Step 1: Information Provided:
Triangles $ \triangle ABC $ and $ \triangle DEF $ are similar, implying proportional corresponding sides.
Given side lengths:
- $ AB = 4 \, \text{cm}$
- $ DE = 6 \, \text{cm}$
- $ EF = 9 \, \text{cm}$
- $ FD = 12 \, \text{cm}$
Objective: Find the perimeter of $ \triangle ABC $.
Step 2: Proportion Setup:
Similarity dictates the proportion of corresponding sides:
\[\frac{AB}{DE} = \frac{BC}{EF} = \frac{CA}{FD}\]Substituting values:
\[\frac{4}{6} = \frac{BC}{9} = \frac{CA}{12}\]Simplified ratio:
\[\frac{2}{3} = \frac{BC}{9} = \frac{CA}{12}\]
Step 3: Determining Side Lengths $BC$ and $CA$:
For $BC$: $\frac{BC}{9} = \frac{2}{3}$\[BC = \frac{2}{3} \times 9 = 6 \, \text{cm}\]For $CA$: $\frac{CA}{12} = \frac{2}{3}$\[CA = \frac{2}{3} \times 12 = 8 \, \text{cm}\]
Step 4: Perimeter Calculation for $ \triangle ABC $:
Perimeter = Sum of sides of $ \triangle ABC $:
\[\text{Perimeter of } \triangle ABC = AB + BC + CA = 4 + 6 + 8 = 18 \, \text{cm}\]
Conclusion:
The perimeter of $ \triangle ABC $ is 18 cm.