To find the height of the rectangle, we can start by calculating the area of the composite shape, which consists of a rectangle and a triangle.
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Let the height of the rectangle be \( h_r \) centimeters.
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The area of the rectangle can be calculated as: \[ \text{Area of rectangle} = \text{base} \times \text{height} = 6 , \text{cm} \times h_r \]
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The triangle has a base equal to the rectangle's base (6 cm) and a height of 6 cm. The area of the triangle is calculated as: \[ \text{Area of triangle} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 6 , \text{cm} \times 6 , \text{cm} = 18 , \text{cm}^2 \]
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The total area of the composite shape is given as 78 square centimeters, which can be expressed as: \[ \text{Area of rectangle} + \text{Area of triangle} = 78 , \text{cm}^2 \]
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Plugging in the values we have: \[ 6 , \text{cm} \times h_r + 18 , \text{cm}^2 = 78 , \text{cm}^2 \]
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Solving for \( h_r \): \[ 6h_r = 78 - 18 \] \[ 6h_r = 60 \] \[ h_r = \frac{60}{6} = 10 , \text{cm} \]
Thus, the height of the rectangle is 10 centimeters.