In triangle PQR ,PS id the busectorbof angleQPR.show that

QS/SR = PQ/PR.

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Solution

Given: A triangle PQR in which PS is the internal bisector of angle P.

R.T.P: QS/SR=PQ/PR

Construction: Draw RE parallel PS to meet QP produced in E.

Proof:
Here we have,
ER || PS
Then,
angle 2 = angle 3.........1 (alternative angles)
and,
angle 1 = angle 4....... 2 (corresponding angles)
Given that PS in an angular bisector
then,we have
angle 1 = angle 2........3
By eq1 , eq2 , eq3 we get,
angle 3 = angle 4
Then,
PR=PE...........4 (sides opposite to equal angles)
In triangle QRE we have,
RE || PS
By basic proportionality theorem we get,
QS/SR = QP/PE

QS/SR = QP/PR

QS/SR = PQ/PR

Hence Proved.

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