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?

Question

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