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

Answer 1

To find the areas of the original triangle and the dilated triangle, we can follow these steps:

1. **Calculate the area of the original triangle.**
The formula for the area \( A \) of a triangle is:
\[
A = \frac{1}{2} \times \text{base} \times \text{height}
\]
Substituting the given base and height:
\[
A = \frac{1}{2} \times 4 \, \text{inches} \times 6 \, \text{inches} = \frac{1}{2} \times 24 \, \text{square inches} = 12 \, \text{square inches}
\]

2. **Apply the dilation.**
When a triangle is dilated by a scale factor \( k \), the area of the new triangle is scaled by \( k^2 \). Here, the scale factor is \( 1.5 \).
Therefore, the area of the dilated triangle will be:
\[
\text{Area of dilated triangle} = k^2 \times \text{Area of original triangle}
\]
Calculating \( k^2 \):
\[
k^2 = (1.5)^2 = 2.25
\]
Now, substituting this into the area equation:
\[
\text{Area of dilated triangle} = 2.25 \times 12 \, \text{square inches} = 27 \, \text{square inches}
\]

3. **Compare the areas.**
- Area of the original triangle: \( 12 \, \text{square inches} \)
- Area of the dilated triangle: \( 27 \, \text{square inches} \)

So, the area of the dilated triangle is greater than the area of the original triangle. In decimal terms, you can say:
- The area increased from \( 12.0 \, \text{square inches} \) to \( 27.0 \, \text{square inches} \).

**Response Summary:** The area of the original triangle is \( 12.0 \) square inches, and the area of the dilated triangle is \( 27.0 \) square inches.