Let's first calculate the area of the original triangle using the formula:
\[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \]
For the original triangle:
\[ \text{Area}_{\text{original}} = \frac{1}{2} \times 10 , \text{cm} \times 15 , \text{cm} = \frac{150}{2} , \text{cm}^2 = 75 , \text{cm}^2 \]
Next, we will determine the dimensions of the dilated triangle. When Kierra dilates the triangle using a scale factor of 45, both the base and the height are multiplied by this factor:
\[ \text{Base}{\text{dilated}} = 10 , \text{cm} \times 45 = 450 , \text{cm} \] \[ \text{Height}{\text{dilated}} = 15 , \text{cm} \times 45 = 675 , \text{cm} \]
Now, we can calculate the area of the dilated triangle using the same area formula:
\[ \text{Area}{\text{dilated}} = \frac{1}{2} \times \text{Base}{\text{dilated}} \times \text{Height}_{\text{dilated}} \] \[ = \frac{1}{2} \times 450 , \text{cm} \times 675 , \text{cm} \]
Calculating this:
\[ \text{Area}_{\text{dilated}} = \frac{1}{2} \times 450 \times 675 = 303375 , \text{cm}^2 \]
Next, we need to find the difference in areas by subtracting the area of the dilated triangle from the area of the original triangle:
However, since the dilated triangle will have a much larger area, Kierra should be subtracting the original area from the dilated area, as generally larger area would be (dilated triangle area – original area).
This question seems to have a mistake; therefore based on the context, we have:
\[ \text{Difference} = \text{Area}{\text{dilated}} - \text{Area}{\text{original}} = 303375 , \text{cm}^2 - 75 , \text{cm}^2 = 303300 , \text{cm}^2 \]
However, if you're specifically looking for how much greater the area of the original triangle than the area of the dilated triangle based on presented options, we should theoretically see \( 75 , cm^2 - 303375 , cm^2 \).
But that is incorrect in evaluating based on your choices:
The original triangle is much less than the dilated area.
Hence confirming, the area of the original triangle is not greater so strictly based upon how you phrased, your original question what we interpret is area of original don't surpass dilated.
Given how choices might be laid out, I advise selecting appropriate responses but none seem fitting your question closely in terms of obtaining a valid interpretation or comparisons directly.
So they are wrong context on what you're asking direct.