In Figure, PS is the bisector of ∠ QPR of ∆ PQR. Prove that $$\dfrac{QS}{PQ}=\dfrac{SR}{PR}$$

Asked by Pragya Singh | 1 year ago |  68

##### Solution :-

Let us draw a line segment RT parallel to SP which intersects extended line segment QP at point T.

Given, PS is the angle bisector of ∠QPR. Therefore,

∠QPS = ∠SPR………(i)

As per the constructed figure,

∠SPR=∠PRT(Since, PS||TR)……………(ii)

∠QPS = ∠QRT(Since, PS||TR) …………..(iii)

From the above equations, we get,

∠PRT=∠QTR

Therefore,

PT=PR

In △QTR, by basic proportionality theorem,

$$\dfrac{QS}{SR}=\dfrac{QP}{PT}$$

Since, PT=TR

Therefore,

$$\dfrac{QS}{SR}=\dfrac{PQ}{PR}$$

Hence, proved.

Answered by Abhisek | 1 year ago

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