To determine the third side length of the corresponding triangle using the side lengths of the original triangle (3, 4, and 6), we can use the SSS (Side-Side-Side) criterion of similarity.
The first triangle has sides of lengths 3, 4, and 6. The other triangle has sides 9 and 12, which correspond to the lengths of 3 and 4, respectively.
To find the scaling factor, we can calculate:
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The scaling factor for side 3 to side 9 is: \[ \frac{9}{3} = 3 \]
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The scaling factor for side 4 to side 12 is: \[ \frac{12}{4} = 3 \]
Both of these side lengths correspond with a scaling factor of 3.
Now, we apply this scaling factor to the third side of the first triangle, which is 6: \[ 6 \times 3 = 18 \]
Thus, the third side of the corresponding triangle is 18.
Therefore, the answer is:
18.