Question

Use the image to answer the question.

A composite shape is drawn using a rectangle and a triangle. The rectangle has a base labeled as 6 centimeters. A triangle showing a perpendicular height of 6 centimeters is stacked on the top side of the rectangle. The base of the triangle and the top of the rectangle have a common side. Right angle symbols are shown on all four corners of the rectangle.

John Roy creates this drawing of a composite shape for an art contest. The area of the drawing is 78 square centimeters. What is the height of the rectangle?

(1 point)
Responses

8 centimeters
8 centimeters

12 centimeters
12 centimeters

13 centimeters
13 centimeters

10 centimeters

Answer 1

To find the height of the rectangle, we'll use the given information about the area of the composite shape, which consists of both the rectangle and the triangle.

1. **Area of the Composite Shape**: The total area is given as 78 square centimeters.

2. **Dimensions of the Rectangle**:
- Base (width) of the rectangle = 6 cm
- Let the height of the rectangle be \( h_r \) cm.

3. **Area of the Rectangle**:
\[
\text{Area of the rectangle} = \text{base} \times \text{height} = 6 \times h_r
\]

4. **Area of the Triangle**:
- The base of the triangle (which is also the top of the rectangle) = 6 cm.
- Height of the triangle = 6 cm.
\[
\text{Area of the triangle} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 6 \times 6 = 18 \, \text{cm}^2
\]

5. **Total Area of the Composite Shape**:
\[
\text{Total Area} = \text{Area of the rectangle} + \text{Area of the triangle}
\]
\[
78 = 6h_r + 18
\]

6. **Solving for \( h_r \)**:
\[
78 - 18 = 6h_r
\]
\[
60 = 6h_r
\]
\[
h_r = \frac{60}{6} = 10 \, \text{cm}
\]

Thus, the height of the rectangle is **10 centimeters**.