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.

Answer 1

To compare the areas of the original triangle and the dilated triangle, we first need to calculate the area of the original triangle.

The formula for the area \( A \) of a triangle is given by:

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

For the original triangle:

- Base = 4 inches
- Height = 6 inches

Plugging in these values:

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

Next, Marisol dilates the triangle by a scale factor of 1.5. The dimensions of the dilated triangle will be:

- New base = \( 1.5 \times 4 = 6 \) inches
- New height = \( 1.5 \times 6 = 9 \) inches

Now, we can compute the area of the dilated triangle:

\[
A_{dilated} = \frac{1}{2} \times \text{new base} \times \text{new height}
\]
\[
A_{dilated} = \frac{1}{2} \times 6 \times 9 = \frac{1}{2} \times 54 = 27 \text{ square inches}
\]

Now we can compare the areas of the original and dilated triangles:

- Area of the original triangle: \( 12 \) square inches
- Area of the dilated triangle: \( 27 \) square inches

To determine the ratio of the areas, we can use the fact that the area of a triangle is proportional to the square of the scale factor. The scale factor is \( 1.5 \), so the area of the dilated triangle will be:

\[
\text{Area scale factor} = (1.5)^2 = 2.25
\]

Thus, we can verify the relationship:

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

In conclusion, the areas of the original triangle and the dilated triangle are:

- The original triangle's area is \( 12 \) square inches.
- The dilated triangle's area is \( 27 \) square inches.

This means the dilated triangle has an area that is \( \frac{27}{12} = 2.25 \) times larger than the area of the original triangle.