First, let's calculate the area of the original triangle using the formula for the area of a triangle:
\[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \]
For the original triangle:
- Base = 10 cm
- Height = 15 cm
\[ \text{Area}_{\text{original}} = \frac{1}{2} \times 10 , \text{cm} \times 15 , \text{cm} = \frac{150}{2} = 75 , \text{cm}^2 \]
Next, we need to find the area of the dilated triangle. When a triangle is dilated by a scale factor \( k \), the area of the triangle changes by a factor of \( k^2 \). Here, the scale factor given is \( \frac{45}{4} \).
Now, we calculate \( k^2 \): \[ k^2 = \left(\frac{45}{4}\right)^2 = \frac{2025}{16} \]
Now, the area of the dilated triangle is: \[ \text{Area}{\text{dilated}} = \text{Area}{\text{original}} \times k^2 = 75 , \text{cm}^2 \times \frac{2025}{16} \]
Calculating this gives us: \[ \text{Area}_{\text{dilated}} = \frac{75 \times 2025}{16} = \frac{151875}{16} \approx 9492.1875 , \text{cm}^2 \]
Now, we find the difference between the original area and the dilated area: \[ \text{Difference} = \text{Area}{\text{original}} - \text{Area}{\text{dilated}} = 75 , \text{cm}^2 - 9492.1875 , \text{cm}^2 \]
Because the dilated area is much larger than the original, we calculate the absolute difference:
\[ \text{Difference} = |75 - 9492.1875| \approx 9417.1875 , \text{cm}^2 \]
In summary, the area of the original triangle is actually less than the area of the dilated triangle. The question asks for how much greater the original area is compared to the dilated area, which results in a negative value showing that the original triangle is not greater, but rather less.
Therefore: \[ \text{The original area is not greater, so there is no valid response from the provided options.} \]
Given the options, if we interpret differently to seek the answer based on absolute areas, we find:
- The area of the original triangle: 75 cm²
- The difference is just for reference; it implies that the original triangle's area is considerably lesser.
However, if strictly following your question:
The correct response from the provided answers does not match; none reflect the absolute difference correctly based on the calculations established.
Thus, none of the provided options are correct, or they might have differently intended interpretations of the relationship between the areas.
1 answer