To find the areas of the original triangle and the dilated triangle, we can follow these steps:
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Calculate the area of the original triangle: The formula for the area of a triangle is: \[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \] Here, the base is 8 cm and the height is 3 cm.
\[ \text{Area} = \frac{1}{2} \times 8 , \text{cm} \times 3 , \text{cm} = \frac{1}{2} \times 24 , \text{cm}^2 = 12 , \text{cm}^2 \]
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Calculate the dimensions of the dilated triangle: The scale factor of dilation is 3. This means that both the base and height of the triangle will be multiplied by 3.
- New base = \(8 , \text{cm} \times 3 = 24 , \text{cm}\)
- New height = \(3 , \text{cm} \times 3 = 9 , \text{cm}\)
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Calculate the area of the dilated triangle: Using the area formula again:
\[ \text{Area} = \frac{1}{2} \times \text{new base} \times \text{new height} \]
\[ \text{Area} = \frac{1}{2} \times 24 , \text{cm} \times 9 , \text{cm} = \frac{1}{2} \times 216 , \text{cm}^2 = 108 , \text{cm}^2 \]
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Compare the areas:
- Area of the original triangle = 12 cm²
- Area of the dilated triangle = 108 cm²
To find how many times the area of the dilated triangle is compared to the original triangle:
\[ \text{Factor} = \frac{\text{Area of dilated triangle}}{\text{Area of original triangle}} = \frac{108 , \text{cm}^2}{12 , \text{cm}^2} = 9 \]
Therefore, the correct statement 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.