Use the image to answer the question.

Question

An illustration shows a rectangle stacked on top of a triangle such that the bottom side of the rectangle and the base of the triangle are the same. The dimensions are as follows: the top side is 12 meters, the right side is 10 meters, and the perpendicular height of the triangle from the apex to the center of the base is

The composite figure has an area of 156 square meters. What is the height of the triangle?

(1 point)
Responses

7 meters
7 meters

3 meters
3 meters

6 meters
6 meters

8 meters

Answer 1

To find the height of the triangle in the composite figure, we start by calculating the area of the rectangle and then use the total area to find the height of the triangle.

1. Area of the Rectangle:

   • The width of the rectangle is 12 meters (the top side).
   • The height of the rectangle is 10 meters (the right side).
   • Area of the rectangle = width × height = 12 m × 10 m = 120 square meters.

2. Total Area of the Composite Figure:

   • Given that the total area of the composite figure (rectangle + triangle) is 156 square meters.

3. Area of the Triangle:

   • Total area of the composite figure = Area of Rectangle + Area of Triangle
   • Area of Triangle = Total Area - Area of Rectangle
   • Area of Triangle = 156 square meters - 120 square meters = 36 square meters.

4. Area of the Triangle Formula:

   • The area of a triangle is given by the formula: \[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height}. \]
   • The base of the triangle is 12 meters (same as the rectangle's width).
   • Let the height of the triangle be \( h \). Thus: \[ 36 = \frac{1}{2} \times 12 \times h. \]

5. Solving for Height \( h \):

   • Rearranging the equation: \[ 36 = 6h, \] \[ h = \frac{36}{6} = 6 \text{ meters}. \]

Thus, the height of the triangle is 6 meters.