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Question

In figure, PS is the bisector of QPR of PQR. Prove that
QSSR=PQPR
1193173_16b8204b027e40e2b9f567ac00f8b64b.PNG

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Solution

In QRT
RT||SP [By construction]
And PS intersects QT and QR at two distinct points P and Q
Therefore,
QT and QR will be divided in the same ratio
QSSR=PQPT......(1)
[Basic proportionality theorem : If a line is drawn parallel to one side of a triangle, intersecting other two sides at distinct points, then other two sides are divided in the same ratio.]
Now,
RT||SP
and PR is the transversal
Therefore,
SPR=PRT....(2) [Alternate interior angles]
and
QPS=PTR....(3) [Corresponding angles]
Also, given the
PS is the bisector of QPR
QPR=SPR
From (2) and (3)
PTR=PRT
Therefore, PT=PR [Sides opposite to equal angles of a trinagle are equal]
Putting PT=PR in (1)
QSSR=PQPT
Hence proved.

1090960_1193173_ans_be15cd9499794674814ea207c6f19259.png

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