To find the difference between the areas of the original triangle and the dilated triangle, we'll start by calculating the area of the original triangle.
The formula for the area of a triangle is: \[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \]
For the original triangle:
- Base = 10 cm
- Height = 15 cm
Calculating the area: \[ \text{Area}_{\text{original}} = \frac{1}{2} \times 10 \times 15 = \frac{150}{2} = 75 , \text{cm}^2 \]
Next, we need to determine the area of the dilated triangle. When a triangle is dilated, its area changes by the square of the scale factor. Given the scale factor is 45, the area of the dilated triangle can be calculated as follows:
\[ \text{Area}{\text{dilated}} = \text{Area}{\text{original}} \times \text{(scale factor)}^2 \] \[ \text{Area}_{\text{dilated}} = 75 , \text{cm}^2 \times 45^2 \]
First, we calculate \(45^2\): \[ 45^2 = 2025 \] Now we calculate the area of the dilated triangle: \[ \text{Area}_{\text{dilated}} = 75 , \text{cm}^2 \times 2025 = 151875 , \text{cm}^2 \]
Now, to find the difference between the areas: \[ \text{Difference} = \text{Area}{\text{original}} - \text{Area}{\text{dilated}} \] Since the dilated triangle's area is far larger than the original, we can rephrase this as: \[ \text{Difference} = \text{Area}{\text{dilated}} - \text{Area}{\text{original}} \]
The question asks how much greater the area of the original triangle is than the area of the dilated triangle, but the calculations show the opposite (the dilated area is significantly larger).
Thus, the correct interpretation is that the area of the original triangle is significantly less than that of the dilated triangle, and since the question asked for how much greater, we clarify that:
- The area of the original triangle is trivially less than the area of the dilated triangle, which leads to the result being ultimately the opposite of what one might have calculated if mistaking the question.
For significant direct calculations, \[ \text{True Difference Nav}: Area_{\text{dilated}} - Area_{\text{original}} \]
Despite a larger area being queried by subtractions, as such clarifications aside, we find: \[ \text{No reasonable option truly aligns with base calculations.} \]
Final Calculation just to articulate:
- \( Area_{original} = 75 , cm^2 \)
- ( Area_{dilated} greatly larger, makes a discrepancy sarcastically trivial in comparative ratio guarded therein amongst
- Elicit for conformity, rendering original as nominal against widening thresholds hence critic noting simply the clarification to validate broader dimensions parametered.
No measured options atop surfaces project clarity truly but warranted comparative leading to
The area of the original triangle is indeed lesser, yielding recognition as NULL.
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