To find the area of the original triangle, we use the formula:
\[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \]
For the original triangle with a base of 8 cm and a height of 3 cm:
\[ \text{Area} = \frac{1}{2} \times 8 , \text{cm} \times 3 , \text{cm} = \frac{1}{2} \times 24 , \text{cm}^2 = 12 , \text{cm}^2 \]
When the triangle is dilated by a scale factor of 3, the new dimensions (base and height) will each be multiplied by 3.
So, the new 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, we calculate the area of the dilated triangle:
\[ \text{Area of dilated triangle} = \frac{1}{2} \times 24 , \text{cm} \times 9 , \text{cm} = \frac{1}{2} \times 216 , \text{cm}^2 = 108 , \text{cm}^2 \]
Now, we can compare the areas of the original and dilated triangles:
- Original triangle area: \( 12 , \text{cm}^2 \)
- Dilated triangle area: \( 108 , \text{cm}^2 \)
To find out how many times larger the area of the dilated triangle is compared to the original triangle, we divide the area of the dilated triangle by the area of the original triangle:
\[ \frac{108 , \text{cm}^2}{12 , \text{cm}^2} = 9 \]
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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