To determine the length of the third side of the second triangle using the SSS (Side-Side-Side) criterion and the properties of similar triangles, we start by setting up a proportion based on the given side lengths of the two triangles.
The first triangle has side lengths of \(1\), \(4\), and \(6\). Let's denote these as:
- \(a_1 = 1\)
- \(b_1 = 4\)
- \(c_1 = 6\)
The second triangle has corresponding sides \(3\) and \(12\). Denote these known lengths as:
- \(a_2 = 3\)
- \(b_2 = 12\)
- Let the length of the unknown third side be \(c_2\).
Since the triangles are similar, the ratios of their corresponding sides must be equal. Therefore, we have:
\[ \frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2} \]
Calculating the ratios gives:
For the first known side: \[ \frac{a_1}{a_2} = \frac{1}{3} \]
For the second known side: \[ \frac{b_1}{b_2} = \frac{4}{12} = \frac{1}{3} \]
Since both ratios are equal, we can use them to find \(c_2\).
From the ratio of the corresponding sides: \[ \frac{c_1}{c_2} = \frac{1}{3} \]
Cross-multiplying gives:
\[ c_1 = \frac{1}{3} c_2 \]
Substituting \(c_1 = 6\) into the equation:
\[ 6 = \frac{1}{3} c_2 \]
Multiplying both sides by 3 to isolate \(c_2\):
\[ c_2 = 6 \times 3 = 18 \]
Thus, the length of the third side of the corresponding triangle is \( \boxed{18} \).
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