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?
Answers
GPT-4o mini
To find the area of Mike's dilated triangle, we start by determining the area of the original triangle and then apply the scale factor to find the area of the dilated triangle.
The formula for the area \( A \) of a triangle is
\[
A = \frac{1}{2} \times \text{base} \times \text{height}
\]
For the original triangle, the base is 6 inches and the height is 4 inches. Therefore, the area of the original triangle is:
\[
A = \frac{1}{2} \times 6 \times 4
\]
Calculating this gives:
\[
A = \frac{1}{2} \times 24 = 12 \text{ square inches}
\]
Next, when the triangle is dilated by a scale factor of 2.5, the area of the dilated triangle is affected by the square of the scale factor. If the scale factor is \( k \), the area of the dilated triangle can be calculated using:
\[
\text{Area of dilated triangle} = k^2 \times \text{Area of original triangle}
\]
In this case, the scale factor \( k = 2.5 \), so:
\[
\text{Area of dilated triangle} = (2.5)^2 \times 12
\]
Calculating \( (2.5)^2 \):
\[
(2.5)^2 = 6.25
\]
Therefore, the area of the dilated triangle is:
\[
\text{Area of dilated triangle} = 6.25 \times 12
\]
Now, calculating this gives:
\[
\text{Area of dilated triangle} = 75 \text{ square inches}
\]
Thus, the area of Mike's dilated triangle is
\[
\boxed{75} \text{ square inches}.
\]
The formula for the area \( A \) of a triangle is
\[
A = \frac{1}{2} \times \text{base} \times \text{height}
\]
For the original triangle, the base is 6 inches and the height is 4 inches. Therefore, the area of the original triangle is:
\[
A = \frac{1}{2} \times 6 \times 4
\]
Calculating this gives:
\[
A = \frac{1}{2} \times 24 = 12 \text{ square inches}
\]
Next, when the triangle is dilated by a scale factor of 2.5, the area of the dilated triangle is affected by the square of the scale factor. If the scale factor is \( k \), the area of the dilated triangle can be calculated using:
\[
\text{Area of dilated triangle} = k^2 \times \text{Area of original triangle}
\]
In this case, the scale factor \( k = 2.5 \), so:
\[
\text{Area of dilated triangle} = (2.5)^2 \times 12
\]
Calculating \( (2.5)^2 \):
\[
(2.5)^2 = 6.25
\]
Therefore, the area of the dilated triangle is:
\[
\text{Area of dilated triangle} = 6.25 \times 12
\]
Now, calculating this gives:
\[
\text{Area of dilated triangle} = 75 \text{ square inches}
\]
Thus, the area of Mike's dilated triangle is
\[
\boxed{75} \text{ square inches}.
\]