To find the difference between the areas of the original triangle and the dilated triangle, we first need to calculate the area of the original triangle and then the area of the dilated triangle.
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Area of the original triangle: The area \( A \) of a triangle is given by the formula: \[ A = \frac{1}{2} \times \text{base} \times \text{height} \]
Plugging in Kierra's triangle measurements: \[ A = \frac{1}{2} \times 10 , \text{cm} \times 15 , \text{cm} = \frac{1}{2} \times 150 , \text{cm}^2 = 75 , \text{cm}^2 \]
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Dimensions of the dilated triangle: Kierra dilates the triangle with a scale factor of \( \frac{4}{5} \). Hence, the new base and height of the dilated triangle will be: \[ \text{New base} = 10 , \text{cm} \times \frac{4}{5} = 8 , \text{cm} \] \[ \text{New height} = 15 , \text{cm} \times \frac{4}{5} = 12 , \text{cm} \]
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Area of the dilated triangle: Now we calculate the area of the dilated triangle: \[ A_{\text{dilated}} = \frac{1}{2} \times 8 , \text{cm} \times 12 , \text{cm} = \frac{1}{2} \times 96 , \text{cm}^2 = 48 , \text{cm}^2 \]
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Finding the difference in areas: Now, we find the difference between the area of the original triangle and the area of the dilated triangle: \[ \text{Difference} = A - A_{\text{dilated}} = 75 , \text{cm}^2 - 48 , \text{cm}^2 = 27 , \text{cm}^2 \]
Thus, the area of the original triangle is 27 cm² greater than the area of the dilated triangle.
Final Answer: 27 cm²