To find the area of the original triangle, we can use the formula for the area of a triangle:
\[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \]
For the original triangle, the base is 8 cm and the height is 3 cm:
\[ \text{Area}_{\text{original}} = \frac{1}{2} \times 8 , \text{cm} \times 3 , \text{cm} = \frac{1}{2} \times 24 , \text{cm}^2 = 12 , \text{cm}^2 \]
Next, we will find the dimensions of the dilated triangle. When a triangle is dilated by a scale factor of 3, both the base and the height are multiplied by that scale factor.
The new dimensions will be:
- New base = \( 8 , \text{cm} \times 3 = 24 , \text{cm} \)
- New height = \( 3 , \text{cm} \times 3 = 9 , \text{cm} \)
Now we can calculate the area of the dilated triangle:
\[ \text{Area}_{\text{dilated}} = \frac{1}{2} \times \text{new base} \times \text{new height} = \frac{1}{2} \times 24 , \text{cm} \times 9 , \text{cm} \]
Calculating that:
\[ \text{Area}_{\text{dilated}} = \frac{1}{2} \times 216 , \text{cm}^2 = 108 , \text{cm}^2 \]
Now let's compare the areas of the original triangle and the dilated triangle:
- Area of the original triangle: \( 12 , \text{cm}^2 \)
- Area of the dilated triangle: \( 108 , \text{cm}^2 \)
We can observe that when a triangle is dilated by a scale factor of \( k \), the area of the triangle is multiplied by a factor of \( k^2 \). In this case, since the scale factor is 3:
\[ k^2 = 3^2 = 9 \]
Thus, the area of the dilated triangle should be \( 9 \times \text{Area}_{\text{original}} \):
\[ 9 \times 12 , \text{cm}^2 = 108 , \text{cm}^2 \]
This confirms our calculation.
In summary:
- Area of the original triangle: \( 12 , \text{cm}^2 \)
- Area of the dilated triangle: \( 108 , \text{cm}^2 \)
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