To find the areas of the original triangle and the dilated triangle and their difference, we can follow these steps:
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Calculate the area of the original triangle: The formula for the area of a triangle is given by: \[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \] For the original triangle, the base is 10 cm and the height is 15 cm. Substituting these values in: \[ \text{Area}_{\text{original}} = \frac{1}{2} \times 10 , \text{cm} \times 15 , \text{cm} = \frac{1}{2} \times 150 , \text{cm}^2 = 75 , \text{cm}^2 \]
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Calculate the dimensions of the dilated triangle: The dilation scale factor is \(\frac{4}{5}\). Thus, the new base and height can be calculated as: \[ \text{Base}{\text{dilated}} = \frac{4}{5} \times 10 , \text{cm} = 8 , \text{cm} \] \[ \text{Height}{\text{dilated}} = \frac{4}{5} \times 15 , \text{cm} = 12 , \text{cm} \]
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Calculate the area of the dilated triangle: Using the same area formula: \[ \text{Area}_{\text{dilated}} = \frac{1}{2} \times 8 , \text{cm} \times 12 , \text{cm} = \frac{1}{2} \times 96 , \text{cm}^2 = 48 , \text{cm}^2 \]
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Find the difference in areas: Now, subtract the area of the dilated triangle from the area of the original triangle: \[ \text{Difference} = \text{Area}{\text{original}} - \text{Area}{\text{dilated}} = 75 , \text{cm}^2 - 48 , \text{cm}^2 = 27 , \text{cm}^2 \]
Thus, the area of the original triangle is greater than the area of the dilated triangle by \(\boxed{27 , \text{cm}^2}\).
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