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:
\(AP = 1 \, \text{cm}\), \(BP = 2 \, \text{cm}\), \(AQ = 1.5 \, \text{cm}\), \(AC = 4.5 \, \text{cm}\), \(BC = 3.6 \, \text{cm}\)

To prove:
\[ \triangle APQ \sim \triangle ABC \]

Proof:
- Since \(P\) lies on \(AB\), we have \(AB = AP + BP = 1 + 2 = 3 \, \text{cm}\).
- We will show the corresponding sides of triangles \(APQ\) and \(ABC\) are proportional.

Calculate ratios:
\[ \frac{AP}{AB} = \frac{1}{3} \] \[ \frac{AQ}{AC} = \frac{1.5}{4.5} = \frac{1}{3} \]
Since \(PQ \parallel BC\), angles \(\angle APQ = \angle ABC\) and \(\angle AQP = \angle ACB\).
Therefore, by the Side-Angle-Side (SAS) similarity criterion,
\[ \triangle APQ \sim \triangle ABC \]

Find length of \(PQ\):
Since corresponding sides of similar triangles are proportional:
\[ \frac{PQ}{BC} = \frac{AP}{AB} = \frac{1}{3} \] \[ PQ = \frac{1}{3} \times BC = \frac{1}{3} \times 3.6 = 1.2 \, \text{cm} \]

Final Answer:
\(\triangle APQ \sim \triangle ABC\)
Length of \(PQ = 1.2 \, \text{cm}\)