To calculate the area of the original triangle, we can use the formula for the area of a triangle:
\[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \]
For the original triangle with a base of 8 cm and a height of 3 cm:
\[ \text{Area}_{\text{original}} = \frac{1}{2} \times 8 , \text{cm} \times 3 , \text{cm} = \frac{1}{2} \times 24 , \text{cm}^2 = 12 , \text{cm}^2 \]
Now, when the triangle is dilated by a scale factor of 3, both the base and the height are multiplied by 3:
\[ \text{New base} = 3 \times 8 , \text{cm} = 24 , \text{cm} \] \[ \text{New height} = 3 \times 3 , \text{cm} = 9 , \text{cm} \]
Now, we can calculate the area of the dilated triangle:
\[ \text{Area}_{\text{dilated}} = \frac{1}{2} \times 24 , \text{cm} \times 9 , \text{cm} = \frac{1}{2} \times 216 , \text{cm}^2 = 108 , \text{cm}^2 \]
Now, let's compare the areas of the original and the dilated triangles.
The original triangle has an area of 12 cm² and the dilated triangle has an area of 108 cm².
To find out how many times larger the area of the dilated triangle is compared to the original triangle, we can divide the area of the dilated triangle by the area of the original triangle:
\[ \text{Ratio} = \frac{\text{Area}{\text{dilated}}}{\text{Area}{\text{original}}} = \frac{108 , \text{cm}^2}{12 , \text{cm}^2} = 9 \]
Therefore, the dilated triangle has an area 9 times the size of the original triangle.
The correct response 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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