Marisol draws a triangle with a 4-inch base and 6-inch height. She dilates the triangle by a scale factor of 1.5.

Question

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
original triangle.

Answer 1

To find the area of the original triangle, we can 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:

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

Next, we need to calculate the dimensions of the dilated triangle. The scale factor of dilation is 1.5, so we multiply the base and height of the original triangle by 1.5:

\[ \text{Base}_{\text{dilated}} = 4 \times 1.5 = 6 , \text{inches} \]

\[ \text{Height}_{\text{dilated}} = 6 \times 1.5 = 9 , \text{inches} \]

Now we can find the area of the dilated triangle:

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

To compare the areas of the dilated triangle to the original triangle, we can calculate the ratio of the areas:

\[ \text{Ratio} = \frac{\text{Area}{\text{dilated}}}{\text{Area}{\text{original}}} = \frac{27}{12} = 2.25 \]

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

So in decimal response form, we can say:

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