Question

In the adjoining figure, \( AP = 1 \, \text{cm}, \ BP = 2 \, \text{cm}, \ AQ = 1.5 \, \text{cm}, \ AC = 4.5 \, \text{cm} \) Prove that \( \triangle APQ \sim \triangle ABC \).
Hence, find the length of \( PQ \), if \( BC = 3.6 \, \text{cm} \).

Hint:

Use corresponding sides and angles to prove similarity, then apply ratios to find unknown lengths.

Answer 1

Given: \[ AP = 1 \text{ cm}, \quad AQ = 1.5 \text{ cm} \] \[ AB = AP + PB = 1 + 2 = 3 \text{ cm}, \quad AC = 4.5 \text{ cm} \] Compare: \[ \frac{AP}{AB} = \frac{1}{3}, \quad \frac{AQ}{AC} = \frac{1.5}{4.5} = \frac{1}{3} \] \[ \angle A \ \text{common} \Rightarrow \triangle APQ \sim \triangle ABC \quad \text{(By SAS criterion)} \] Now use similarity: \[ \frac{PQ}{BC} = \frac{AP}{AB} = \frac{1}{3} \Rightarrow PQ = \frac{1}{3} \times 3.6 = 1.2 \, \text{cm} \]