Step 1: Problem Statement:
Given that \( \triangle ABC \sim \triangle DEF \), meaning they are similar triangles. The sides of similar triangles are proportional. The following side lengths are provided:
- \( AB = 4 \, \text{cm} \)
- \( DE = 6 \, \text{cm} \)
- \( EF = 9 \, \text{cm} \)
- \( FD = 12 \, \text{cm} \)
The objective is to determine the perimeter of \( \triangle ABC \).
Step 2: Apply Similarity Properties:
Due to triangle similarity, corresponding sides are proportional. This yields the following ratios:
\[\frac{AB}{DE} = \frac{BC}{EF} = \frac{CA}{FD}\]
Substituting the given values:
\[\frac{4}{6} = \frac{BC}{9} = \frac{CA}{12}\]
Simplifying the initial ratio:
\[\frac{2}{3} = \frac{BC}{9} = \frac{CA}{12}\]
Step 3: Calculate Missing Side Lengths:
The unknown sides of \( \triangle ABC \) are solved using the established proportions.
For side \( BC \):
\[\frac{BC}{9} = \frac{2}{3}\]
Solving for \( BC \) via cross-multiplication:
\[BC = 9 \times \frac{2}{3} = 6 \, \text{cm}\]
For side \( CA \):
\[\frac{CA}{12} = \frac{2}{3}\]
Solving for \( CA \) via cross-multiplication:
\[CA = 12 \times \frac{2}{3} = 8 \, \text{cm}\]
Step 4: Compute Perimeter of \( \triangle ABC \):
The perimeter is the sum of the lengths of \( \triangle ABC \)'s sides:
\[\text{Perimeter of } \triangle ABC = AB + BC + CA = 4 + 6 + 8 = 18 \, \text{cm}\]
Conclusion:
The perimeter of \( \triangle ABC \) is \( 18 \, \text{cm} \).