To find the height of the rectangle in the composite shape that consists of a rectangle and a triangle, we need to break down the areas.
The area of the rectangle can be calculated using the formula:
\[ \text{Area of rectangle} = \text{base} \times \text{height} = 6 , \text{cm} \times h \]
where \( h \) is the height of the rectangle.
The area of the triangle can be calculated using the formula:
\[ \text{Area of triangle} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 6 , \text{cm} \times 6 , \text{cm} \]
Calculating the area of the triangle:
\[ \text{Area of triangle} = \frac{1}{2} \times 6 \times 6 = 18 , \text{square centimeters} \]
Now, let's denote the height of the rectangle as \( h \). The total area of the composite shape (rectangle + triangle) is given as 78 square centimeters:
\[ \text{Area of rectangle} + \text{Area of triangle} = 78 \]
Substituting the area of the triangle:
\[ 6h + 18 = 78 \]
Now, we can solve for \( h \):
\[ 6h = 78 - 18 \] \[ 6h = 60 \] \[ h = \frac{60}{6} = 10 , \text{cm} \]
Thus, the height of the rectangle is 10 centimeters.