To solve the problem, we first calculate the area of the original triangle and then the area of the dilated triangle.
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Original Triangle Area Calculation: The formula for the area of a triangle is: \[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \] Given the base \( b = 8 \) cm and height \( h = 3 \) cm: \[ \text{Area} = \frac{1}{2} \times 8 \times 3 = \frac{1}{2} \times 24 = 12 \text{ cm}^2 \]
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Dilated Triangle Dimensions: The triangle is dilated by a scale factor of 3, which means:
- New base = \( 3 \times 8 = 24 \) cm
- New height = \( 3 \times 3 = 9 \) cm
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Dilated Triangle Area Calculation: Using the same area formula: \[ \text{Area} = \frac{1}{2} \times \text{new base} \times \text{new height} \] \[ \text{Area} = \frac{1}{2} \times 24 \times 9 = \frac{1}{2} \times 216 = 108 \text{ cm}^2 \]
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Comparison of Areas:
- Area of the original triangle: \( 12 \) cm²
- Area of the dilated triangle: \( 108 \) cm²
To find how many times larger the area of the dilated triangle is compared to the original triangle: \[ \text{Factor} = \frac{108}{12} = 9 \]
Thus, 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.
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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