To find the area of the original triangle, we use the formula for the area of a triangle:
\[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \]
For the original triangle:
- Base = 8 cm
- Height = 3 cm
Calculating the area:
\[ \text{Area} = \frac{1}{2} \times 8 \times 3 = \frac{1}{2} \times 24 = 12 , \text{cm}^2 \]
Next, we calculate the area of the dilated triangle. When a triangle is dilated by a scale factor \(k\), the area scales by \(k^2\). In this case, the scale factor is 3.
\[ k = 3 \] \[ \text{Area of dilated triangle} = k^2 \times \text{Area of original triangle} = 3^2 \times 12 = 9 \times 12 = 108 , \text{cm}^2 \]
Now we compare the areas of the original and the dilated triangles:
- Area of original triangle = 12 cm²
- Area of dilated triangle = 108 cm²
To see how many times larger the dilated triangle is compared to the original triangle, we calculate:
\[ \frac{\text{Area of dilated triangle}}{\text{Area of original triangle}} = \frac{108}{12} = 9 \]
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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