To find out how much greater the area of the original triangle is than the area of the dilated triangle, we'll go through the following steps:
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Calculate the area of the original triangle.
The area \( A \) of a triangle is given by the formula:
\[ A = \frac{1}{2} \times \text{base} \times \text{height} \]
For the original triangle, with a base of 10 cm and a height of 15 cm:
\[ A_{\text{original}} = \frac{1}{2} \times 10 \times 15 = 75 , \text{cm}^2 \] -
Determine the dimensions of the dilated triangle.
The scale factor for the dilation is \( \frac{4}{5} \).
Therefore, the new base and height after dilation will be:
\[ \text{Base}{\text{dilated}} = 10 \times \frac{4}{5} = 8 , \text{cm} \]
\[ \text{Height}{\text{dilated}} = 15 \times \frac{4}{5} = 12 , \text{cm} \] -
Calculate the area of the dilated triangle.
Using the same area formula:
\[ A_{\text{dilated}} = \frac{1}{2} \times \text{Base}{\text{dilated}} \times \text{Height}{\text{dilated}} \]
\[ A_{\text{dilated}} = \frac{1}{2} \times 8 \times 12 = 48 , \text{cm}^2 \] -
Find the difference between the areas.
Now, we subtract the area of the dilated triangle from the area of the original triangle:
\[ \text{Difference} = A_{\text{original}} - A_{\text{dilated}} = 75 , \text{cm}^2 - 48 , \text{cm}^2 = 27 , \text{cm}^2 \]
Thus, the area of the original triangle is 27 cm² greater than the area of the dilated triangle.
Final answer: 27 cm²