To find the area of the original triangle, we use the formula for the area of a triangle:
\[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \]
Substituting the values for the original triangle:
\[ \text{Area} = \frac{1}{2} \times 8 , \text{cm} \times 3 , \text{cm} = \frac{1}{2} \times 24 , \text{cm}^2 = 12 , \text{cm}^2 \]
Next, we dilate the triangle by a scale factor of 3. The new dimensions (base and height) will be:
- New base = \(8 , \text{cm} \times 3 = 24 , \text{cm}\)
- New height = \(3 , \text{cm} \times 3 = 9 , \text{cm}\)
Now, calculate the area of the dilated triangle:
\[ \text{Area of dilated triangle} = \frac{1}{2} \times \text{new base} \times \text{new height} \] \[ \text{Area of dilated triangle} = \frac{1}{2} \times 24 , \text{cm} \times 9 , \text{cm} = \frac{1}{2} \times 216 , \text{cm}^2 = 108 , \text{cm}^2 \]
The original triangle has an area of \(12 , \text{cm}^2\) and the dilated triangle has an area of \(108 , \text{cm}^2\).
To compare the areas: \[ \text{Factor increase} = \frac{108 , \text{cm}^2}{12 , \text{cm}^2} = 9 \]
Thus, the dilated triangle has an area 9 times the size of the original triangle.
The correct 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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