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

1 answer

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