To find how much greater the area of the original triangle is compared to the area of the dilated triangle, we first need to calculate the area of the original triangle, and then the area of the dilated triangle.
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Calculate the area of the original triangle: \[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \] Base = 10 cm, Height = 15 cm. \[ \text{Area} = \frac{1}{2} \times 10 , \text{cm} \times 15 , \text{cm} = \frac{1}{2} \times 150 , \text{cm}^2 = 75 , \text{cm}^2 \]
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Calculate the dimensions of the dilated triangle: Since Kierra dilates the triangle with a scale factor of 45, the dimensions of the dilated triangle will be: \[ \text{Dilated base} = 10 , \text{cm} \times 45 = 450 , \text{cm} \] \[ \text{Dilated height} = 15 , \text{cm} \times 45 = 675 , \text{cm} \]
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Calculate the area of the dilated triangle: \[ \text{Area of dilated triangle} = \frac{1}{2} \times \text{dilated base} \times \text{dilated height} \] \[ \text{Area of dilated triangle} = \frac{1}{2} \times 450 , \text{cm} \times 675 , \text{cm} \] \[ = \frac{1}{2} \times 303750 , \text{cm}^2 = 151875 , \text{cm}^2 \]
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Find the difference in areas: \[ \text{Difference} = \text{Area of original triangle} - \text{Area of dilated triangle} \] \[ \text{Difference} = 75 , \text{cm}^2 - 151875 , \text{cm}^2 = -151800 , \text{cm}^2 \]
The area of the original triangle is actually much less than the area of the dilated triangle, so the original triangle is not greater. Thus the final answer regarding how much greater the area of the original triangle is than the area of the dilated triangle is:
The original triangle is not greater, actually it's less by 151800 cm².
If the options only allow for comparisons as stated (greater than for original), then no positive response applies based on the phrasing of the question. However, strictly based on calculations, the area of the original is less.
If you simply need to match responses, none apply directly as all provided options are for a difference favoring the original triangle.
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