To find the height of the rectangle, we first need to understand the total area of the composite shape, which is composed of the rectangle and the triangle.
Given:
- The base of the rectangle = 6 cm
- The height of the triangle = 6 cm
- The total area of the composite shape = 78 cm²
The area of the triangle can be calculated using the formula:
\[ \text{Area of triangle} = \frac{1}{2} \times \text{base} \times \text{height} \]
For the triangle:
- Base = 6 cm (which is also the width of the rectangle)
- Height = 6 cm
So, the area of the triangle is: \[ \text{Area of triangle} = \frac{1}{2} \times 6 \times 6 = 18 \text{ cm}² \]
Next, let the height of the rectangle be \( h \) cm. The area of the rectangle is: \[ \text{Area of rectangle} = \text{base} \times \text{height} = 6 \times h \text{ cm}² \]
Now, the total area of the composite shape is the sum of the areas of the rectangle and the triangle:
\[ \text{Total Area} = \text{Area of rectangle} + \text{Area of triangle} \] \[ 78 = 6h + 18 \]
To find \( 6h \), we subtract 18 from both sides:
\[ 6h = 78 - 18 \] \[ 6h = 60 \]
Now we can solve for \( h \):
\[ h = \frac{60}{6} = 10 \text{ cm} \]
Therefore, the height of the rectangle is 10 centimeters.