The diagonals of trapezium PQRS where PQ||RS intersect at point O are given in the figure. Length of OP and OQ is 5 and 4 units, respectively. Find the length of OS, if the length of OR is 2 units.

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Solution

Given, PQ||RS, OP = 5, OQ = 4 and OR = 2

Let us take $△ O R Q$ and $△ O P S$ .

$∠ Q O R = ∠ S O P$ (opposite angles)
$∠ R Q O = ∠ P S O$ (QR || PS)
$∠ Q R O = ∠ S P O$ (QR || PS)

Hence, by AAA similarity,
$△ O R Q \sim △ O P S$

$⟹ \frac{O R}{O P} = \frac{O S}{O Q}$

Given,
$O P = 5$
$O Q = 4$
$O R = 2$

$⟹ \frac{2}{5} = \frac{O S}{4}$

$⟹ O S = 1.6$ units

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