Use the image to answer the question.

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

12 centimeters
12 centimeters

13 centimeters

Answer 1

To find the height of the rectangle, we first need to determine the area of both the rectangle and the triangle that make up the composite shape.

1. The area of the composite shape is given as 78 square centimeters.
2. The base of the rectangle is 6 centimeters.
3. The triangle on top has a base of 6 centimeters (the same as the rectangle) and a height of 6 centimeters.

Now, we can calculate the area of the triangle:

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

Next, let the height of the rectangle be \( h \) centimeters. The area of the rectangle is given by:

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

Now we can set up the equation for the area of the composite shape:

\[ \text{Area of rectangle} + \text{Area of triangle} = 78 \] \[ 6h + 18 = 78 \]

Now, subtract 18 from both sides:

\[ 6h = 60 \]

Next, divide both sides by 6:

\[ h = 10 \]

Thus, the height of the rectangle is \( \boxed{10} \) centimeters.