To find the areas of the original triangle and the dilated triangle, we start with the formula for the area of a triangle:
\[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \]
For the original triangle:
- Base = 8 cm
- Height = 3 cm
So the area is:
\[ \text{Area}_{\text{original}} = \frac{1}{2} \times 8 \times 3 = 12 \text{ cm}^2 \]
Next, let’s find the dimensions of the dilated triangle. Since the triangle is dilated by a scale factor of 3, the new base and height will be:
- New Base = 3 × 8 cm = 24 cm
- New Height = 3 × 3 cm = 9 cm
Now, we calculate the area of the dilated triangle:
\[ \text{Area}_{\text{dilated}} = \frac{1}{2} \times 24 \times 9 = \frac{1}{2} \times 216 = 108 \text{ cm}^2 \]
Now, we compare the areas:
- The original triangle has an area of 12 cm².
- The dilated triangle has an area of 108 cm².
To find how many times larger the dilated area is compared to the original, we divide the area of the dilated triangle by the area of the original triangle:
\[ \frac{\text{Area}{\text{dilated}}}{\text{Area}{\text{original}}} = \frac{108}{12} = 9 \]
Thus, the correct comparison is:
The original triangle has an area of 12 cm² and the dilated triangle has an area of 108 cm²; therefore, the dilated triangle has an area 9 times the size of the original triangle.
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