Question

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?

60
75
150
24

Answer 1

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

The area \( A \) of a triangle can be calculated using the formula:

\[
A = \frac{1}{2} \times \text{base} \times \text{height}
\]

For the original triangle, we have:
- Base = 6 inches
- Height = 4 inches

Calculating the area of the original triangle:

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

When a triangle is dilated by a scale factor \( k \), the area of the dilated triangle will be scaled by a factor of \( k^2 \). For a scale factor of 2.5:

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

Now, we multiply the area of the original triangle by the scale factor squared to find the area of the dilated triangle:

\[
\text{Area of dilated triangle} = \text{Area of original triangle} \times k^2 = 12 \times 6.25 = 75 \text{ square inches}
\]

Thus, the area of Mike's dilated triangle is \(\boxed{75}\) square inches.