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 with a base of 8 cm and a height of 3 cm:
\[ \text{Area} = \frac{1}{2} \times 8 , \text{cm} \times 3 , \text{cm} = \frac{1}{2} \times 24 , \text{cm}^2 = 12 , \text{cm}^2 \]
Next, when the triangle is dilated by a scale factor of 3, the new base and height are:
\[ \text{New base} = 8 , \text{cm} \times 3 = 24 , \text{cm} \] \[ \text{New height} = 3 , \text{cm} \times 3 = 9 , \text{cm} \]
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, let's compare the areas:
- Original triangle area: \(12 , \text{cm}^2\)
- Dilated triangle area: \(108 , \text{cm}^2\)
To find how many times larger the area of the dilated triangle is compared to 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 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.
4 answers