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

10 centimeters
10 centimeters

13 centimeters
13 centimeters

12 centimeters
12 centimeters

Answer 1

To find the height of the rectangle in the composite shape that consists of a rectangle and a triangle, we need to break down the areas.

The area of the rectangle can be calculated using the formula:

\[
\text{Area of rectangle} = \text{base} \times \text{height} = 6 \, \text{cm} \times h
\]

where \( h \) is the height of the rectangle.

The area of the triangle can be calculated using the formula:

\[
\text{Area of triangle} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 6 \, \text{cm} \times 6 \, \text{cm}
\]

Calculating the area of the triangle:

\[
\text{Area of triangle} = \frac{1}{2} \times 6 \times 6 = 18 \, \text{square centimeters}
\]

Now, let's denote the height of the rectangle as \( h \). The total area of the composite shape (rectangle + triangle) is given as 78 square centimeters:

\[
\text{Area of rectangle} + \text{Area of triangle} = 78
\]

Substituting the area of the triangle:

\[
6h + 18 = 78
\]

Now, we can solve for \( h \):

\[
6h = 78 - 18
\]
\[
6h = 60
\]
\[
h = \frac{60}{6} = 10 \, \text{cm}
\]

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