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Question

In Fig., PS is the bisector of QPR of ΔPQR.
Prove that QSSR=PQPR.


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Solution

Draw a line parallel to PS through R
It intersects QP produced at T.

By using angle bisector properties
In Δ QPR
QPS=SPR (PS is bisector of QPR) .....(1)

By using alternate interior angle property For parallel lines PS and RT and transversal PR
PRT=SPR (Alternate interior angles) ..........(2)

By using corresponding angle property
For parallel lines PS and RT and transversal QT
QPS=PTR (Corresponding angles) ........(3)
From (1), (2) and (3)
PTR=PRT

Sides opposite to the equal angles are equal.
PR=PT........(4)

Apply BPT theorem in Δ QTR
In ΔQTR
PS||RT
QSSR=QPPT
From equation (4)
QSSR=QPPR
Hence, proved.

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