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

Step 1: Understand the given information:
We are given that the triangles $ \triangle ABC $ and $ \triangle DEF $ are similar, which means that the corresponding sides of the triangles are proportional.
We are also given the following side lengths:
- $ AB = 4 \, \text{cm}$
- $ DE = 6 \, \text{cm}$
- $ EF = 9 \, \text{cm}$
- $ FD = 12 \, \text{cm}$
We need to find the perimeter of $ \triangle ABC $.

Step 2: Set up the proportion:
Since the triangles are similar, the corresponding sides are proportional. Therefore, we can write the following proportion based on the sides of the two triangles:
\[ \frac{AB}{DE} = \frac{BC}{EF} = \frac{CA}{FD} \] Substitute the given values:
\[ \frac{4}{6} = \frac{BC}{9} = \frac{CA}{12} \] Simplify the ratios:
\[ \frac{2}{3} = \frac{BC}{9} = \frac{CA}{12} \]

Step 3: Find the lengths of $BC$ and $CA$:
From the proportion $\frac{BC}{9} = \frac{2}{3}$, we can solve for $BC$:
\[ BC = \frac{2}{3} \times 9 = 6 \, \text{cm} \] Next, from the proportion $\frac{CA}{12} = \frac{2}{3}$, we can solve for $CA$:
\[ CA = \frac{2}{3} \times 12 = 8 \, \text{cm} \]

Step 4: Calculate the perimeter of $ \triangle ABC $:
The perimeter of $ \triangle ABC $ is the sum of its sides:
\[ \text{Perimeter of } \triangle ABC = AB + BC + CA = 4 + 6 + 8 = 18 \, \text{cm} \]

Conclusion:
The perimeter of $ \triangle ABC $ is 18 cm.