To calculate the area of the original triangle, we can use the formula:
\[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \]
Substituting the values for the original triangle:
\[ \text{Area} = \frac{1}{2} \times 8 , \text{cm} \times 3 , \text{cm} = \frac{1}{2} \times 24 , \text{cm}^2 = 12 , \text{cm}^2 \]
Now, we apply the scale factor of 3 to the dimensions of the triangle. The new base and height after dilation will be:
\[ \text{New base} = 3 \times 8 , \text{cm} = 24 , \text{cm} \] \[ \text{New height} = 3 \times 3 , \text{cm} = 9 , \text{cm} \]
Now we calculate the area of the dilated triangle:
\[ \text{Area} = \frac{1}{2} \times 24 , \text{cm} \times 9 , \text{cm} = \frac{1}{2} \times 216 , \text{cm}^2 = 108 , \text{cm}^2 \]
Now, we can compare the areas of the original triangle and the dilated triangle:
- Original Triangle Area: \( 12 , \text{cm}^2 \)
- Dilated Triangle Area: \( 108 , \text{cm}^2 \)
To find out how many times larger the area of the dilated triangle is compared to the original, we can calculate:
\[ \text{Ratio} = \frac{\text{Area of dilated triangle}}{\text{Area of original triangle}} = \frac{108 , \text{cm}^2}{12 , \text{cm}^2} = 9 \]
Thus, the dilated triangle has an area 9 times the size of the original triangle.
The correct answer is:
C. 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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