Question

The composite figure is a rectangle with a base of 12 m and a side length of 10 m. The triangle part has a base of 12 m. What is the height of the triangle if the area of the whole composite is 156 m².

Answer 1

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

1. **Calculate the area of the rectangle**:
The area \( A \) of a rectangle is given by the formula:
\[
A = \text{base} \times \text{height}
\]
Here, the base of the rectangle is 12 m and the side length (height) is 10 m.
\[
A_{\text{rectangle}} = 12 \, \text{m} \times 10 \, \text{m} = 120 \, \text{m}^2
\]

2. **Determine the area of the triangle**:
We know that the area of the whole composite figure (rectangle + triangle) is 156 m². Thus, the area of the triangle can be calculated as follows:
\[
A_{\text{triangle}} = A_{\text{composite}} - A_{\text{rectangle}}
\]
Substituting the known values:
\[
A_{\text{triangle}} = 156 \, \text{m}^2 - 120 \, \text{m}^2 = 36 \, \text{m}^2
\]

3. **Calculate the height of the triangle**:
The area \( A \) of a triangle is given by the formula:
\[
A = \frac{1}{2} \times \text{base} \times \text{height}
\]
Here, the base of the triangle is 12 m, and we need to find the height \( h \) of the triangle. Setting up the equation, we have:
\[
36 \, \text{m}^2 = \frac{1}{2} \times 12 \, \text{m} \times h
\]
Simplifying, we find:
\[
36 \, \text{m}^2 = 6 \, \text{m} \times h
\]

To isolate \( h \), we divide both sides by 6 m:
\[
h = \frac{36 \, \text{m}^2}{6 \, \text{m}} = 6 \, \text{m}
\]

Therefore, the height of the triangle is \( \boxed{6 \, \text{m}} \).