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?

a. 60
b. 24
c. 75
d. 150

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.

The area \( A \) of a triangle is given by the formula:

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

For the original triangle, the base is 6 inches and the height is 4 inches.

Calculating the original area:

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

The scale factor for the dilation is 2.5. When a shape is dilated by a scale factor \( k \), the area of the new shape is increased by a factor of \( k^2 \).

Calculating the new area after dilation:

\[
A_{\text{dilated}} = A_{\text{original}} \times k^2 = 12 \times (2.5^2)
\]

Calculating \( 2.5^2 \):

\[
2.5^2 = 6.25
\]

Now we can find the area of the dilated triangle:

\[
A_{\text{dilated}} = 12 \times 6.25 = 75 \text{ square inches}
\]

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