Given: In trapezium PQRS, side PQ ∥ side SR, AR = 5AP, AS = 5AQ then prove that, SR = 5PQ.
Figure: The diagonals PR and QS of trapezium PQRS intersect each other at point A.
Answer: \(\triangle APQ \sim \triangle ARS\) (SAS test), so \(\frac{PQ}{RS} = \frac{1}{5}\), i.e. SR = 5PQ.
From the given data,
\[AR = 5AP \Rightarrow \frac{AP}{AR} = \frac{1}{5}\] \[AS = 5AQ \Rightarrow \frac{AQ}{AS} = \frac{1}{5}\]In \(\triangle APQ\) and \(\triangle ARS\):
\(\therefore \triangle APQ \sim \triangle ARS\) (SAS test of similarity).
Corresponding sides of similar triangles are in proportion, so
\[\frac{PQ}{RS} = \frac{AP}{AR} = \frac{1}{5}\]\(\therefore\) SR = 5PQ. Hence proved.
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