To find the perimeter of the dilated triangle, you first need to apply the scale factor of \( \frac{1}{3} \) to each of the original side lengths of the triangle.
The side lengths of the original triangle are:
- 11 inches
- 9 inches
- 28 inches
Now, we can calculate the side lengths of the dilated triangle by multiplying each original side length by the scale factor \( \frac{1}{3} \):
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For the side length of 11 inches: \[ 11 \times \frac{1}{3} = \frac{11}{3} \text{ inches} \]
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For the side length of 9 inches: \[ 9 \times \frac{1}{3} = 3 \text{ inches} \]
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For the side length of 28 inches: \[ 28 \times \frac{1}{3} = \frac{28}{3} \text{ inches} \]
Now we have the new side lengths:
- \( \frac{11}{3} \) inches
- \( 3 \) inches
- \( \frac{28}{3} \) inches
Next, let's find the perimeter of the dilated triangle by summing the new side lengths:
\[ \text{Perimeter} = \frac{11}{3} + 3 + \frac{28}{3} \]
First, convert \( 3 \) into a fraction with a common denominator of 3: \[ 3 = \frac{9}{3} \]
Now add them 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.