Question

In ∆PQR ,PS is the bisector of angle P meets QR in S . Prove that area of triangle (PQS) / area of triangle (PRS ) =PQ×PQ/ PR×PR​

Answer 1

Answer:

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Step-by-step explanation:

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

SR

QS

=

PT

PQ

......(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)

SR

QS

=

PT

PQ

Hence proved.