Question

If $\triangle ABC \sim \triangle DEF$ and $AB = 4$ cm, $DE = 6$ cm, $EF = 9$ cm, and $FD = 12$ cm, find the perimeter of $\triangle ABC$.

Answer 1

Using the similarity ratio:
\[\frac{AB}{DE} = \frac{BC}{EF} = \frac{AC}{FD} = \frac{4}{6} = \frac{2}{3}\]
Let the sides of $\triangle ABC$ be $4 \, \text{cm}$, $x$, $y$, corresponding to $\triangle DEF$ sides $6 \, \text{cm}$, $9 \, \text{cm}$, $12 \, \text{cm}$.
Solve:
\[x = \frac{2}{3} \cdot 9 = 6 \, \text{cm}, \quad y = \frac{2}{3} \cdot 12 = 8 \, \text{cm}.\]
Perimeter of $\triangle ABC = 4 + 6 + 8 = 18 \, cm$.
Correct Answer: 18 cm