To determine the height of the rectangle in the composite shape, we start by using the information provided about the areas of the rectangle and triangle.
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Area of the Composite Shape: The total area of the composite shape is 78 square centimeters.
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Base of the Rectangle: The base of the rectangle is 6 centimeters.
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Height of the Triangle: The triangle has a height of 6 centimeters, and its base is also the same as the base of the rectangle, which is 6 centimeters.
Now, we calculate the area of the triangle:
\[ \text{Area of the Triangle} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 6 \times 6 = 18 \text{ square centimeters} \]
Next, we denote the height of the rectangle as \( h \). The area of the rectangle can be calculated as:
\[ \text{Area of the Rectangle} = \text{base} \times \text{height} = 6 \times h \]
Now we can express the total area of the composite shape:
\[ \text{Total Area} = \text{Area of the Rectangle} + \text{Area of the Triangle} \]
Substituting the areas we calculated:
\[ 78 = 6h + 18 \]
To find \( h \), we first isolate the term involving \( h \):
\[ 78 - 18 = 6h \] \[ 60 = 6h \]
Now, divide both sides by 6:
\[ h = \frac{60}{6} = 10 \text{ centimeters} \]
Therefore, the height of the rectangle is:
\[ \boxed{10 \text{ centimeters}} \]