First, we identify the sides of the original triangle, which measures 3, 4, and 5. The longest side is 5.
We are tasked with finding the lengths of the sides of a similar triangle where the longest side measures 20. Since the triangles are similar, the sides are proportional.
To find the scale factor, we divide the corresponding sides of the triangles:
\[ \text{Scale Factor} = \frac{\text{Longest side of the new triangle}}{\text{Longest side of the original triangle}} = \frac{20}{5} = 4 \]
Now, we will multiply each side of the original triangle by the scale factor (4):
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The shortest side (3): \[ 3 \times 4 = 12 \]
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The next side (4): \[ 4 \times 4 = 16 \]
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The longest side (5): \[ 5 \times 4 = 20 \]
Since we are looking for the length of the shortest side of the similar triangle, that is:
\[ \boxed{12} \]