To find the difference in area between the original triangle and the dilated triangle, we first need to calculate the area of the original triangle.
The formula for the area of a triangle is:
\[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \]
Using the original triangle's measurements:
\[ \text{Area}_{\text{original}} = \frac{1}{2} \times 10 , \text{cm} \times 15 , \text{cm} = \frac{1}{2} \times 150 , \text{cm}^2 = 75 , \text{cm}^2 \]
Next, since Kierra dilates the triangle using a scale factor, we need that scale factor to find the area of the dilated triangle. However, it looks like the scale factor value was not provided in your question.
Assuming Kierra dilates the triangle by a scale factor \( k \), the area of the dilated triangle can be expressed as:
\[ \text{Area}{\text{dilated}} = k^2 \times \text{Area}{\text{original}} \]
This means:
\[ \text{Area}_{\text{dilated}} = k^2 \times 75 , \text{cm}^2 \]
To find the difference in areas:
\[ \text{Difference} = \text{Area}{\text{original}} - \text{Area}{\text{dilated}} = 75 , \text{cm}^2 - (k^2 \times 75 , \text{cm}^2) \]
\[ \text{Difference} = 75 , \text{cm}^2 (1 - k^2) \]
Now, if we had the scale factor \( k \), we could substitute it in and calculate the specific difference in area.
However, for finding a numerical answer among the choices provided:
- If \( k = 1 \) (no dilation), the difference is \( 0 , \text{cm}^2 \).
- If \( k = 0.5 \), \( (1 - k^2) = 0.75 \), so the difference would be \( 75 , \text{cm}^2 \times 0.75 = 56.25 , \text{cm}^2 \).
- If \( k = 0.66 \) (approximately), the difference could fall around \( 54 , \text{cm}^2 \) depending on the exact calculation.
If you can provide the scale factor, I can help you pinpoint the exact difference in areas resulting in a clearer answer from the multiple choices you provided.
For the given choices (48 cm², 27 cm², 75 cm², 54 cm²), the closest value can point towards k values approximating those areas. If you assume \( k \) about 0.4, the area difference could be more closely resolved.
If you're unsure, using the \( k \) values above should help narrow it down.