points are marked with dimensions.

2. First, calculate the area of rectangle A. Using the area formula for a rectangle: Area = Length × Width. Therefore, the area of rectangle A is 7 inches × 2 inches = 14 square inches.

3. Then, calculate the area of rectangle B. Similarly, using the area formula for a rectangle: Area = Length × Width. Therefore, the area of rectangle B is 2 inches × 4 inches = 8 square inches.

4. Add the areas of the two rectangles to obtain the area of the entire shape: 14 square inches (rectangle A) + 8 square inches (rectangle B) = 22 square inches.

5. Through the above reasoning, the final answer is 22 square inches.

Definition of Rectangle

<note>In the figure of this problem, each interior angle of the quadrilateral is a right angle (90 degrees)</note>, and <note>the opposite sides are parallel and equal in length</note>. Therefore, the quadrilateral in the figure is two <note>rectangles</note>.

Formula for the Area of a Rectangle

<note>Using the formula for the area of a rectangle: Area = Length × Width</note>. The area of rectangle A is calculated as <note>7 inches × 2 inches</note> = 14 square inches. The area of rectangle B is calculated as <note>2 inches × 4 inches</note> = 8 square inches.


Question:
What is m $\angle $ G?

m $\angle $ G= $\Box$ °

Answer: m \$\angle \$ G=82°

1. Given that quadrilateral FGHI is a cyclic quadrilateral, according to the theorem (corollary 3 of the inscribed angle theorem) that the opposite angles of a cyclic quadrilateral are supplementary, in quadrilateral FGHI, ∠I and ∠G are supplementary. This is because the sum of the opposite angles of a cyclic quadrilateral is 180°.

2. According to the given conditions, ∠I = 98°.

3. According to the theorem (corollary 3 of the inscribed angle theorem) that the opposite angles of a cyclic quadrilateral are supplementary, specifically in this problem: ∠I + ∠G = 180°.

4. Substituting the given condition ∠I = 98°, we get the equation: 98° + ∠G = 180°.

5. Solving the equation, we get ∠G = 180° - 98°.

6. Calculating, we get ∠G = 82°.

7. Through the above reasoning, the final answer is 82°.

Cyclic Quadrilateral

A quadrilateral is called a cyclic quadrilateral if and only if all four vertices lie on the circumference of a single circle. This circle is referred to as the circumcircle of the quadrilateral.

<note>The four vertices F, G, H, and I of quadrilateral FGHI are all on the same circle</note>. This circle is called the circumcircle of quadrilateral FGHI. Therefore, <note>quadrilateral FGHI is a cyclic quadrilateral</note>. According to the properties of cyclic quadrilaterals, it can be concluded that <note>the sum of opposite angles is equal to 180 degrees, i.e., ∠I + ∠G = 180 degrees, ∠F + ∠H = 180 degrees</note>.

Corollary 3 of the Inscribed Angle Theorem: Diagonal Supplementary Theorem for Cyclic Quadrilateral

In a cyclic quadrilateral (a quadrilateral inscribed in a circle), the sum of each pair of opposite angles is equal to 180°. Additionally, any exterior angle is equal to its interior opposite angle.

<note>In the figure of this problem, in the cyclic quadrilateral FGHI, the vertices F, G, H, and I are all on the circle.</note> According to the Diagonal Supplementary Theorem for Cyclic Quadrilateral, <note>the sum of each pair of opposite angles of the quadrilateral FGHI is equal to 180°.</note> Specifically, <note>∠I + ∠G = 180°; ∠F + ∠H = 180°.</note> In this problem, <note>it is known that ∠I = 98°, substituting it in, we get ∠G = 180° - 98° = 82°.</note>


Question:
Find WX and the area of △XYZ.

Write your answer as an integer or as a decimal rounded to the nearest tenth.
WX= $\Box$  ydArea= $\Box$  yd ^ 2

Answer: WX=36.9 ydArea=738.0 yd ^ 2

1. By observing the triangle △WXZ, it is found to be a right triangle, where ∠W is a right angle.

2. In the right triangle △WXZ, it is known that side XZ=44 yd, ∠XZW = 57°, and we need to find side WX.

3. According to the definition of the sine function, sin(∠XZW) = opposite side WX / hypotenuse XZ.

4. Substitute the known values into the formula: sin(57°) = WX / 44.

5. Find the approximate value of sin(57°), getting sin(57°) ≈ 0.8386.

6. Substitute into the calculation: 0.8386 = WX / 44, solving for WX gives WX = 0.8386 * 44.

7. Calculate: WX ≈ 36.9015 yd.

8. Now, calculate the area of △XYZ.

9. According to the formula for the area of a triangle, Area = 1/2 * base * height.

10. In △XYZ, YZ is the base, WX is the height, so Area = 1/2 * YZ * WX.

11. Substitute the known values, YZ = 40 yd, WX ≈ 36.9015 yd, Area = 1/2 * 40 * 36.9015.

12. Calculate: Area ≈ 738.0300 yd².

13. Round the answers to one decimal place, WX is approximately 36.9 yd, Area is approximately 738.0 yd².

14. Through the above reasoning, the final answers are WX=36.9 yd, Area=738.0 yd².

Definition of Right Triangle

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

In the figure of this problem, triangle <note>△WXZ</note> has <note>angle ∠W as a right angle (90 degrees)</note>, so triangle <note>△WXZ is a right triangle</note>. <note>Side WX and side WZ are the legs</note>, <note>side XZ is the hypotenuse</note>.

Definition of Sine Function

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

In the right triangle △WXZ, angle ∠XZW is an acute angle, side WX is the opposite side of angle ∠XZW, and side XZ is the hypotenuse. According to the definition of the sine function, the sine value of angle ∠XZW is equal to the ratio of the opposite side WX to the hypotenuse XZ, that is, sin(∠XZW) = WX / XZ.

Area Formula of a Triangle

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

<note>In triangle XYZ</note>, <note>side YZ is the base, segment WX is the height on this base</note>, so <note>the area of triangle XYZ</note> is equal to <note>base YZ multiplied by height WX divided by 2</note>, that is, <note>area = (YZ * WX) / 2</note>.


Question:
What is the area?

 $\Box$  square miles

Answer: 324 square miles

1. Based on the information in the figure, we know this is a square, meaning all sides are equal in length and each angle is a right angle.

2. Observing the side lengths given in the figure, each side of the square is 18 mi.

3. To find the area of the square, we need to use the square area formula: Area = side length × side length.

4. Substitute the known side length value of 18 mi: Area = 18 mi × 18 mi.

5. Calculate to get the area as 324 square mi.

6. All relevant calculations are in miles, so the unit of area is square miles.

7. Through the above reasoning, the final answer is 324 square mi.

Definition of Square

The original text: <note>All sides are 18 miles long</note>, and <note>all four angles are right angles (90 degrees)</note>, so this is a square.

Area Formula for Square

Original: <note>Side length is 18 miles</note>, so the area = <note>18 miles × 18 miles</note> = 324 square miles.


Question:
Find the area of △UVW.

Write your answer as an integer or as a decimal rounded to the nearest tenth.
 $\Box$  cm ^ 2

Answer: 86.4 cm ^ 2

1. Given that the measure of angle V is 20°, the measure of angle W is 42°, and the length of side WU = 10 cm. To find the area of triangle △UVW, we need to know the lengths of two sides and the measure of the included angle.

2. First, find the measure of angle U. According to the triangle angle sum theorem, the sum of the three interior angles of a triangle is 180°. Therefore, the measure of angle U can be found using the formula ∠U = 180° - ∠V - ∠W = 180° - 20° - 42° = 118°.

3. Next, use the Law of Sines to find the length of side UV. The Law of Sines states that in any triangle, the ratio of the length of a side to the sine of its opposite angle is equal to the ratio of the lengths of the other sides to the sines of their opposite angles. Thus, WU/sin(∠V) = UV/sin(∠W). Substituting the known values gives 10/sin(20°) = UV/sin(42°).

4. Calculate sin(20°) ≈ 0.3420 and sin(42°) ≈ 0.6691, then substitute to get 10/0.3420 = UV/0.6691.

5. Continue solving the equation to get UV = (10 × 0.6691) / 0.3420 = 19.564 cm.

6. We now have the length of side UV = 19.564 cm, the length of side WU = 10 cm, and the measure of the included angle U is 118°.

7. Use the triangle area formula A = 1/2 × a × b × sin(C), where a and b are the lengths of two sides, and C is the measure of the included angle. Therefore, the area of △UVW is A = 1/2 × 19.564 × 10 × sin(118°).

8. Calculate sin(118°) ≈ 0.8829, then substitute to get the area A = 1/2 × 19.564 × 10 × 0.8829 = 86.3704.

9. Round the result 86.3704 to the nearest tenth, finally obtaining the area of triangle △UVW as 86.4 square centimeters.

Definition of an Angle

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

In the diagram of this problem, <note>∠V is a geometric figure formed by rays UV and VW</note>, <note>these two rays have a common endpoint V</note>. <note>This common endpoint V is called the vertex of ∠V</note>, and <note>rays UV and VW are called the sides of ∠V</note>. <note>∠W is a geometric figure formed by rays UW and VW</note>, <note>these two rays have a common endpoint W</note>. <note>This common endpoint W is called the vertex of ∠W</note>, and <note>rays UW and VW are called the sides of ∠W</note>. <note>∠U is a geometric figure formed by rays UV and UW</note>, <note>these two rays have a common endpoint U</note>. <note>This common endpoint U is called the vertex of ∠U</note>, and <note>rays UV and UW are called the sides of ∠U</note>.

Definition of Sine Function

<note>In triangle UVW, angles V, W, and U are 20°, 42°, and 118° respectively</note>. According to the definition of the sine function, <note>sin(20°) ≈ 0.3420</note>, <note>sin(42°) ≈ 0.6691</note>, <note>sin(118°) ≈ 0.8829</note>.

Triangle Angle Sum Theorem

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

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

Sine Theorem

In any triangle, the ratio of the length of each side to the sine of its opposite angle is constant and equal to the diameter of the circumcircle, i.e., \( \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = 2r = D \) (where \( r \) is the radius of the circumcircle, and \( D \) is its diameter).

In <note>△UVW</note>, the sides <note>UV, VW, and WU</note> correspond to <note>∠W, ∠U, and ∠V</note> respectively. According to the <note>Sine Theorem</note>, the ratio of the lengths of the sides to the sine values of their opposite angles is equal and is equal to the diameter of the circumscribed circle, i.e., <note>UV/sin(∠W) = VW/sin(∠U) = WU/sin(∠V) = 2r = D</note> (where <note>r is the radius of the circumscribed circle, D is the diameter</note>). In this problem, using the Sine Theorem we have <note>UV/sin(∠W) = WU/sin(∠V)</note>, substituting the known values gives <note>10/sin(20°) = UV/sin(42°)</note>. By calculation, the side <note>UV ≈ 19.564 cm</note>.

Triangle Area Formula (Using Sine Function)

The area \( S \) of any triangle can be expressed as \( S = \frac{1}{2} \cdot a \cdot b \cdot \sin(C) \), where \( a \) and \( b \) are the lengths of two sides, and \( C \) is the angle between these two sides.

<note>In the figure of this problem, in triangle UVW, side WU and side UV are u and w respectively</note>, <note>angle WUV is the included angle C between these two sides</note>. According to the triangle area formula, <note>the area S of triangle UVW can be expressed as S = (1/2) * u * w * sin(WUV)</note>, that is, <note>S = (1/2) * WU * VU * sin(WUV)</note>.


Question:
Find the area of △XYZ.

Write your answer as an integer or as a decimal rounded to the nearest tenth.
 $\Box$  m ^ 2

Answer: 2.8 m ^ 2

1. Given m ∠ Y = 26°, m ∠ X = 132°. According to the triangle angle sum theorem, the sum of the interior angles of a triangle is 180°, thus m ∠ Y + m ∠ X + m ∠ Z = 180°.

2. Substituting the given angles, we get 26° + 132° + m ∠ Z = 180°.

3. Solving the above equation, we get 158° + m ∠ Z = 180°, therefore m ∠ Z = 22°.

4. According to the sine rule, calculate the length of side ZX |ZX| = y. The sine rule formula is a/sin(A) = b/sin(B) = c/sin(C), where a, b, c are the sides of the triangle, and A, B, C are the angles opposite to these sides.

5. Apply the given conditions and substitute into the sine rule x/sin(X) = y/sin(Y).

6. Given |YZ| = 5 m, which is the side opposite ∠ X, corresponding angle ∠ X = 132° and ∠ Y = 26°.

7. Substitute the values: 5/sin(132°) = y/sin(26°).

8. Calculate sin(132°) ≈ 0.7431 and sin(26°) ≈ 0.4384.

9. Rearrange to get 5/0.7431 = y/0.4384.

10. Solve the equation to get y = 5 * (0.4384/0.7431) ≈ 2.9498.

11. Use the triangle area formula (using sine function) to calculate ∆XYZ, the area formula is (1/2)ab⋅sin(C), where a, b are the known sides, and C is the included angle.

12. In this problem, a = |YZ| = 5 m, b ≈ 2.9498 m, and the included angle C is ∠ Z = 22°.

13. Substitute into the area formula, we get Area = (1/2) * 5 * 2.9498 * sin(22°).

14. Calculate sin(22°) ≈ 0.3746.

15. Calculate Area = 0.5 * 5 * 2.9498 * 0.3746.

16. Perform the calculation to get Area ≈ 2.7625.

17. Round the result to one decimal place, Area ≈ 2.8 m².

18. Through the above reasoning, the final answer is 2.8 m².

Definition of an Angle

In the figure of this problem, the angle <note>∠XYZ is a geometric figure formed by rays XY and YZ, which share a common endpoint Y</note>. This common endpoint Y is called the <note>vertex of angle ∠XYZ</note>, and rays XY and YZ are called the <note>sides of angle ∠XYZ</note>. The angle <note>∠YXZ is a geometric figure formed by rays YX and XZ, which share a common endpoint X</note>. This common endpoint X is called the <note>vertex of angle ∠YXZ</note>, and rays YX and XZ are called the <note>sides of angle ∠YXZ</note>. The angle <note>∠XZY is a geometric figure formed by rays XZ and YZ, which share a common endpoint Z</note>. This common endpoint Z is called the <note>vertex of angle ∠XZY</note>, and rays XZ and YZ are called the <note>sides of angle ∠XZY</note>.

Triangle Angle Sum Theorem

In the figure of this problem, in triangle XYZ, <note>angles ∠X, ∠Y, and ∠Z are the three interior angles of triangle XYZ</note>, according to the Triangle Angle Sum Theorem, <note>∠X + ∠Y + ∠Z = 180°</note>.

Sine Theorem

<note>In any triangle △XYZ, sides XZ, YZ, and XY correspond to angles ∠Y, ∠X, and ∠Z respectively</note>. According to the Sine Theorem, the ratio of the lengths of the sides to the sine of their opposite angles is equal to the diameter of the circumscribed circle, that is: <note>XZ/sin(∠Y)=YZ/sin(∠X)=XY/sin(∠Z) = 2r = D (where r is the radius of the circumscribed circle, D is the diameter)</note>. In this problem, it is known that the opposite side <note>YZ=5 m, corresponding angle ∠X=132°</note>; the opposite side <note>XZ=y corresponding angle ∠Y=26°</note>, applying the Sine Theorem we get <note>5/sin(132°)=y/sin(26°)</note>.

<note>Side ZY and side XZ are respectively a and b</note>, <note>angle XZY is the included angle C between these two sides</note>. According to the triangle area formula, the area S of triangle XYZ can be expressed as S = (1/2) * a * b * sin(C), that is, <note>S = (1/2) * ZY * XZ * sin(XZY)</note>.


Question:
What is the area of this square?

 $\Box$  square millimeters

Answer: 625 square millimeters

1. Given that the side length of the square is 25 mm.

2. According to the definition of a square, a square is a quadrilateral with four equal sides and four right angles (90 degrees).

3. Therefore, the formula for the area of a square is side×side, which is the square of the side length.

4. Applying the formula, the area of the square = 25 mm × 25 mm.

5. The calculation yields 625 square mm.

6. Through the above reasoning, the final answer is 625 square mm.

Definition of Square

In the diagram of this problem, we <note>have a square</note>, <note>its side length is 25 millimeters</note>, <note>each side is 25 millimeters long</note>, and <note>all four interior angles are 90 degrees</note>.

Area Formula for Square

The original text: <note>边长为25毫米</note>, therefore applying the area formula, the area of the square = <note>25毫米</note> × <note>25毫米</note> = 625 square millimeters.


Question:
What is the area of this figure?

 $\Box$  square inches

Answer: 84 square inches

1. Observe the given figure and find that it can be divided into two rectangles by a horizontal line: Rectangle A and Rectangle B.

2. Confirm the dimensions of Rectangle A: the length of the base is 8 inches, and the height is 9 inches.

3. According to the rectangle area formula, the area of a rectangle equals the base length multiplied by the height. Calculate the area of Rectangle A: 8 inches × 9 inches = 72 square inches.

4. Confirm the dimensions of Rectangle B: the length of the base is 4 inches, and the height is 3 inches.

5. Apply the rectangle area formula again to calculate the area of Rectangle B: 4 inches × 3 inches = 12 square inches.

6. Add the areas of Rectangle A and Rectangle B to get the total area of the original figure: 72 square inches + 12 square inches = 84 square inches.

7. Through the detailed reasoning steps above, the final area of the original figure is 84 square inches.

Definition of Rectangle

<note>Divide the figure into two rectangles, Rectangle A and Rectangle B, using a horizontal line</note>. <note>The base length of Rectangle A is 8 inches, and the height is 9 inches</note>. <note>The base length of Rectangle B is 4 inches, and the height is 3 inches</note>. <note>The interior angles of Rectangle A and Rectangle B are right angles (90 degrees)</note>, and the opposite sides are parallel and equal in length.

Formula for the Area of a Rectangle

<note>The length of the base of rectangle A is 8 inches, and the height is 9 inches</note>, applying the formula for the area of a rectangle: <note>The area of rectangle A = 8 inches × 9 inches = 72 square inches</note>. <note>The length of the base of rectangle B is 4 inches, and the height is 3 inches</note>, applying the formula for the area of a rectangle: <note>The area of rectangle B = 4 inches × 3 inches = 12 square inches</note>.


Question:
What is the area?

 $\Box$  square miles

Answer: 323 square miles

1. Given that the length of the rectangle is 19 miles and the width is 17 miles.

2. According to the formula for calculating the area of a rectangle: Area = Length × Width, we can obtain the area as Area = 19 × 17.

4. Since the units of length and width are miles, the unit of area is square miles.

5. Based on the above reasoning, the final answer is 323 square miles.

Definition of Rectangle

<note>The two opposite sides of the quadrilateral are 19 miles and 17 miles respectively</note>, <note>each interior angle is a 90-degree right angle</note>, <note>the opposite sides are parallel and equal in length</note>.

Formula for the Area of a Rectangle

<note>In a rectangle, the length is 19 miles and the width is 17 miles</note>, so the area of the rectangle = 19 * 17 = 323 square miles.


Question:
What is the area?

 $\Box$  square yards

Answer: 960 square yards

1. The problem provides a rectangle, and we need to find the area of this rectangle.

2. According to the definition of a rectangle, the area of the rectangle is equal to its base length multiplied by its height, i.e., A = base * height.

3. From the problem-solving steps, we know the base length of the rectangle is 32 yards, denoted as base = 32.

4. Similarly, from the problem-solving steps, we know the height of the rectangle is 30 yards, denoted as height = 30.

5. Substituting into the above formula, the area A = base * height = 32 * 30.

6. Performing the multiplication, 32 * 30 = 960.

7. Since the unit of length in the problem is yards, the unit of area should be square yards.

8. Therefore, the area of the rectangle is 960 square yards.

Definition of Rectangle

<note>A quadrilateral is a rectangle</note>, with all its interior angles being <note>right angles (90 degrees)</note>, and <note>opposite sides are parallel and equal in length</note>. According to the problem's provided <note>base length of 32 yards and height of 30 yards</note>, the area of the rectangle is 960 square yards.

Formula for the Area of a Rectangle

<note>The length of the base of the rectangle is 32 yards</note>, <note>the height is 30 yards</note>. According to the formula for the area of a rectangle, <note>the area of the rectangle A = base length * height = 32 yards * 30 yards = 960 square yards</note>.


Question:
What is the area of this square?

 $\Box$  square feet

Answer: 4 square feet

1. First, observe the side length marked in the figure. We find that one side length of the square is 2 ft.

2. According to the definition of a square, all four sides of a square are equal. Therefore, in this problem, each side length of the square is 2 ft.

3. The formula to calculate the area of a square is: S = side length × side length.

4. Substitute the known side length into the formula: S = 2 ft × 2 ft.

5. Calculate to get: S = 4 square ft.

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

Definition of Square

In the diagram of this problem, we observe a <note>square with a side length of 2 feet</note>. According to the definition of a square, this square's <note>four sides are equal, each with a length of 2 feet</note>, and <note>all four angles are right angles (90°)</note>.

Area Formula for Square

<note>The length of side AB is 2 feet</note>. According to the definition of a square, <note>sides BC, CD, and DA are also 2 feet</note>. Use the square area formula to calculate the area: <note>S = AB × BC = 2 feet × 2 feet = 4 square feet</note>.


Question:
What is the area?

 $\Box$  square miles

Answer: 88 square miles

1. From the figure provided in the problem, it can be seen that the given figure is a rectangle.

2. In a rectangle, the formula for calculating the area is: Area = Length × Width.

3. From the figure, it can be seen that the length of the rectangle is 11 mi and the width is 8 mi.

4. Substitute the given length and width into the area formula, resulting in Area = 11 mi × 8 mi.

5. Perform the multiplication, 11 mi × 8 mi = 88 square miles.

6. Since the unit of area is square miles, the final area of the rectangle is 88 square miles.

Definition of Rectangle

In the figure of this problem, <note>the quadrilateral is a rectangle</note>, <note>its interior angles are all right angles (90 degrees)</note>, and the side lengths are 11 miles and 8 miles. <note>The opposite sides of the rectangle are parallel and equal in length</note>, which conforms to the definition of a rectangle.

Formula for the Area of a Rectangle

In the figure of this problem, <note>the length of the rectangle is 11 miles, and the width is 8 miles</note>. According to the formula for the area of a rectangle, the area of the rectangle is equal to the product of its length and width, that is, <note>Area = 11 miles × 8 miles = 88 square miles</note>.


Question:
What is the area of this square?

 $\Box$  square meters

Answer: 1156 square meters

1. Given that the side length of a square is 34 meters. According to the definition of a square, all sides are equal, so the side length of this square is 34 meters.

2. The formula for calculating the area of a square is the square of the side length, which is the side length multiplied by the side length.

3. Substitute the side length value into the formula: Area = 34 meters × 34 meters.

4. By calculation, 34 × 34 = 1,156.

5. The unit of area is the square of the unit of side length, so the area of this square is 1,156 square meters.

6. Therefore, through the above reasoning, the final answer is: 1,156 square meters.

Definition of Square

The original text: <note>All four sides are 34 meters</note>, and <note>all four interior angles are right angles (90 degrees)</note>, therefore the quadrilateral is a square.

Area Formula for Square

Original text: <note>The side length of the square is 34 meters</note>, according to this formula, area = <note>34 meters × 34 meters</note>.


Question:
What is the area of this figure?

 $\Box$  square millimeters

Answer: 96 square millimeters

1. According to the given figure, it is observed that the figure can be divided into two rectangles, labeled as rectangle A and rectangle B.

2. Determine the dimensions of rectangle A. According to the description in the problem, the width of rectangle A is 6 mm and the height is 4 mm.

3. Using the rectangle area formula: Area = Length × Width, calculate the area of rectangle A as 6 mm × 4 mm = 24 square mm.

4. Determine the dimensions of rectangle B. According to the description in the problem, the width of rectangle B is 8 mm and the height is 9 mm.

5. Using the rectangle area formula: Area = Length × Width, calculate the area of rectangle B as 8 mm × 9 mm = 72 square mm.

6. The two rectangles form the entire figure, so the area of the entire figure is the sum of the areas of rectangle A and rectangle B.

7. Add the areas of rectangle A and rectangle B: 24 square mm + 72 square mm = 96 square mm.

8. Through the above reasoning, the final answer is 96 square mm.

Definition of Rectangle

In the figure of this problem, the two marked rectangles are <note>Rectangle A and Rectangle B</note>. <note>Rectangle A has a width of 6 mm and a height of 4 mm</note>, <note>Rectangle B has a width of 8 mm and a height of 9 mm</note>. Each rectangle conforms to the <note>definition of a rectangle</note>, which means <note>having four right angles</note> and <note>opposite sides of equal length</note>.

Formula for the Area of a Rectangle

<note>The area of rectangle A</note> is calculated according to the formula: <note>length 6 mm</note> multiplied by <note>width 4 mm</note>, resulting in 24 square millimeters. <note>The area of rectangle B</note> is calculated according to the formula: <note>length 8 mm</note> multiplied by <note>width 9 mm</note>, resulting in 72 square millimeters. Finally, <note>adding the areas of the two rectangles</note> results in the total area of the entire figure being 96 square millimeters.


Question:
What is the area of this figure?

 $\Box$  square meters

Answer: 70 square meters

1. Divide the given figure into two rectangles, referred to as Rectangle A and Rectangle B.

2. Observe Rectangle A, its width is 8 meters and its height is 2 meters.

3. According to the formula for the area of a rectangle, Area = Width × Height, we can find the area of Rectangle A: 8 meters × 2 meters = 16 square meters.

4. Next, check the dimensions of Rectangle B, its width is 9 meters and its height is 6 meters.

5. Using the same area formula, calculate the area of Rectangle B: 9 meters × 6 meters = 54 square meters.

6. Add the areas of the two rectangles to get the total area: 16 square meters + 54 square meters = 70 square meters.

7. Through the above reasoning, the final answer is 70 square meters.

Definition of Rectangle

<note>Rectangle A</note> and <note>Rectangle B</note> both meet the definition of a rectangle. <note>Rectangle A's four interior angles are all right angles (90 degrees)</note>, with side lengths of <note>8 meters and 2 meters</note>, and <note>opposite sides are parallel and equal in length</note>. <note>Rectangle B's four interior angles are also right angles (90 degrees)</note>, with side lengths of <note>9 meters and 6 meters</note>, and <note>opposite sides are parallel and equal in length</note>.

Formula for the Area of a Rectangle

<note>The width of rectangle A is 8 meters</note>, <note>the height is 2 meters</note>, so <note>the area of rectangle A = 8 meters × 2 meters = 16 square meters</note>. <note>The width of rectangle B is 9 meters</note>, <note>the height is 6 meters</note>, so <note>the area of rectangle B = 9 meters × 6 meters = 54 square meters</note>. Total area = <note>area of rectangle A + area of rectangle B = 16 square meters + 54 square meters = 70 square meters</note>.


Question:
The diagonals of this rhombus are 10 feet and 2 feet.

What is the area of the rhombus?
 $\Box$  square feet

Answer: 10 square feet

1. Given that the diagonals of the rhombus are 10 feet and 2 feet. The diagonals of the rhombus are perpendicular to each other and bisect each other. According to the properties of the diagonals, the diagonals of the rhombus divide the rhombus into four congruent right triangles.

2. Let the lengths of the diagonals be d1=10 feet, d2=2 feet. Since the diagonals bisect each other, according to the properties of the diagonals, each diagonal is divided into two parts of 5 feet and 1 foot respectively.

3. Consider one of the right triangles formed by the two diagonals. The two legs of the right triangle are 5 feet and 1 foot respectively.

4. According to the formula for the area of a triangle (A=0.5×base×height), the area of one right triangle is 0.5×5×1=2.5 square feet.

5. Since the diagonals divide the rhombus into four identical right triangles, the total area of the rhombus A=4×2.5=10 square feet.

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

Definition of Rhombus

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

<note>In the diagram of this problem, the diagonals of the quadrilateral are 10 feet and 2 feet (d1=10 feet, d2=2 feet)</note>, according to the <note>definition of the properties of rhombus diagonals</note>, <note>these two diagonals are perpendicular to each other and bisect each other</note>, each diagonal is divided into <note>two parts of 5 feet and 1 foot</note>. <note>Each diagonal divides the rhombus into four congruent right triangles</note>.

Area Formula of a Triangle

In the figure of this problem, in the triangle, <note>the 5ft long side is the base</note>, <note>the 1ft long side is the height on that base</note>, so the area of the triangle is equal to the base 5 multiplied by the height 1 divided by 2, that is, Area = (5 * 1) / 2.


Question:
The diagonals of this rhombus are 3 millimeters and 10 millimeters.

What is the area of the rhombus?
 $\Box$  square millimeters

Answer: 15 square millimeters

1. According to the properties of the diagonals of a rhombus, the diagonals of a rhombus are perpendicular to each other and bisect each other. Therefore, in this rhombus, the known diagonals are d1=3 mm and d2=10 mm.

2. Because the diagonals are perpendicular and bisect each other, the rhombus is divided into four congruent right triangles by the diagonals.

3. Using the area formula of a rhombus: the area of a rhombus is equal to half the product of its diagonals. That is, A=1/2×d1×d2.

4. Substitute the known values into the formula: A=1/2×3 mm×10 mm.

5. Calculate: A=1/2×30=15.

6. Through the above reasoning, the final area of the rhombus is 15 square millimeters.

Definition of Rhombus

<note>All sides are equal</note>, so the quadrilateral is a rhombus. In addition, <note>the diagonals of the quadrilateral bisect each other at right angles</note>, that is, the diagonals intersect at <note>point O</note>, and <note>angle ∠AOB is a right angle (90 degrees)</note>, and <note>AO=OC</note> and <note>BO=OD</note>.

Rhombus Area Formula

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

<note>In a rhombus, diagonals d1 and d2 are the two diagonals of the rhombus</note>. According to the rhombus area formula, the area of the rhombus can be determined by its diagonals, i.e., Area = (d1 × d2) / 2. Therefore, the area of the rhombus = <note>(3 × 10) / 2</note>.


Question:
What is the area?

 $\Box$  square miles

Answer: 1080 square miles

1. Given that the length of the rectangle is 40 miles and the width is 27 miles.

2. According to the formula for the area of a rectangle: Area = Length × Width.

3. Substitute the given length and width into the formula: Area = 40 miles × 27 miles.

4. Calculate to get: Area = 1080 square miles.

5. Therefore, the area of the rectangle is 1080 square miles.

Definition of Rectangle

<note>Quadrilateral ABCD is a rectangle</note>, <note>its interior angles ∠A, ∠B, ∠C, and ∠D are all right angles (90 degrees)</note>, and <note>sides AB and CD are parallel and equal in length</note>, <note>sides AD and BC are parallel and equal in length</note>. The length of the rectangle is 40 miles, corresponding to <note>sides AB and CD</note>; the width of the rectangle is 27 miles, corresponding to <note>sides AD and BC</note>.

Formula for the Area of a Rectangle

<note>In a rectangle, the side length of 40 miles and the side width of 27 miles are the length and width of the rectangle</note>, so <note>the area of the rectangle = 40 miles * 27 miles = 1080 square miles</note>.


Question:
The diagonals of this rhombus are 1 kilometer and 4 kilometers.

What is the area of the rhombus?
 $\Box$  square kilometers

Answer: 2 square kilometers

1. Given that the rhombus ABCD has two diagonals of 1 km and 4 km respectively. Assume these diagonals are diagonal AC and diagonal BD respectively.

2. The area of the rhombus is equal to half the product of the lengths of its two diagonals, the formula is Area = 1/2 × AC × BD.

3. Substituting the given diagonal lengths: Area = 1/2 × 1 km × 4 km = 2 km².

4. Through the above reasoning, the final answer is 2 km².

Definition of Rhombus

<note>All sides of a rhombus are equal in length</note>, and <note>the diagonals are perpendicular bisectors of each other</note>. That is, <note>the diagonals intersect at point O</note>, and form a <note>right angle (90 degrees)</note>.

Properties of the Diagonals of a Rhombus

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

In the diagram of this problem, in rhombus ABCD, <note>the diagonals AC and BD bisect each other and are perpendicular to each other</note>. Specifically, <note>point O is the intersection of diagonals AC and BD</note>, and <note>OA = OC = 0.5 kilometers, OB = OD = 2 kilometers</note>. Additionally, <note>angles AOB, BOC, COD, and DOA are all right angles (90 degrees)</note>, so <note>the diagonals AC and BD are perpendicular to each other</note>.

Rhombus Area Formula

<note>The two diagonals of the rhombus are 1 kilometer and 4 kilometers respectively</note>. According to the <note>Rhombus Area Formula</note>, the area of the rhombus is equal to half the product of the two diagonals, i.e., Area = (<note>1 kilometer</note> * <note>4 kilometers</note>) / 2 = <note>2 square kilometers</note>.


Question:
The diagonals of this rhombus are 4 meters and 2 meters.

What is the area of the rhombus?
 $\Box$  square meters

Answer: 4 square meters

1. Given that the diagonals of the rhombus are 4 meters and 2 meters respectively.

2. According to the properties of the rhombus: the diagonals of the rhombus are perpendicular to each other and bisect each other. Let the diagonals of the rhombus be AC and BD, intersecting at point O. Therefore, OA=OC=2 meters, OB=OD=1 meter.

3. Thus, the rhombus is divided into four congruent right triangles by the diagonals: triangle AOB, triangle BOC, triangle COD, triangle DOA.

4. Consider one of the right triangles, for example, triangle AOB, the right angle is at point O, and AO=2 meters, BO=1 meter.

5. According to the area formula of a right triangle: Area = (1/2) × product of the lengths of the legs, the area of triangle AOB is (1/2) × 2 meters × 1 meter = 1 square meter.

6. Since these right triangles have equal areas, and there are four of them, the total area of the rhombus is 4 × 1 square meter = 4 square meters.

7. Through the above reasoning, the final answer is 4 square meters.

Definition of Rhombus

<note>In the figure of this problem, all sides of the quadrilateral are equal</note>, so the quadrilateral is a rhombus. <note>Additionally, the diagonals of the rhombus are perpendicular bisectors of each other</note>, meaning the diagonals intersect at the center point.

Definition of Right Triangle

<note>All four triangles in the rhombus are right triangles</note>. <note>The four angles in the middle are all right angles (90 degrees)</note>, therefore these triangles are all right triangles.

Area Formula of a Triangle

The original text: <note>2m long side is the base</note>, <note>1m long side is the height on that base</note>, so the area of the triangle is equal to the base 2 multiplied by the height 1 divided by 2, i.e., area = <note>(2 * 1) / 2</note>.


Question:
The radius of a circle is 6 kilometers. What is the area of a sector bounded by a 180° arc?

Give the exact answer in simplest form.
 $\Box$ 𝜋/ square kilometers

Answer: 18𝜋 square kilometers

1. Given that the radius of the circle is 6 kilometers, first use the formula for the area of a circle: Area A = 𝜋r², where r is the radius of the circle.

2. Substitute the radius r = 6 into the formula to get A = 𝜋 × 6² = 36𝜋 square kilometers. Therefore, the area of the entire circle is 36𝜋 square kilometers.

3. The problem requires determining the area of the sector defined by a 180° arc. Given that the corresponding central angle θ of the circle is 180°.

4. According to the formula for the area of a sector of a circle, sector area K = (θ/360) × A, where θ is the central angle and A is the area of the entire circle.

5. Substitute θ = 180° and A = 36𝜋 square kilometers into the formula to get: K = (180/360) × 36𝜋 = 0.5 × 36𝜋 = 18𝜋 square kilometers.

6. Through the above reasoning, the final answer is 18𝜋 square kilometers.

Definition of Radius

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

<note>In the diagram of this problem, the center of the circle is the center point</note>, <note>any point on the circumference is a circumference point</note>, <note>the length of the line segment from the center to the circumference point is 6 kilometers</note>. The radius of the circle refers to the length of the line segment from the center to any point on the circumference.

Area Formula of a Circle

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

<note>The radius of the circle is 6 kilometers</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 6</note>, which is A = π × 6² = 36π square kilometers.

Definition of Central Angle

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

The <note>central angle θ</note> is 180°, formed by the <note>line segment connecting two points on the circumference from the center</note>.

Formula for the Area of a Sector

The area \( X \) of a sector can be calculated using the formula \( X = \frac{\theta}{360} \times \pi \times r^2 \), where \( \theta \) is the measure of the central angle in degrees, and \( r \) is the radius length.

In the given diagram, in the sector, <note>the central angle is θ degrees</note>, <note>the radius is r</note>. According to the formula for the area of a sector, the area A of the sector can be calculated using the formula A = (<note>θ/360</note>) * π * <note>r²</note>, where <note>θ is the central angle in degrees</note>, <note>r is the radius</note>. Therefore, <note>the area of the sector A = (θ/360) * π * r²</note>.


Question:
Find UX and the area of △UVW.

Write your answer as an integer or as a decimal rounded to the nearest tenth.
UX= $\Box$  ydArea= $\Box$  yd ^ 2

Answer: UX=19.9 ydArea=258.7 yd ^ 2

1. In △UWX, according to the given conditions, it is known that UW=24 yd, ∠UWX=56°.

2. To find the length of UX, since ∠UXW is a right angle, according to the definition of a right triangle, △UWX is a right triangle. According to the definition of the sine function, sin∠UWX=opposite side (UX)/hypotenuse (UW).

3. According to the definition of the sine function, sin(∠UWX)=UX/UW, therefore: sin(56°)=UX/24.

4. Calculate sin(56°), we get sin(56°)≈0.8290.

5. Substitute the sine value into the equation: 0.8290=UX/24.

6. Solve the equation UX=0.8290×24, we get UX=19.896 yd.

7. Round to the nearest tenth, we get UX≈19.9 yd.

8. Next, calculate the area of △UVW, using the formula for the area of a triangle: Area = 1/2 × base × height.

9. Choose VW as the base, according to the definition of height, UX is the height, therefore the area is 1/2 × VW × UX.

10. It is known that VW=26 yd, therefore the area=1/2 × 26 × 19.9.

11. Calculate: Area=258.7 square yards.

12. Through the above reasoning, the final results are UX=19.9 yd, and the area of △UVW is 258.7 square yards.

Definition of Right Triangle

In the figure of this problem, in triangle UWX, <note>angle UXW is a right angle (90 degrees)</note>, therefore <note>triangle UWX is a right triangle</note>. <note>Side UX and side WX are the legs</note>, <note>side UW is the hypotenuse</note>.

Definition of Sine Function

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

Area Formula of a Triangle

In <note>triangle UVW</note>, <note>side VW is the base</note>, <note>segment UX is the height on this base</note>, so the area of triangle UVW is equal to the base <note>VW multiplied by the height UX</note> divided by 2, i.e., area = (<note>VW * UX</note>) / 2.

Definition of Altitude

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

In the figure of this problem, the <note>vertex U</note> is perpendicular to the <note>opposite side VW</note> (or its extension), and the <note>segment UX</note> is the <note>altitude from vertex U</note>. The <note>segment UX forms a right angle (90 degrees) with side VW (or its extension)</note>, which indicates that the <note>segment UX is the perpendicular distance from vertex U to the opposite side VW (or its extension)</note>.


Question:
The radius of a circle is 7 centimeters. What is the area of a sector bounded by a 180° arc?

Give the exact answer in simplest form.
 $\Box$ 𝜋/ square centimeters

Answer: 49/2𝜋 square centimeters

1. According to the problem, the radius of the circle is 7 cm.

2. According to the formula for the area of a circle A=πr², substituting r=7 gives: A=π×7²=49π square cm.

3. It is known that the radian measure of the sector is 180°, so the area of the sector is 180°/360°=1/2 of the entire circle.

4. The area of the sector K is: K=49π×(1/2)=49π/2 square cm.

5. Through the above reasoning, the final answer is 49π/2.

Area Formula of a Circle

In the figure of this problem, <note>the radius of the circle is 7 cm</note>, according to the area formula of a circle, the area A of the circle is equal to <note>pi π multiplied by the square of the radius 7, that is, A = π × 7² = 49π square cm</note>.

Formula for the Area of a Sector

<note>In the diagram of this problem</note>, <note>in the sector, the central angle is 180°</note>, <note>the radius length is 7</note>. According to the formula for the area of a sector, the area A of the sector can be calculated using the formula A = (θ/360) * π * r², <note>where θ is the central angle in degrees</note>, <note>r is the radius length</note>. Therefore, <note>the area of the sector A = (180/360) * π * 7²</note>.


Question:
The radius of a circle is 3 kilometers. What is the area of a sector bounded by a 180° arc?

Give the exact answer in simplest form.
 $\Box$ 𝜋/ square kilometers

Answer: 9/2𝜋 square kilometers

1. Given that the radius of the circle is 3 km, and the length of the radius r is 3 km.

2. According to the formula for the area of a circle, the area of the circle A=πr^2, we can calculate the total area of the circle as: A=π×(3)^2=9π square kilometers.

3. The sector to be calculated is defined by an arc with a radian of 180°, and this arc occupies a proportion of 180°/360°=1/2 in the circumference of the circle.

4. According to the formula for the area of a sector A = (θ/360°)×πr^2, where θ is the angle of the arc, substituting θ=180° into the formula we get: A = (180°/360°)×π(3)^2.

5. Further simplification gives the area of this sector A = 1/2×9π=9π/2 square kilometers.

6. Through the above reasoning, the final answer is 9π/2 square kilometers.

Definition of Radius

In the figure of this problem, in the circle, <note>the point is the center of the circle</note>, <note>the distance to any point on the circumference is 3 kilometers</note>, <note>the line segment is a line segment from the center to any point on the circumference</note>, therefore <note>the line segment is the radius of the circle</note>.

Formula for the Area of a Sector

<note>In the figure of this problem, in the sector, the degree of the central angle is θ, and the length of the radius is r</note>. According to the formula for the area of a sector, the area A of the sector can be calculated using the formula A = (θ/360) * π * r², where <note>θ is the degree of the central angle</note>, <note>r is the length of the radius</note>. Therefore, the area of the sector A = (θ/360) * π * r².


Question:
What is the area of the trapezoid?

 $\Box$ square inches

Answer: 24square inches

1. Given that the upper base of the trapezoid is 3 inches, the lower base is 9 inches, and the height is 4 inches.

2. The area of the trapezoid can be calculated using the formula A = 1/2 * (b1 + b2) * h, where b1 and b2 are the bases of the trapezoid, the upper base and the lower base respectively, and h is the height of the trapezoid.

3. Substitute the given values into the formula to calculate A = 1/2 * (3 + 9) * 4.

4. Calculate the sum of the upper base and the lower base: 3 + 9 = 12.

5. Substitute the result into the area formula to calculate A = 1/2 * 12 * 4.

7. Multiply by 1/2, finally obtaining the area as A = 24.

8. Through the above reasoning, the final answer is 24.

Definition of Trapezoid

A quadrilateral is a trapezoid if and only if it has exactly one pair of parallel sides.

<note>The sides of 3 inches and 9 inches are parallel</note>, while the sides of 5 inches and 5 inches are not parallel. Therefore, according to the definition of a trapezoid, the quadrilateral is a trapezoid because <note>it has exactly one pair of parallel sides</note>.

Trapezoid Area Formula

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

<note>The upper base of the trapezoid b1 = 3 inches</note>, <note>the lower base b2 = 9 inches</note>, <note>the height h = 4 inches</note>. Using the formula, <note>A = 1/2 * (b1 + b2) * h = 1/2 * (3 + 9) * 4</note>.


Question:
What is the area?

 $\Box$  square kilometers

Answer: 20 square kilometers

1. Given that the problem requires the area of a rectangular region, and the figure indicates that the length of the rectangle is 5 km and the width is 4 km.

2. According to the area formula for a rectangle: Area = Length × Width, we can substitute the known length and width into this formula.

3. Substituting the length of 5 km and the width of 4 km into the rectangle area formula, we get: Area = 5 km × 4 km.

4. The calculation result is: Area = 20 square km.

5. Through the above reasoning, the final answer is 20 square km.

Definition of Rectangle

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

<note>The length of the rectangle is 5 kilometers</note>, <note>the width is 4 kilometers</note>. All four corners of the rectangle are <note>right angles</note>, and <note>opposite sides are parallel and equal in length</note>. <note>The opposite sides of the rectangle are 5 kilometers and 4 kilometers respectively</note>.

Formula for the Area of a Rectangle

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

Original text: <note>The length of the rectangle is 5 kilometers</note>, <note>the width is 4 kilometers</note>. According to the formula for the area of a rectangle, substitute the length and width into the formula to get: Area = <note>5 kilometers × 4 kilometers</note> = <note>20 square kilometers</note>.


Question:
The diagonals of this rhombus are 10 feet and 2 feet.

What is the area of the rhombus?
 $\Box$  square feet

Answer: 10 square feet

1. Given that the lengths of the diagonals of the rhombus are 10 ft and 2 ft respectively, let the intersection point of the diagonals be O, and the rhombus be ABCD, with the horizontal diagonal being AC and the vertical diagonal being BD.

2. According to the formula for the area of a rhombus: The area of a rhombus can be determined by its diagonals, i.e., Area = (d1 × d2) / 2, where d1 and d2 are the two diagonals of the rhombus.

3. Substitute d1 = 10 and d2 = 2 into the formula to calculate, and get Area = (10 × 2) / 2 = 10.

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

Definition of Rhombus

In the figure of this problem, <note>quadrilateral ABCD</note> has all sides <note>AB, BC, CD, DA equal</note>, thus <note>quadrilateral ABCD is a rhombus</note>. Additionally, <note>the diagonals AC and BD of quadrilateral ABCD are perpendicular bisectors of each other</note>, meaning that the diagonals <note>AC and BD intersect at point O</note>, and <note>angle AOB is a right angle (90 degrees)</note>, and <note>AO=CO and BO=DO</note>.

Rhombus Area Formula

<note>The area of a rhombus is equal to half the product of its two diagonals</note>, i.e., Area = <note>(10 * 2) / 2 = 10 square feet</note>.


Question:
The diagonals of this rhombus are 8 centimeters and 3 centimeters.

What is the area of the rhombus?
 $\Box$  square centimeters

Answer: 12 square centimeters

1. Given that the lengths of the diagonals of the rhombus are 8 cm and 3 cm.

2. According to the formula for the area of a rhombus, Area = (d1 × d2) / 2, where d1 and d2 are the two diagonals of the rhombus, i.e., 8 cm and 3 cm.

3. Therefore, the area of the rhombus = (8 cm * 3 cm) / 2 = 12 square cm.

4. Through the above reasoning, the final answer is 12.

<note>The two diagonals of the rhombus are 8 cm and 3 cm respectively</note>, so the area = (8 cm * 3 cm) / 2 = 12 square cm.


Question:
What is the area of the trapezoid?

 $\Box$ square inches

Answer: 24square inches

1. According to the information given in the problem, the trapezoid has the upper base b1=3 inches, the lower base b2=9 inches, and the height h=4 inches.

2. The formula for the area of a trapezoid is A=1/2*(b1+b2)*h.

3. Substitute the known values of b1, b2, and h into the area formula, yielding A=1/2×(3+9)×4.

4. Calculate the sum inside the parentheses: 3+9=12.

5. Substitute the result of the sum into the formula, transforming it to A=(1/2)×12×4.

6. Continue with the multiplication: 12×4=48.

7. Calculate A=1/2×48, resulting in A=24.

8. Through the above reasoning, the final area of the trapezoid is found to be 24 square inches.

Definition of Trapezoid

<note>The upper base of the quadrilateral is 3 inches, and the lower base is 9 inches</note>, <note>the two legs are 5 inches each</note>. According to the definition of a trapezoid, <note>a quadrilateral has one pair of parallel sides (upper base and lower base)</note>, therefore the quadrilateral is a trapezoid.

Trapezoid Area Formula

<note>Top base b1=3 inches</note>, <note>Bottom base b2=9 inches</note>, <note>Height h=4 inches</note>, so the area of the trapezoid is (3 + 9) * 4 / 2.


Question:
What is the area of the trapezoid?

 $\Box$ square miles

Answer: 7square miles

1. Given the two bases of the trapezoid are b1=5 miles and b2=2 miles, and the height of the trapezoid is h=2 miles.

2. According to the trapezoid area formula A=1/2*(b1+b2)*h, the area of the trapezoid can be calculated.

3. Substitute the known values into the formula, where b1=5 miles, b2=2 miles, h=2 miles:

4. First, calculate the sum of the two bases: 5+2=7.

5. Then, calculate 7 multiplied by 2 to get 14.

6. Finally, calculate 14 multiplied by 1/2 to get 7.

7. Through the above reasoning, the final answer is 7 square miles.

Definition of Trapezoid

<note>The parallel sides of the trapezoid are the base of 5 miles and the top base of 2 miles</note>. <note>The height of 2 miles is perpendicular to these two parallel sides</note>.

Trapezoid Area Formula

<note>The upper base of the trapezoid is 2 miles</note>, <note>the lower base is 5 miles</note>, <note>the height is 2 miles</note>, so the area of the trapezoid is <note>(2 + 5) * 2 / 2</note>.


Question:
The diagonals of this rhombus are 6 inches and 8 inches.

What is the area of the rhombus?
 $\Box$  square inches

Answer: 24 square inches

1. Given that the diagonals of the rhombus are 6 inches and 8 inches respectively, according to the properties of the rhombus's diagonals, they are perpendicular to each other and bisect each other.

2. Auxiliary line: Let the diagonals of the rhombus intersect at point O, then the midpoints of diagonals AC and BD are point O, and ∠AOB, ∠BOC, ∠COD, ∠DOA are all right angles, i.e., 90°.

3. The area of the rhombus can be calculated by taking half the product of the diagonals. According to the area formula of the rhombus: Area = 1/2 * AC * BD, where AC and BD are the lengths of the two diagonals of the rhombus.

4. Substitute the given diagonal lengths into the formula, and calculate the area = 1/2 * 6 * 8.

5. Continue to simplify to get the area = 1/2 * 48.

6. Calculate to get the final area of 24 square inches.

7. Through the above reasoning, the final answer is 24 square inches.

Definition of Rhombus

In the figure of this problem, <note>the four sides of the rhombus are equal</note> and <note>the diagonals are perpendicular and bisect each other</note>. The diagonals are <note>6 inches and 8 inches</note>, <note>the intersection point of the diagonals is point O</note>, and <note>∠AOB, ∠BOC, ∠COD, ∠DOA are all right angles</note>.

Definition of Diagonal

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

<note>The diagonals of the rhombus are 6 inches and 8 inches respectively</note>. The diagonal is the line segment connecting one vertex of the rhombus to the non-adjacent vertex. Therefore, <note>the line segments AC and BD are the diagonals of the rhombus</note>. <note>The diagonals AC and BD are perpendicular to each other and bisect each other, measuring 6 inches and 8 inches respectively</note>. <note>Point O is the intersection of the diagonals and also the midpoint of the diagonals</note>.

Rhombus Area Formula

<note>The diagonals of the rhombus are 6 inches and 8 inches respectively</note>. According to the rhombus area formula, the area of the rhombus is equal to half the product of the two diagonals, that is, <note>Area = (6 * 8) / 2 = 24 square inches</note>.

Original text: <note>The diagonals of the rhombus are 6 inches and 8 inches</note>. <note>The diagonals bisect each other and are perpendicular to each other</note>. Specifically, <note>point O is the intersection point of the diagonals</note>, and <note>AO=OC=3 inches</note>, <note>BO=OD=4 inches</note>. At the same time, <note>angle AOB, angle BOC, angle COD, and angle DOA are all right angles (90 degrees)</note>, so <note>the diagonals are perpendicular to each other</note>.


Question:
The radius of a circle is 10 inches. What is the area of a sector bounded by a 90° arc?

Give the exact answer in simplest form.
 $\Box$ 𝜋/ square inches

Answer: 25𝜋 square inches

1. Let the center of the circle be O, and points A and B be points on the circle (from top to bottom). Given: the radius of the circle r = 10 inches, and the angle corresponding to the arc length of the sector is 90°.

2. According to the formula for the area of a circle, A = πr², calculate the area of the entire circle. Substitute r = 10 to get A = π * 10² = 100π square inches.

3. The calculation of the area K of the sector depends on the relationship between the central angle m and the area A of the entire circle. According to the formula for the area of a sector K = A * (m/360), where m is the central angle in degrees.

4. Substitute the known values into the formula for the area of the sector: A = 100π and m = 90°, i.e., K = 100π * (90/360).

5. Calculate K = 100π * (1/4) = 25π square inches.

6. Through the above reasoning, the final answer is 25π square inches.

Definition of Radius

<note>The center of the circle is the vertex of the orange sector in the figure</note>, <note>any point on the circumference is any point on the circumference</note>, <note>the length of the line segment from the center to any point on the circumference is 10 inches</note>, therefore <note>the line segment r = 10 inches is the radius of the circle</note>.

Definition of Central Angle

<note>The center of the circle is point O</note>, <note>the angle ∠AOB formed by the lines OA and OB from points A and B on the circle to the center O</note> is called the central angle. In this problem, <note>the degree of the central angle ∠AOB is 90°</note>.

Definition of Sector

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

In the figure of this problem, the sector is composed of <note>two radii and the arc between them</note>, <note>with radii of r = 10 inches</note>, <note>the central angle is 90°</note>, <note>the arc length is one-fourth of the circumference</note>.

Area Formula of a Circle

<note>The radius of the circle is 10 inches</note>. According to the area formula of a circle, the area A of the circle is equal to pi π multiplied by the square of the radius 10, that is, <note>A = π * 10²</note>.

Formula for the Area of a Sector

In this problem, the formula for the area of a sector <note>K = A * (m/360)</note>. Given <note>the area of the circle A = 100π</note>, <note>the central angle m = 90°</note>, substitute into the formula to get <note>K = 100π * (90/360) = 25π</note>.


Question:
The radius of a circle is 3 centimeters. What is the area of a sector bounded by a 90° arc?

Give the exact answer in simplest form.
 $\Box$ 𝜋/ square centimeters

Answer: 9/4𝜋 square centimeters

1. Given the radius r = 3 cm, let the center of the circle be O, OA be the upper radius of the circle, OB be the lower radius of the circle, and the sector be AOB. First, we calculate the area of the entire circle using the area formula A = πr².

2. Substitute the radius r = 3 into the formula, we get A = π(3)² = 9π square centimeters.

3. According to the area formula of the sector K = A * (θ/360°), where θ represents the central angle of the sector. It is given in the problem that θ = 90°.

4. Substitute A = 9π and θ = 90° into the sector area formula, we get K = 9π * (90/360).

5. Calculate 90/360 = 1/4, substitute it into the formula to get K = 9π * 1/4 = 9π/4.

6. Through the above reasoning, the final answer is 9π/4 square centimeters.

Definition of Circle

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

In the diagram of this problem, <note>the radius of the circle is r = 3 cm</note>, <note>the center of the circle is the central point of the circle</note>. <note>All points in the diagram that are at a distance of 3 cm from the center of the circle are on the circle</note>.

Definition of Sector

<note>In a sector, the radius r = 3 cm and the radius r = 3 cm are two radii of the circle</note>, <note>the arc is the arc enclosed by these two radii</note>, so according to <note>the definition of sector</note>, the figure formed by these two radii and the arc they enclose is a sector.

Area Formula of a Circle

<note>The radius of the circle is 3 centimeters</note>, according to the area formula of a circle, <note>the area A of the circle is equal to the circumference π multiplied by the square of the radius 3</note>, which is A = π(3)².

Formula for the Area of a Sector

In the <note>sector AOB</note>, the degree of the <note>central angle AOB</note> is θ, and the length of the <note>radius OA</note> is r. According to the formula for the area of a sector, the area A of the sector can be calculated using the formula A = (θ/360) * π * r², where <note>θ is the degree of the central angle</note> and <note>r is the length of the radius</note>. Therefore, <note>the area of sector AOB A = (θ/360) * π * r²</note>.


Question:
The radius of a circle is 4 centimeters. What is the area of a sector bounded by a 45° arc?

Give the exact answer in simplest form.
 $\Box$ 𝜋/ square centimeters

Answer: 2𝜋 square centimeters

1. Let the center of the circle be O, the left radius be OA, the right radius be OB, and the sector be AOB. Given that the radius of the circle is 4 cm, according to the formula for the area of a circle A=πr^2, substituting r=4, we get the area of the circle as 16π square centimeters.

2. The problem states that the central angle corresponding to the arc length of the sector is 45°, according to the formula for the area of a sector K=(θ/360)×A, where θ is the central angle in degrees, and A is the area of the circle.

3. Substituting A=16π and θ=45° into the formula for the area of the sector, K=(45/360)×16π, simplifying we get K=(1/8)×16π.

4. Further calculation gives K=2π.

5. Through the above reasoning, the final answer is 2π square centimeters.

Definition of Circle

In the figure of this problem, <note>the center of the circle is point O</note>, <note>the radius length is 4 centimeters</note>. All points in the figure that are <note>4 centimeters away from point O are on the circle</note>.

Definition of Radius

<note>The radius length of the circle r = 4 cm</note>, <note>point O is the center of the circle</note>, <note>the distance from any point on the circumference to the center O is 4 cm</note>, therefore <note>the distance of the line segment from the center O to any point on the circumference is 4 cm</note>.

Definition of Central Angle

The <note>central angle</note> is 45°, and its two sides extend to the circumference forming an arc.

Definition of Sector

<note>Sector</note> consists of <note>radius r=4 cm</note> and <note>another radius r=4 cm</note> and the <note>arc (45° arc)</note> between them. According to the <note>definition of sector</note>, the <note>figure formed by these two radii</note> and the <note>arc</note> between them is a <note>sector</note>.

Area Formula of a Circle

<note>The radius of the circle is 4 centimeters</note>. According to the area formula of a circle, the area A of the circle is equal to the circumference π multiplied by <note>the square of radius 4</note>, that is <note>A = π*4²</note>.

Formula for the Area of a Sector

In the diagram of this problem, <note>sector AOB</note>, <note>the central angle AOB measures θ degrees</note>, <note>the radius OA has a length of r</note>. According to the formula for the area of a sector, the area A of the sector can be calculated using the formula A = (θ/360) * π * r², where θ is the measure of the central angle and r is the length of the radius. Therefore, <note>the area of sector AOB A = (θ/360) * π * r²</note>.


Question:
What is the area of the trapezoid?

 $\Box$ square miles

Answer: 30square miles

1. According to the figure, the upper base of the trapezoid is 3 miles, the lower base is 9 miles, the length of the slant side is 7 miles, and the height is 5 miles.

2. The formula for the area of the trapezoid is: Area = 1/2 × (upper base + lower base) × height.

3. Substitute the known data, upper base = 3 miles, lower base = 9 miles, height = 5 miles. We get Area = 1/2 × (3 + 9) × 5.

4. Calculate the sum inside the parentheses, 3 + 9 = 12.

5. Calculate the formula: Area = 1/2 × 12 × 5.

6. Further calculation, 1/2 × 12 = 6.

7. Final step calculation, Area = 6 × 5 = 30.

8. Through the above reasoning, the area of the trapezoid is 30 square miles.

Definition of Trapezoid

<note>In the diagram of this problem</note>, in the quadrilateral, <note>the upper base is 3 miles, the lower base is 9 miles, and the two sides are parallel</note>, the height is 5 miles, and the length of the slanted side is 7 miles. Therefore, <note>according to the definition of trapezoid</note>, this quadrilateral is a <note>trapezoid because it has only one pair of parallel sides</note>.

Trapezoid Area Formula

<note>The upper base of the trapezoid is 3 miles</note>, <note>the lower base is 9 miles</note>, <note>the height is 5 miles</note>. According to the trapezoid area formula, substituting the known data, the calculation is as follows: Area = 1/2 × (3 + 9) × 5, therefore, <note>the area of the trapezoid is 30 square miles</note>.


Question:
What is the area of the trapezoid?

 $\Box$ square millimeters

Answer: 70square millimeters

1. Given that the length of the top base of the trapezoid is 4 mm, and the length of the bottom base is 10 mm.

2. Given that one side height of the trapezoid is 10 mm. According to the accompanying diagram, a vertical line indicates the perpendicular distance from the top base to the bottom base, which is the height of the trapezoid.

3. The area of the trapezoid can be calculated using the trapezoid area formula: A = 1/2 * (b1 + b2) * h, where b1 and b2 are the lengths of the top and bottom bases respectively, and h is the height of the trapezoid.

4. Substitute the given values into the area formula: A = 1/2 * (4 mm + 10 mm) * 10 mm.

5. Calculate the sum inside the parentheses: 4 mm + 10 mm = 14 mm.

6. Substitute the result into the formula: A = 1/2 * 14 mm * 10 mm.

7. Continue with the multiplication: 1/2 * 14 mm = 7 mm.

8. Perform the final multiplication: 7 mm * 10 mm = 70 square mm.

9. Through the above reasoning, the final answer is 70 square mm.

Definition of Trapezoid

In the quadrilateral, the upper and lower bases of the trapezoid are parallel, with lengths of 4 millimeters and 10 millimeters respectively. The two sides of the trapezoid are not parallel. Therefore, according to the definition of trapezoid, this quadrilateral is a trapezoid because it has only one pair of parallel sides.

Trapezoid Area Formula

In the figure of this problem, in the trapezoid, <note>the upper base and the lower base are two parallel sides</note>, with lengths of <note>a and b</note>, and the height between them is <note>h</note>, so the area of the trapezoid is <note>(a + b) * h / 2</note>.


Question:
The diagonals of this rhombus are 9 miles and 6 miles.

What is the area of the rhombus?
 $\Box$  square miles

Answer: 27 square miles

1. Given that the lengths of the two diagonals of the rhombus are 9 miles and 6 miles respectively.

2. According to the properties of the diagonals of a rhombus, the diagonals are perpendicular bisectors of each other. Therefore, add auxiliary lines so that the two diagonals intersect at point O, dividing the rhombus into four congruent right triangles.

3. Let the diagonal AC be 9 miles, BD be 6 miles, and O be the intersection point of the diagonals. By the bisector property, we know: AO = OC = 9/2 = 4.5 miles, BO = OD = 6/2 = 3 miles.

4. Select one right triangle AOB for calculation. According to the triangle area formula, triangle area = 1/2 × base × height. For the right triangle AOB, base AO = 4.5 miles, height BO = 3 miles.

5. Use the triangle area formula to calculate the area of AOB: area = 1/2 × 4.5 × 3 = 6.75 square miles.

6. Since the rhombus is divided into 4 congruent right triangles, the total area of the rhombus is 4 times the area of a single triangle. Rhombus area = 4 × 6.75 = 27 square miles.

7. Through the above reasoning, the final answer is 27 square miles.

Definition of Rhombus

<note>Quadrilateral ABCD is a rhombus</note>, in which <note>sides AB, BC, CD, and AD are all equal</note>, and <note>diagonals AC and BD are perpendicular bisectors of each other</note>. Therefore, <note>sides AB = BC = CD = AD</note>, and <note>diagonals AC and BD intersect at point O and form a 90-degree angle</note>.

Definition of Diagonal

In the figure of this problem, in the polygon, vertex A and vertex C are non-adjacent vertices, <note>line segment AC is the line segment connecting vertex A and vertex C</note>, therefore <note>line segment AC is a diagonal of polygon ABCD</note>. Similarly, vertex B and vertex D are non-adjacent vertices, <note>line segment BD is the line segment connecting vertex B and vertex D</note>, therefore <note>line segment BD is also a diagonal of polygon ABCD</note>.

Definition of Right Triangle

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

<note>Angle AOB is a right angle (90 degrees)</note>, therefore triangle AOB is a right triangle.

In the figure of this problem, in rhombus ABCD, <note>diagonals AC and BD bisect each other and are perpendicular to each other</note>. Specifically, <note>point O is the intersection point of diagonals AC and BD</note>, and <note>AO=CO</note> and <note>BO=DO</note>. At the same time, <note>angles AOB and BOC are right angles (90 degrees)</note>, so <note>diagonals AC and BD are perpendicular to each other</note>.

Area Formula of a Triangle

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

<note>In triangle AOB, side AO is the base, and segment BO is the height on this base</note>, so <note>the area of triangle AOB equals the base AO multiplied by the height BO divided by 2</note>, i.e., area = (4.5 * 3) / 2.


Question:
What is m $\overset{\frown}{FH}$ ?

m $\overset{\frown}{FH}$ = $\Box$ °

Answer: m \$\overset{\frown}{FH}\$ =90°

1. According to the angle properties of a circle, the total radians of a circle's circumference is 360 degrees.

2. The circumference of the circle corresponds to 360°, so in the circle, the sum of the degrees of all central angles is 360°.

3. The arcs given in the problem are arc FG, arc GH, and arc FH, with the degrees of arc FG = 150° and the degrees of arc GH = 120°.

4. Based on the conclusion in step 2, the sum of the degrees of arc FG, arc GH, and arc FH is 360°, so we can write the following equation: arc FG + arc GH + arc FH = 360°.

5. Substitute the known data, i.e., 150° + 120° + arc FH = 360°.

6. Add 150° and 120°, resulting in 270° + arc FH = 360°.

7. To find arc FH, subtract 270° from both sides, resulting in arc FH = 360° - 270°.

8. After calculation, arc FH = 90°.

9. Through the above reasoning, the final answer is 90°.

Angle Properties of a Circle

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

In the diagram of this problem, <note>arc GH, arc HF, and arc GF around the circle, the sum of the radians is 360 degrees</note>.


Question:
What is m $\overset{\frown}{GI}$ ?

m $\overset{\frown}{GI}$ = $\Box$ °

Answer: m \$\overset{\frown}{GI}\$ =130°

1. According to the angle properties of a circle, ∡GH + ∡HI + ∡GI = 360°, because they form a complete circle.

2. Given μ(∡GH) = 120°, μ(∡HI) = 110°, substitute these values into the equation: 120° + 110° + μ(∡GI) = 360°.

3. Calculate the sum on the right side, 120° + 110° = 230°.

4. Move 230° to the other side of the equation to get μ(∡GI) = 360° - 230°.

5. Perform the subtraction to get μ(∡GI) = 130°.

6. Through the above reasoning, the final answer is 130°.

<note>Arc GH</note><note>Arc HI</note><note>Arc IG</note> around the circle, the sum of the arc degrees is 360 degrees, and the corresponding <note>sum of the circumferential angles is also 360 degrees</note>, i.e., <note>Angle GOH</note>+<note>Angle HOI</note>+<note>Angle IOG</note>=360 degrees, then <note>Angle IOG</note>=130 degrees.


Question:
What is m $\angle $ EDG?

m $\angle $ EDG= $\Box$ °

Answer: m \$\angle \$ EDG=140°

1. Given ∠FDG, ∠EDG, and ∠EDF are angles around point D, their sum is 360°.

2. According to the problem statement, ∠FDG=100°, ∠EDF=120°.

3. Based on the properties of angles in a circle, we have ∠FDG + ∠EDG + ∠EDF = 360°.

4. Substituting the known angles, we get: 100° + ∠EDG + 120° = 360°.

5. Simplifying the above equation: ∠EDG + 220° = 360°.

6. Subtracting 220° from both sides to solve for ∠EDG, we get ∠EDG = 140°.

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

In the figure of this problem, <note>arc FG, arc GE, arc EF around the circle, the sum of the radians is 360 degrees</note>, <note>the sum of their corresponding angles at the circumference is also 360 degrees</note>, that is, <note>∠FDG + ∠EDG + ∠EDF = 360°</note>, then <note>∠EDG = 140°</note>.


Question:
What is m $\overset{\frown}{FG}$ ?

m $\overset{\frown}{FG}$ = $\Box$ °

Answer: m \$\overset{\frown}{FG}\$ =90°

1. Given that the measure of arc GH is 120°, and the measure of arc FH is 150°, let the center of the circle be O. The central angles corresponding to these arcs are also 120° and 150°, respectively.

2. To find the measure of arc FG, which is the measure of its corresponding central angle, we have 360° - 150° - 120° = 90°.

Definition of Central Angle

In the diagram of this problem, <note>circle O</note>, <note>point F and point G are two points on the circle</note>, <note>the center of the circle is point O</note>. <note>The angle ∠FOG formed by the lines OF and OG is called the central angle</note>. <note>Point F and point H are two points on the circle</note>, <note>the center of the circle is point O</note>. <note>The angle ∠FOH formed by the lines OF and OH is called the central angle</note>. <note>Point G and point H are two points on the circle</note>, <note>the center of the circle is point O</note>. <note>The angle ∠GOH formed by the lines OG and OH is called the central angle</note>.

Angle Properties of a Circle

<note>Arc FH, arc HG, and arc GF make a complete circle, with a total of 360 degrees</note>, and their corresponding <note>circumferential angles also sum up to 360 degrees</note>, that is, <note>angle FOH + angle HOG + angle GOF = 360 degrees</note>.


Question:
The radius of a circle is 4 inches. What is the area of a sector bounded by a 135° arc?

Give the exact answer in simplest form.
 $\Box$ 𝜋/ square inches

Answer: 6𝜋 square inches

1. Given the radius of the circle is 4 inches, use the formula for the area of a circle A=πr^2 to calculate the area of the entire circle.

2. Substitute r=4 to calculate the area of the circle A=π×(4)^2=16π square inches.

3. The problem states that the central angle of the circle is 135°, which is the central angle of the sector.

4. The formula for the area of a sector is: K=(A·θ)/360, where A is the area of the circle and θ is the central angle in degrees.

5. Substitute A=16π and θ=135 to calculate the area of the sector: K=(16π×135)/360.

6. Simplify the fraction: (16×135)/(360)=2160/360=6.

7. The area of the sector is 6π square inches.

8. Through the above reasoning, the final answer is 6π square inches.

Definition of Radius

<note>Radius of a circle</note> in this problem diagram, <note>the radius of the circle is r=4 inches</note>, the distance from the center of the circle to the circumference is <note>4 inches</note>, this distance <note>is represented in the diagram by the line segment drawn from the center to the circumference</note>.

Definition of Central Angle

In the figure of this problem, <note>the central angle is 135°</note>, <note>the vertex is at the center of the circle</note>, representing the sector's <note>two sides extending from the center to the circumference</note>. The central angle's <note>two sides are respectively two radii extending from the center to the circumference</note>.

Definition of Sector

The <note>sector</note> is formed by <note>two radii of 4 inches</note> and the <note>arc</note> between them, with a central angle of <note>135°</note>.

Area Formula of a Circle

<note>The radius of the circle is 4 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 4</note>, that is <note>A = π4²</note>.

Formula for the Area of a Sector

<note>The area of the circle A = 16π square inches</note>, <note>the central angle θ = 135°</note>. Substitute into the formula to calculate the area of the sector: K = (16π × 135) / 360.


Question:
What is m $\angle $ PSQ?

m $\angle $ PSQ= $\Box$ °

Answer: m \$\angle \$ PSQ=140°

1. In the problem, point S is the center of the circle, ∠PSR and ∠QSR are central angles formed by line segments PS, SR, and SQ, given that ∠PSR = 140° and ∠QSR = 80°.

2. According to the properties of angles in a circle, the sum of the central angles corresponding to the arcs of the same center in a circle is equal to 360°. Therefore, the sum of ∠PSQ, ∠PSR, and ∠QSR is 360°.

3. Substituting the known angles, we get the equation: ∠PSQ + ∠PSR + ∠QSR = 360°. That is, ∠PSQ + 140° + 80° = 360°.

4. Combining like terms, we get ∠PSQ + 220° = 360°.

5. Subtracting 220° from both sides, we get ∠PSQ = 140°.

6. Through the above reasoning, the final answer is ∠PSQ = 140°.

Angle Properties of a Circle

In the diagram of this problem, <note>arc PR, arc PQ, arc RQ around the circle</note>, the sum of the radians is <note>360 degrees</note>, and the corresponding <note>sum of the central angles is also 360 degrees</note>, that is, <note>angle PSR + angle QSR + angle PSQ = 360 degrees</note>, then <note>angle PSR = 140 degrees</note>.


Question:
The diagonals of this rhombus are 10 kilometers and 7 kilometers.

What is the area of the rhombus?
 $\Box$  square kilometers

Answer: 35 square kilometers

1. Given that the lengths of the diagonals of a rhombus are 10 km and 7 km respectively.

2. According to the properties of the diagonals of a rhombus, the diagonals are perpendicular to each other and bisect each other. Therefore, the two diagonals divide the rhombus into four congruent right triangles, each with the two legs being half of the diagonals.

3. The diagonals are 10 km and 7 km respectively, so each right triangle has legs of 5 km (i.e., 10/2) and 3.5 km (i.e., 7/2).

4. According to the triangle area formula, the area of a triangle is equal to 1/2 multiplied by the product of the two legs.

5. Calculate the area of one right triangle: 1/2 × 5 × 3.5 = 1/2 × 17.5 = 8.75 square km.

6. Since the four right triangles are congruent, the area of the entire rhombus is 4 times the area of one right triangle.

7. Calculate the area of the rhombus: 4 × 8.75 = 35 square km.

8. In conclusion, the area of the rhombus is 35 square km.

Definition of Rhombus

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

<note>All sides of the quadrilateral are equal</note>, therefore the quadrilateral is a rhombus. Additionally, the quadrilateral's <note>diagonals are perpendicular bisectors of each other</note>, meaning <note>the diagonals intersect at a point and form right angles (90 degrees)</note>.

Definition of Diagonal

<note>In a rhombus, a diagonal is a line segment connecting the vertices</note>. Therefore, <note>the two line segments of 10 kilometers and 7 kilometers respectively</note> are the diagonals of the rhombus.

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

In the problem diagram, <note>the diagonals of the rhombus are 10 km and 7 km respectively</note>, <note>the diagonals are perpendicular to each other and bisect each other</note>. Specifically, <note>point O is the intersection point of the diagonals</note>, and <note>AO = OC = 5 km</note> and <note>BO = OD = 3.5 km</note>. At the same time, <note>∠AOB and ∠COD are both right angles (90 degrees)</note>, so <note>the diagonals are perpendicular to each other</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 problem diagram, due to <note>four small triangles being congruent</note>, each area is 8.75 square kilometers. Specifically, <note>the diagonals of the rhombus are 10 kilometers and 7 kilometers respectively</note>, <note>the intersection point divides the rhombus into four congruent right triangles</note>. Each right triangle's <note>legs are 5 kilometers and 3.5 kilometers respectively</note>, <note>corresponding sides are equal</note>, <note>corresponding angles are equal</note>.

Area Formula of a Triangle

The area of a <note>triangle</note> is equal to 1/2 multiplied by the <note>product of the two perpendicular sides</note>.


Question:
What is m $\angle $ QPR?

m $\angle $ QPR= $\Box$ °

Answer: m \$\angle \$ QPR=70°

1. Property of central angles: In a circle, the central angle of the same arc is equal to 360°.

2. First, observe the figure, point P is the center of the circle, and there are two line segments PR and PS passing through this point.

3. According to the given conditions, m∠RPS = 150°, m∠QPS = 140°, we need to find the degree of m∠QPR.

4. Based on the property of central angles, m∠RPS + m∠QPR + m∠QPS = 360°.

5. Substitute the known angles into the equation, we get 150° + m∠QPR + 140° = 360°.

6. Add the known degrees, we get 150° + 140° = 290°.

7. Simplify the equation to m∠QPR + 290° = 360°.

8. Subtract 290° from both sides to solve the equation, m∠QPR = 360° - 290°.

9. After subtraction, we get m∠QPR = 70°.

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

Definition of Central Angle

<note>Point R</note> and <note>Point S</note> are two points on the circle, <note>the center of the circle is Point P</note>. The angle <note>∠RPS</note> formed by the lines <note>PR</note> and <note>PS</note> is called a central angle. Similarly, <note>Point Q</note> and <note>Point S</note> are two points on the circle, <note>the center of the circle is Point P</note>. The angle <note>∠QPS</note> formed by the lines <note>PQ</note> and <note>PS</note> is called a central angle. <note>Point Q</note> and <note>Point R</note> are two points on the circle, <note>the center of the circle is Point P</note>. The angle <note>∠QPR</note> formed by the lines <note>PQ</note> and <note>PR</note> is called a central angle.


Question:
What is the area of the trapezoid?

 $\Box$ square centimeters

Answer: 18square centimeters

1. According to the conditions of the problem, let the trapezoid be ABCD (vertices are taken counterclockwise, A is the top left vertex), AD is the top base, BC is the bottom base, the top base of the trapezoid is AD = 4 cm, the bottom base is BC = 8 cm, and the height is h = 3 cm.

2. The area formula of the trapezoid is: \( A = \frac{1}{2} (b_1 + b_2) \times h \), where b1 and b2 are the lengths of the two parallel sides of the trapezoid, and h is the height.

3. Substitute the known lengths of the top and bottom bases into the formula: \( A = \frac{1}{2} (4 + 8) \times 3 \).

4. Calculate the value inside the parentheses: \( 4 + 8 = 12 \).

5. Substitute the result from the previous step into the formula and continue calculating: \( A = \frac{1}{2} \times 12 \times 3 \).

6. First calculate the product: \( 12 \times 3 = 36 \).

7. Find the area: \( A = \frac{1}{2} \times 36 = 18 \).

8. Through the above reasoning, the final answer is (unit square centimeters):

Definition of Trapezoid

<note>In the diagram of this problem, in quadrilateral ABCD, sides AD and BC are parallel</note>, while sides AB and CD are not parallel. Therefore, according to the definition of a trapezoid, <note>quadrilateral ABCD is a trapezoid because it has exactly one pair of parallel sides</note>.

Trapezoid Area Formula

<note>The length of the upper base AD of the trapezoid is b1 = 4 cm</note>, <note>the length of the lower base BC is b2 = 8 cm</note>, <note>the height h = 3 cm</note>. According to the trapezoid area formula A = \( \frac{1}{2} (b_1 + b_2) \times h \), substituting the known values gives the area A = \( \frac{1}{2} (4 + 8) \times 3 = 18 \) square centimeters.


Question:
What is the area of the trapezoid?

 $\Box$ square centimeters

Answer: 28square centimeters

1. Given that the upper base of the trapezoid b1 = 4 cm, the lower base of the trapezoid b2 = 10 cm, and the height h = 4 cm.

2. The area of the trapezoid can be calculated using the following formula: A = 1/2 * (b1 + b2) * h.

3. Substitute the given conditions into the formula to get A = 1/2 * (4 + 10) * 4.

4. Calculate the sum inside the parentheses to get A = 1/2 * 14 * 4.

5. First, calculate 1/2 * 14 to get the result 7.

6. Then, calculate 7 * 4 to get the result 28.

7. Through the above reasoning, the final answer is 28 square centimeters.

Definition of Trapezoid

The original text: <note>The upper base of the quadrilateral is 4 cm, the lower base is 10 cm, and the height is 4 cm</note>. According to the definition of a trapezoid, a trapezoid refers to a quadrilateral that <note>has one pair of parallel sides</note>. <note>In this figure, the upper base and the lower base are parallel</note>, while the other two sides are not parallel. Therefore, this quadrilateral is a trapezoid.

Trapezoid Area Formula

<note>The upper base of the trapezoid is 4 cm</note>, <note>the lower base is 10 cm</note>, <note>the height is 4 cm</note>, so the area of the trapezoid is (4 + 10) * <note>4 / 2</note> = <note>28 square centimeters</note>.


Question:
What is the area of the trapezoid?

 $\Box$ square inches

Answer: 40square inches

1. According to the problem statement, the two bases of the trapezoid are b1=10 inches and b2=6 inches.

2. The height h of the trapezoid is perpendicular from the top base to the bottom base, and the height is 5 inches.

3. Select a point, and draw a perpendicular line from the right end of the top base to the bottom base, forming a right triangle (let the right triangle be ABC, AB=5, BC is another leg, AC is the hypotenuse).

4. The formula for calculating the area of the trapezoid is: A=1/2 * (b1+b2) * h.

5. Substitute the known values into the formula: A = 1/2 * (10 + 6) * 5.

6. Calculate the sum inside the parentheses: b1+b2=16 inches.

7. Substitute the result inside the parentheses into the formula: A = 1/2 * 16 * 5.

8. Perform the multiplication: A = 8 * 5.

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

Definition of Trapezoid

<note>The upper base of the trapezoid is 6 inches, and the lower base is 10 inches</note>; the other two sides are not parallel and are two slanted sides. Therefore, <note>according to the definition of a trapezoid, this quadrilateral is a trapezoid because it has exactly one pair of parallel sides</note>.

Definition of Right Triangle

Select a point, perpendicular from the right end of the upper base to the lower base, forming a <note>right triangle</note>. The <note>right angle is at the intersection of the perpendicular line and the lower base</note>. The two <note>legs of the right triangle are the height of the trapezoid (5 inches) and BC</note>, and the <note>hypotenuse is the segment from the right end of the upper base to the right end of the lower base</note>.

Trapezoid Area Formula

<note>The sides of 6 inches and 10 inches are two parallel sides</note>, <note>The side of 5 inches is the height between them</note>, so the area of the trapezoid is (6 + 10) * 5 / 2.


Question:
What is m $\overset{\frown}{RS}$ ?

m $\overset{\frown}{RS}$ = $\Box$ °

Answer: m \$\overset{\frown}{RS}\$ =90°

1. Given a circle, let the center be O, the degree of arc RT is 120°, and the degree of arc ST is 150°. Find the degree of arc RS.

2. According to the properties of angles in a circle, the total degree of the circumference is 360°, and the sum of the central angles corresponding to the entire circumference is 360°.

3. Therefore, arc RT + arc ST + arc RS = 360°

4. Rearranging, we get arc RS = 360° - 120° - 150° = 90°

5. Through the above reasoning, the final answer is 90°.

Definition of Circle

In the figure of this problem, <note>the center of the circle is the center point</note>, <note>the radius is the distance from the center to any point on the circumference, such as the distance from the center to points R, S, T</note>. <note>All points in the figure that are at a distance equal to the radius from the center point are on this circle</note>.

Definition of Arc

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

In the figure of this problem, there are <note>three points R, S, and T</note> on the circle, <note>arc RT is the curve connecting point R and point T</note>, <note>arc ST is the curve connecting point S and point T</note>, <note>arc RS is the curve connecting point R and point S</note>. According to the definition of arc, <note>arcs RT, ST, and RS are all segments of curves between two points on the circle</note>.

Angle Properties of a Circle

Arc <note>RT</note>, arc <note>ST</note>, and arc <note>RS</note> make a complete circle, with a total radian measure of <note>360 degrees</note>. The sum of their corresponding central angles is also <note>360 degrees</note>, that is, <note>angle ROT</note> + <note>angle SOT</note> + <note>angle ROS</note> = <note>360 degrees</note>.


Question:
What is m $\angle $ HGJ?

m $\angle $ HGJ= $\Box$ °

Answer: m \$\angle \$ HGJ=100°

1. Given that the central angle of arc IJ is ∠IGJ = 120°, and the central angle of arc IH is ∠IGH = 140°

2. According to the properties of angles in a circle, the total radians of the entire circle is 360 degrees, and the sum of the central angles corresponding to the entire circle is 360 degrees.

3. It can be deduced that ∠HGJ = 360° - ∠IGJ - ∠IGH

4. After simplification, ∠HGJ = 360° - 120° - 140° = 100°

5. Therefore, through the above reasoning, the final answer is: ∠HGJ = 100°.

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, in the circle, <note>point G is the center of the circle</note>, <note>points H, I, and J are any points on the circle</note>, <note>line segments GH, GI, and GJ are the line segments from the center G to any point on the circle</note>, therefore <note>line segments GH, GI, and GJ are the radii of the circle</note>.

Definition of Central Angle

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

<note>In a circle, point I and point J are two points on the circle, and the center of the circle is point G</note>. The angle ∠IGJ formed by the lines <note>GI and GJ is called the central angle</note>. Similarly, the angle ∠HGI formed by the lines <note>GH and GI</note>, and the angle ∠HGJ formed by the lines <note>GH and GJ are also called central angles</note>.

Angle Properties of a Circle

In the figure of this problem, <note>arc HI, arc IJ, and arc JH around the circle sum up to 360 degrees</note>, <note>the sum of their corresponding angles at the circumference is also 360 degrees, that is, angle IGJ + angle HGI + angle JGH = 360 degrees</note>.