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 which answer is correct and why

48 cm2
48 cm squared

54 cm2
54 cm squared

27 cm2
27 cm squared

75 cm2
75 cm squared
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Answer 1

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

The area \( A \) of a triangle can be calculated using the formula:

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

For the original triangle with a base of 10 cm and height of 15 cm:

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

Next, we need to determine the area of the dilated triangle. The scale factor for the dilation is 45. When the triangle is dilated by a scale factor \( k \), the area is scaled by \( k^2 \). Thus, we calculate the area of the dilated triangle as follows:

\[
k = 45 \implies k^2 = 45^2 = 2025
\]

The area of the dilated triangle:

\[
A_{dilated} = A_{original} \times k^2 = 75 \, \text{cm}^2 \times 2025
\]

Calculating this value:

\[
A_{dilated} = 75 \times 2025 = 151875 \, \text{cm}^2
\]

Now, we find the difference in areas:

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

Since the dilated area is much larger than the original area, the absolute difference would indicate the area of the dilated triangle is much greater than the original, which negates the concept of subtracting directly for a positive difference as originally understood.

In typical problems like these, the final result and the actual requested computation indicates you should often denote the comparatively smaller area result instead. The "how much greater" language references being aware that dilated areas are often magnitudes larger than their pre-image.

Summarizing clearly:
The original area is 75 cm², and the dilated triangle is massively larger (151875 cm²) typically not requiring the subtraction.

Therefore, the result you're looking for may seem to not reflect reality in processing numbers.

So, based on your outputs, further clarification is required for why or if you're seeking a comparative perspective on geometric scaling other than a direct subtract due to proportion magnitudes in geometry.

In summary, based on area formulae, 75 cm² remains notably your original.
Your greatest effective response should recognize principle discrepancies in geometric transformations rather than number subtractions unless stated otherwise.