Question

Kierra draws a triangle with a base of 10 cm and a height of 15 cm. She dilates the triangle using a scale factor of 45
. Then, Kierra finds the difference between the two areas by subtracting. How much greater is the area of the original triangle than the area of the dilated triangle?(1 point)
Responses

48 cm2
48 cm squared

54 cm2
54 cm squared

27 cm2
27 cm squared

75 cm2
75 cm squared

Answer 1

To find the difference in areas between the original triangle and the dilated triangle, we start by calculating the area of the original triangle.

The formula for the area of a triangle is given by:

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

For the original triangle:
- Base = 10 cm
- Height = 15 cm

Calculating the area:

\[
\text{Area}_{\text{original}} = \frac{1}{2} \times 10 \times 15 = \frac{1}{2} \times 150 = 75 \, \text{cm}^2
\]

Next, we calculate the area of the dilated triangle. Since Kierra dilates the triangle using a scale factor of 45, both the base and the height will be multiplied by this scale factor.

The new base and height after dilation will be:
- New base = \(10 \times 45 = 450 \, \text{cm}\)
- New height = \(15 \times 45 = 675 \, \text{cm}\)

Now, we calculate the area of the dilated triangle:

\[
\text{Area}_{\text{dilated}} = \frac{1}{2} \times 450 \times 675
\]

Calculating this:

\[
\text{Area}_{\text{dilated}} = \frac{1}{2} \times (450 \times 675) = \frac{1}{2} \times 303750 = 151875 \, \text{cm}^2
\]

Now, we find the difference in areas between the original triangle and the dilated triangle:

\[
\text{Difference} = \text{Area}_{\text{original}} - \text{Area}_{\text{dilated}} = 75 \, \text{cm}^2 - 151875 \, \text{cm}^2
\]

Thus, the difference is:

\[
\text{Difference} = -151800 \, \text{cm}^2
\]

This shows that the area of the original triangle is significantly less than the area of the dilated triangle. Therefore, considering the choices provided, it seems there is a misunderstanding in the problem since it asks for "how much greater is the area of the original triangle than the area of the dilated triangle?".

The correct interpretation is that the area of the original triangle is not greater, but significantly less. Hence, no valid option aligns with our findings based on the original question.

The answer to the question about how much greater is the original triangle area compared to the dilated triangle is negative. The original triangle does not surpass the area of the dilated triangle by any positive measure.