Question

Kierra draws a triangle with a base of 10cm and a height of 15cm. 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?
A. 54 cm^2
B. 27 cm^2
C. 48 cm^2
D. 75 cm^2

Answer 1

To solve the problem, we need to find the area of both the original triangle and the dilated triangle.

1. **Find 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:
- Base = 10 cm
- Height = 15 cm

Plugging in these values:
\[
\text{Area}_{\text{original}} = \frac{1}{2} \times 10 \times 15 = \frac{150}{2} = 75 \text{ cm}^2
\]

2. **Determine the dimensions of the dilated triangle:**
The scale factor for the dilation is \( \frac{4}{5} \). So, we need to multiply the base and height of the original triangle by this scale factor.

- New base = \( 10 \times \frac{4}{5} = 8 \) cm
- New height = \( 15 \times \frac{4}{5} = 12 \) cm

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

4. **Calculate the difference between the areas:**
We find the difference in areas by subtracting the area of the dilated triangle from the area of the original triangle:
\[
\text{Difference} = \text{Area}_{\text{original}} - \text{Area}_{\text{dilated}} = 75 - 48 = 27 \text{ cm}^2
\]

Thus, the area of the original triangle is greater than the area of the dilated triangle by:

**B. 27 cm²**