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Question
In the given figure, PS is the bisector of ∠QPR of ΔPQR. Prove that `(QS)/(SR) = (PQ)/(PR)`
Solution
Let us draw a line segment RT parallel to SP which intersects extended line segment QP at point T.
Given that, PS is the angle bisector of ∠QPR.
∠QPS = ∠SPR … (1)
By construction,
∠SPR = ∠PRT (As PS || TR) … (2)
∠QPS = ∠QTR (As PS || TR) … (3)
Using these equations, we obtain
∠PRT = ∠QTR
∴ PT = PR
By construction,
PS || TR
By using basic proportionality theorem for ΔQTR,
`(QS)/(SR) = (QP)/(PT)`
`=>(QS)/(SR) = (PQ)/(PR) (∵ PT = PR)`
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