To find the areas of the triangles, we first calculate the area of the original triangle, followed by the area of the dilated triangle.
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Area of the original triangle: The formula for the area of a triangle is given by: \[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \] For the original triangle:
- Base = 8 cm
- Height = 3 cm
Substituting the values: \[ \text{Area} = \frac{1}{2} \times 8 \times 3 = \frac{1}{2} \times 24 = 12 \text{ cm}^2 \]
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Area of the dilated triangle: The scale factor of dilation is 3. When a triangle is dilated by a scale factor \(k\), the area of the new triangle is \(k^2\) times the area of the original triangle. Here, \(k = 3\): \[ \text{Area}{\text{dilated}} = k^2 \times \text{Area}{\text{original}} = 3^2 \times 12 = 9 \times 12 = 108 \text{ cm}^2 \]
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Comparison of the areas: The original triangle has an area of 12 cm² and the dilated triangle has an area of 108 cm². To find how many times larger the dilated triangle is compared to the original, we calculate: \[ \frac{\text{Area}{\text{dilated}}}{\text{Area}{\text{original}}} = \frac{108}{12} = 9 \]
Based on the calculations, the original triangle has an area of 12 cm², and the dilated triangle has an area of 108 cm², making the dilated triangle 9 times larger than the original.
Thus, 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.