Similar Triangles: Calculate the Ratio Between ABC (12,9,6) and KLT (4,3,2)

To determine if the triangles $\triangle ABC$ and $\triangle KLT$ are similar, we apply the Side-Side-Side (SSS) similarity criterion. This requires that the ratios of corresponding sides are equal.

We are given the side lengths: $BC = 12$, $AB = 9$, $CA = 6$ for $\triangle ABC$, and $LK = 2$, $KT = 3$, $LT = 4$ for $\triangle KLT$.

First, find the ratio for each pair of corresponding sides:

• Compare $BC$ and $LT$: $\frac{BC}{LT} = \frac{12}{4} = 3$
• Compare $CA$ and $LK$: $\frac{CA}{LK} = \frac{6}{2} = 3$
• Compare $AB$ and $KT$: $\frac{AB}{KT} = \frac{9}{3} = 3$

Since $\frac{BC}{LT} = \frac{CA}{LK} = \frac{AB}{KT} = 3$, all sides maintain a constant ratio. Hence, the triangles are similar.

The similarity ratio is $3$, indicating $\triangle ABC \sim \triangle KLT$ with a ratio of 3:1.

The correct choice, as given in the options, is:

^{Yes, similarity ratio:}
$\frac{BC}{LT}=\frac{CA}{LK}=\frac{AB}{KT}$