PQRS is a trapezium with PQ y SR. If M and N are two points on the non-parallel sides PS and QR respectively, such that MN is parallel to PQ, then show that

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Solution

Q.
PQRS is a trapezium with PQ y SR. If M and N are two points on the non-parallel sides PS and QR respectively, such that MN is parallel to PQ, then show that $\frac{P M}{M S} = \frac{Q N}{N R} .$

Ans.
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PQ $∥$ SR and MN $∥$ PQ ⇒ MN $∥$ SR

In $Δ$ PSR, by Basic proportionality theorem, we have

$\frac{P M}{M S} = \frac{P O}{O R}$ ...(i)

Similarly, In $Δ$ PQR,

$\frac{P O}{O R} = \frac{Q N}{N R}$ ...(ii)

From (i) and (ii), $\frac{P M}{M S} = \frac{Q N}{N R}$

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