Question

Kierra draws a triangle with a base of 10 cm and a height of 15 cm. She dilates the triangle using a scale factor of 4/5. Then, Kierra finds the difference between the two areas by subtracting.
How much greater is the area of the original triangle than the area of the dilated triangle?

Answer 1

To find the areas of the original triangle and the dilated triangle and their difference, we can follow these steps:

1. **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:
\[
\text{Area}_{\text{original}} = \frac{1}{2} \times 10 \, \text{cm} \times 15 \, \text{cm} = \frac{1}{2} \times 150 \, \text{cm}^2 = 75 \, \text{cm}^2
\]

2. **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:
\[
\text{Base}_{\text{dilated}} = \frac{4}{5} \times 10 \, \text{cm} = 8 \, \text{cm}
\]
\[
\text{Height}_{\text{dilated}} = \frac{4}{5} \times 15 \, \text{cm} = 12 \, \text{cm}
\]

3. **Calculate the area of the dilated triangle:**
Using the same area formula:
\[
\text{Area}_{\text{dilated}} = \frac{1}{2} \times 8 \, \text{cm} \times 12 \, \text{cm} = \frac{1}{2} \times 96 \, \text{cm}^2 = 48 \, \text{cm}^2
\]

4. **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 \(\boxed{27 \, \text{cm}^2}\).