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 find the area of both the original triangle and the dilated triangle, we start with the formula for the area of a triangle:

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

### Step 1: Calculate the area of the original triangle.
The base of the original triangle is 4 inches and the height is 6 inches.

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

### Step 2: Determine the dimensions of the dilated triangle.
Marisol dilates the triangle by a scale factor of 1.5. This means both the base and height will increase by this factor.

\[
\text{New base} = 4 \times 1.5 = 6 \text{ inches}
\]
\[
\text{New height} = 6 \times 1.5 = 9 \text{ inches}
\]

### Step 3: Calculate the area of the dilated triangle.
Now, we can use the new dimensions to find the area of the dilated triangle.

\[
\text{Area}_{\text{dilated}} = \frac{1}{2} \times 6 \times 9 = \frac{1}{2} \times 54 = 27 \text{ square inches}
\]

### Step 4: Compare the areas.
We want to find out how many times larger the area of the dilated triangle is compared to the original triangle.

\[
\text{Factor of increase} = \frac{\text{Area}_{\text{dilated}}}{\text{Area}_{\text{original}}} = \frac{27}{12} = \frac{9}{4} = 2.25
\]

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

The final answer is:

\[
\text{The area of the dilated triangle is } 2.25 \text{ times the area of the original triangle.}
\]