Question
A trapezoid is decomposed into a rectangle and two right triangles, the top is 10 inches, the bottom is 12 inches, and the height of both triangles is 8 inches, what’s the area
Answers
GPT-4o mini
To find the area of the trapezoid decomposed into a rectangle and two right triangles, we first need to calculate the area of the trapezoid using its formula, and then we'll verify the decomposition.
Given:
- Top base (shorter base) \( b_1 = 10 \) inches
- Bottom base (longer base) \( b_2 = 12 \) inches
- Height \( h = 8 \) inches
The formula for the area \( A \) of a trapezoid is:
\[
A = \frac{1}{2} \times (b_1 + b_2) \times h
\]
Substituting the values we have:
\[
A = \frac{1}{2} \times (10 + 12) \times 8
\]
\[
A = \frac{1}{2} \times 22 \times 8
\]
\[
A = 11 \times 8 = 88 \text{ square inches}
\]
Now, let's break down the trapezoid:
1. **Rectangle**: The rectangle sits between the heights of the top and bottom bases. The width of the rectangle equals the length of the shorter base, which is 10 inches, and the height is the height of the trapezoid, which is 8 inches.
\[
\text{Area of Rectangle} = \text{width} \times \text{height} = 10 \times 8 = 80 \text{ square inches}
\]
2. **Triangles**: There are two right triangles, each having a height of 8 inches. The base of each triangle can be calculated as follows:
- The difference between the top base and bottom base is \( 12 - 10 = 2 \) inches.
- This means each triangle has a base of \( \frac{2}{2} = 1 \) inch because both triangles share the excess width of 2 inches equally.
The area of one triangle can be calculated using:
\[
\text{Area of one Triangle} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 1 \times 8 = 4 \text{ square inches}
\]
The total area for the two triangles is:
\[
\text{Area of two triangles} = 2 \times 4 = 8 \text{ square inches}
\]
Adding up the areas of the rectangle and the triangles:
\[
\text{Total Area} = \text{Area of Rectangle} + \text{Area of two triangles} = 80 + 8 = 88 \text{ square inches}
\]
Thus, the area of the trapezoid is indeed \( \boxed{88} \) square inches.
Given:
- Top base (shorter base) \( b_1 = 10 \) inches
- Bottom base (longer base) \( b_2 = 12 \) inches
- Height \( h = 8 \) inches
The formula for the area \( A \) of a trapezoid is:
\[
A = \frac{1}{2} \times (b_1 + b_2) \times h
\]
Substituting the values we have:
\[
A = \frac{1}{2} \times (10 + 12) \times 8
\]
\[
A = \frac{1}{2} \times 22 \times 8
\]
\[
A = 11 \times 8 = 88 \text{ square inches}
\]
Now, let's break down the trapezoid:
1. **Rectangle**: The rectangle sits between the heights of the top and bottom bases. The width of the rectangle equals the length of the shorter base, which is 10 inches, and the height is the height of the trapezoid, which is 8 inches.
\[
\text{Area of Rectangle} = \text{width} \times \text{height} = 10 \times 8 = 80 \text{ square inches}
\]
2. **Triangles**: There are two right triangles, each having a height of 8 inches. The base of each triangle can be calculated as follows:
- The difference between the top base and bottom base is \( 12 - 10 = 2 \) inches.
- This means each triangle has a base of \( \frac{2}{2} = 1 \) inch because both triangles share the excess width of 2 inches equally.
The area of one triangle can be calculated using:
\[
\text{Area of one Triangle} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 1 \times 8 = 4 \text{ square inches}
\]
The total area for the two triangles is:
\[
\text{Area of two triangles} = 2 \times 4 = 8 \text{ square inches}
\]
Adding up the areas of the rectangle and the triangles:
\[
\text{Total Area} = \text{Area of Rectangle} + \text{Area of two triangles} = 80 + 8 = 88 \text{ square inches}
\]
Thus, the area of the trapezoid is indeed \( \boxed{88} \) square inches.