To find the perimeter of the dilated triangle using a scale factor of \( \frac{1}{3} \), you first need to dilate each side length of the original triangle.
The original side lengths of the triangle are:
- 11 inches
- 9 inches
- 28 inches
Now, we apply the scale factor of \( \frac{1}{3} \) to each side length:
-
Dilation of the first side: \[ 11 \times \frac{1}{3} = \frac{11}{3} \text{ inches} \]
-
Dilation of the second side: \[ 9 \times \frac{1}{3} = 3 \text{ inches} \]
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Dilation of the third side: \[ 28 \times \frac{1}{3} = \frac{28}{3} \text{ inches} \]
Now, we calculate the perimeter of the dilated triangle by summing the dilated side lengths:
\[ \text{Perimeter} = \frac{11}{3} + 3 + \frac{28}{3} \]
Convert 3 into a fraction with the same denominator:
\[ 3 = \frac{9}{3} \]
Now, adding the fractions together:
\[ \text{Perimeter} = \frac{11}{3} + \frac{9}{3} + \frac{28}{3} = \frac{11 + 9 + 28}{3} = \frac{48}{3} = 16 \text{ inches} \]
Thus, the perimeter of the dilated triangle is \( \boxed{16} \) inches.