** **In trapezium PQRS, side PQ || side SR, AR = 5AP, AS = 5AQ then prove that, SR = 5PQ

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#### Solution

Given:

side PQ || side SR

AR = 5AP,

AS = 5AQ

To prove: SR = 5PQ

Proof: In ∆APQ and ∆ARS

∠PAQ = ∠RAS (Vertically Opposite angles)

∠PQA = ∠RSA (Alternate angles, side PQ || side SR and QS is a transversal line)

By AA test of similarity

∆APQ ~ ∆ARS

\[\frac{PQ}{SR} = \frac{AP}{AR} \left( \text{ Corresponding sides are proportional } \right)\]

\[ \Rightarrow \frac{PQ}{SR} = \frac{1}{5} \left( AR = 5AP \right)\]

\[ \Rightarrow SR = 5PQ\]

\[ \Rightarrow \frac{PQ}{SR} = \frac{1}{5} \left( AR = 5AP \right)\]

\[ \Rightarrow SR = 5PQ\]

Hence proved.

Concept: Properties of Ratios of Areas of Two Triangles

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