Mike draws a triangle and dilates it by a scale factor of 2.5. If the triangle’s original dimensions are 4 inches high and 6 inches along its base, what is the area in square inches of Mike’s dilated triangle?

Question

(1 point)
Responses

24
24

150
150

60
60

75

Answer 1

To find the area of the dilated triangle, we first calculate the area of the original triangle and then apply the scale factor for the area.

1. Calculate the area of the original triangle: The formula for the area of a triangle is: \[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \]

   For the original triangle:

   • Base = 6 inches
   • Height = 4 inches

   \[ \text{Area} = \frac{1}{2} \times 6 \times 4 = \frac{1}{2} \times 24 = 12 \text{ square inches} \]

2. Apply the scale factor to find the area of the dilated triangle: When a shape is dilated by a scale factor of \( k \), the area of the new shape is scaled by \( k^2 \). Here, the scale factor \( k = 2.5 \).

   \[ k^2 = (2.5)^2 = 6.25 \]

   Thus, the area of the dilated triangle is: \[ \text{Dilated Area} = \text{Original Area} \times k^2 = 12 \times 6.25 = 75 \text{ square inches} \]

So, the area of Mike’s dilated triangle is 75 square inches.