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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.