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Question
Through the vertex S of a parallelogram PQRS, a line is drawn to intersect the sides Qp and QR produced at M and N respectively. Prove that `"SP"/"PM" = "MQ"/"QN" = "MR"/"SR"`
Solution
In ΔPMS and ΔMQN
∠PMS = ∠NMQ ...(vertcally oppe angles)
∠SPM = ∠MQN ...(alternate angles, ssince PS || QN)
Therefore, ΔPMS ∼ ΔMQN
∴ `"SP"/"PM" = "MQ"/"QN"` ........(i)
In ΔPMS and ΔMSR
∠PMS = ∠MSR ...(alternate angles, since PM || SR)
SM = SM
Therefore, ΔPMS ∼ ΔMRS
∴ `"SP"/"PM" = "MR"/"SR"` ........(ii)
From (i) and (ii)
∴ `"SP"/"PM" = "MQ"/"QN" = "MR"/"SR"`.
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