Question

Marisol draws a triangle with a 4-inch base and 6-inch height. She dilates the triangle by a scale factor of 1.5. Compare the areas of the dilated triangle and the original triangle. Use a decimal response if necessary.
The area of the dilated triangle is __ times the area of the original triangle

Answer 1

To compare the areas of the dilated triangle and the original triangle, we start by calculating the area of the original 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, with a base of 4 inches and a height of 6 inches, the area is:

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

Next, we dilate the triangle by a scale factor of 1.5. When a triangle is dilated by a scale factor \( k \), the area of the dilated triangle is scaled by \( k^2 \).

Calculating the scale factor for the area:

\[
k^2 = (1.5)^2 = 2.25
\]

Thus, the area of the dilated triangle \( A_{\text{dilated}} \) is:

\[
A_{\text{dilated}} = A_{\text{original}} \times k^2 = 12 \times 2.25 = 27 \text{ square inches}
\]

Now, we compare the areas:

The area of the dilated triangle is \( \frac{A_{\text{dilated}}}{A_{\text{original}}} \):

\[
\text{Comparison} = \frac{27}{12} = 2.25
\]

Therefore, the area of the dilated triangle is **2.25 times** the area of the original triangle.

So, the final answer is:

The area of the dilated triangle is **2.25** times the area of the original triangle.