To determine the third side length of the second triangle that corresponds to the first triangle with side lengths of 3, 4, and 6, we can use the principle of similarity in triangles based on the SSS (Side-Side-Side) criterion.
Given:
- First triangle side lengths: \( a_1 = 3 \), \( b_1 = 4 \), \( c_1 = 6 \)
- Corresponding side lengths in the second triangle: \( a_2 = 9 \), \( b_2 = 12 \)
We need to calculate the unknown side length \( c_2 \) of the second triangle that corresponds to \( c_1 = 6 \).
First, we find the ratio of similarity between the corresponding sides of the triangles. We can use any of the known sides:
For \( a_1 \) and \( a_2 \): \[ k = \frac{a_2}{a_1} = \frac{9}{3} = 3 \]
For \( b_1 \) and \( b_2 \): \[ k = \frac{b_2}{b_1} = \frac{12}{4} = 3 \]
Since the ratio of the corresponding sides is consistent, we can conclude that the triangles are indeed similar with a scale factor \( k = 3 \).
Now, we can find the corresponding side \( c_2 \) of the second triangle by using the scale factor: \[ c_2 = k \cdot c_1 = 3 \cdot 6 = 18 \]
Thus, the third side length of the corresponding triangle is \[ \boxed{18}. \]
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