To solve this problem, we first need to calculate the area of the original triangle. The area \( A \) of a triangle can be calculated using the formula:
\[ A = \frac{1}{2} \times \text{base} \times \text{height} \]
For the original triangle with a base of 8 cm and a height of 3 cm, we have:
\[ A = \frac{1}{2} \times 8 \times 3 = \frac{1}{2} \times 24 = 12 \text{ cm}^2 \]
Next, we need to find the area of the dilated triangle. When a shape is dilated by a scale factor \( k \), the area of the new shape is scaled by a factor of \( k^2 \). In this case, the scale factor is 3.
Calculating the new area:
\[ \text{Area of the dilated triangle} = k^2 \times \text{Area of the original triangle} = 3^2 \times 12 \text{ cm}^2 = 9 \times 12 \text{ cm}^2 = 108 \text{ cm}^2 \]
Now, we can compare the areas:
- Area of the original triangle: \( 12 \text{ cm}^2 \)
- Area of the dilated triangle: \( 108 \text{ cm}^2 \)
- The dilated triangle has an area which is \( \frac{108}{12} = 9 \) times the size of the original triangle.
Thus, the correct statement is:
a. The original triangle has an area of 12 cm\(^2\) and the dilated triangle has an area of 108 cm\(^2\); therefore, the dilated triangle has an area 9 times the size of the original triangle.
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