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

27 cm2
27 cm squared

75 cm2
75 cm squared

54 cm2
54 cm squared

48 cm2

Answer 1

To determine how much greater the area of the original triangle is compared to the dilated triangle, we can follow these steps:

1. **Calculate the area of the original triangle:**
The formula for the area of a triangle is:
\[
\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}
\]
For the original triangle with a base of 10 cm and a height of 15 cm:
\[
\text{Area} = \frac{1}{2} \times 10 \, \text{cm} \times 15 \, \text{cm} = \frac{1}{2} \times 150 \, \text{cm}^2 = 75 \, \text{cm}^2
\]

2. **Calculate the area of the dilated triangle:**
When a figure is dilated by a scale factor \( k \), the area of the figure is multiplied by \( k^2 \). Here, the scale factor \( k = 45 \):
\[
\text{Area of Dilated Triangle} = \text{Original Area} \times k^2 = 75 \, \text{cm}^2 \times 45^2
\]

First, compute \( 45^2 \):
\[
45^2 = 2025
\]

Now calculate the area:
\[
\text{Area of Dilated Triangle} = 75 \, \text{cm}^2 \times 2025 = 151875 \, \text{cm}^2
\]

3. **Find the difference between the original and the dilated triangle's area:**
Since the dilated triangle has a much greater area than the original triangle, we actually need to note the greater area belongs to the dilated triangle:
\[
\text{Difference} = \text{Area of Dilated Triangle} - \text{Original Area} = 151875 \, \text{cm}^2 - 75 \, \text{cm}^2
\]

However, the problem specifically asks for "how much greater is the area of the original triangle than the area of the dilated triangle." Given that the dilated area is vastly larger, it appears there may have been a misunderstanding.

Therefore, we'd clarify that:
\[
\text{Original Triangle Area} = 75 \, \text{cm}^2
\]
\[
\text{Dilated Triangle Area} = 151875 \, \text{cm}^2
\]

The difference is such that:
\[
\text{Original Area} < \text{Dilated Area}
\]

In conclusion, the original triangle is not greater than the dilated triangle; in fact, it is smaller. The question appears to have a mismatch in expectation. If seeking between only values presented (which irrelevant here), those numbers aren't combinable to get such:

Correctly, the area of the dilated triangle is excessively greater than the original triangle. Therefore, if you really need to denote numbers within your choices accurately, re-evaluate the statement in full context.