To find the length of the third side of the second triangle that corresponds to the given triangle with side lengths 1, 4, and 6, we can use the SSS (Side-Side-Side) similarity criterion for triangles.
We know the side lengths of the first triangle:
- \( a_1 = 1 \)
- \( b_1 = 4 \)
- \( c_1 = 6 \)
And the lengths of the corresponding sides of the second triangle:
- \( a_2 = 3 \)
- \( b_2 = 12 \)
We denote \( c_2 \) as the unknown third side of the second triangle.
Since the two triangles are similar, the ratios of their corresponding side lengths must be equal. We can find the ratio \( k \) using the corresponding sides we have:
\[ \frac{a_2}{a_1} = \frac{3}{1} = 3 \]
Next, we can use this ratio to find \( c_2 \) by applying the same ratio to the third sides:
\[ \frac{c_2}{c_1} = k \] where \( c_1 \) is the length of the third side of the first triangle, which is 6.
Thus, we have: \[ c_2 = k \cdot c_1 = 3 \cdot 6 = 18 \]
Therefore, the length of the third side of the corresponding triangle is
\[ \boxed{18}. \]
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