To determine the third side length of the corresponding triangle using the SSS (Side-Side-Side) similarity criterion, we first need to find the ratio of similarity between the triangles.
The first triangle has sides of length 3, 4, and 6. The second triangle has two corresponding side lengths of 9 and 12.
Let's identify the ratio for the corresponding sides:
- The side length 3 corresponds to the side length 9.
- The side length 4 corresponds to the side length 12.
Now, we can calculate the ratio of their lengths:
For the side lengths 3 and 9: \[ \text{Ratio} = \frac{9}{3} = 3 \]
For the side lengths 4 and 12: \[ \text{Ratio} = \frac{12}{4} = 3 \]
Both ratios are equal, confirming that the two triangles are similar with a scaling factor of 3.
Now, we need to find the length of the corresponding third side. The third side of the first triangle is 6.
To find the corresponding third side length (let's call it \( x \)) of the larger triangle, we use the ratio: \[ x = 6 \times \text{Ratio} = 6 \times 3 = 18 \]
Therefore, the third side length of the corresponding triangle is:
\[ \boxed{18} \]
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