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.

(1 point)
The area of the dilated triangle is
times the area of the original triangle.

Answer 1

To find the area of the original triangle, we use the formula for the area of a triangle:

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

For the original triangle with a base of 4 inches and a height of 6 inches, the area is:

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

Next, we consider the effect of dilating the triangle by a scale factor of 1.5. When a triangle is dilated, its area scales by the square of the scale factor. Therefore, the area of the dilated triangle can be calculated as follows:

\[
\text{Area}_{\text{dilated}} = \text{Area}_{\text{original}} \times (\text{Scale Factor})^2
\]

Calculating the area of the dilated triangle:

\[
\text{Area}_{\text{dilated}} = 12 \times (1.5)^2 = 12 \times 2.25 = 27 \text{ square inches}
\]

Now, we compare the area of the dilated triangle to the area of the original triangle by examining the factor by which the area has increased:

\[
\text{Factor} = \frac{\text{Area}_{\text{dilated}}}{\text{Area}_{\text{original}}} = \frac{27}{12}
\]

To convert to a decimal, we can perform the division:

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

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

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