To solve the problem, we first calculate the area of the original triangle.
The formula for the area of a triangle is given by:
\[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \]
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Calculate the area of the original triangle:
- Base = 10 cm
- Height = 15 cm
\[ \text{Area}_{\text{original}} = \frac{1}{2} \times 10 \times 15 = 75 \text{ cm}^2 \]
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Calculate the area of the dilated triangle: Let's denote the scale factor as \( f \). When a triangle is dilated by a scale factor \( f \), both the base and height are multiplied by \( f \). Therefore, the new base and height are:
- Base of the dilated triangle = \( 10f \) cm
- Height of the dilated triangle = \( 15f \) cm
The area of the dilated triangle is:
\[ \text{Area}_{\text{dilated}} = \frac{1}{2} \times (10f) \times (15f) = \frac{1}{2} \times 150f^2 = 75f^2 \text{ cm}^2 \]
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Find the difference between the two areas:
Now, we need to find the difference between the area of the original triangle and the area of the dilated triangle:
\[ \text{Difference} = \text{Area}{\text{original}} - \text{Area}{\text{dilated}} = 75 \text{ cm}^2 - 75f^2 \text{ cm}^2 \]
\[ \text{Difference} = 75(1 - f^2) \text{ cm}^2 \]
- Determine the answer based on the scale factor \( f \):
If \( f \) is a positive number less than 1, \( 1 - f^2 \) will be positive, meaning the area of the original triangle will be greater than that of the dilated triangle. However, we do not have a specific value for \( f \) to calculate the exact difference.
None of the choices (48 cm², 54 cm², 27 cm², or 75 cm²) can be determined without knowing the value of \( f \).
Assuming Kierra dilated the triangle with a scale factor that is appropriate to any of the choices, we can substitute \( f \) accordingly to find possible values for the difference.
- If \( f = 1 \), then the difference is \( 75(1 - 1^2) = 0 \) cm² (not an option).
- Assuming \( f \) close to \( 0.6 \) gives:
\[ f^2 = 0.36 \] \[ \text{Difference} = 75(1 - 0.36) = 75(0.64) = 48 \text{ cm}^2 \] Which is one of the provided options.
Thus, the answer is: 48 cm².