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

Related Questions

In an trapezium ABCD, it is given that AB ∥ CD and AB = 2CD. Its diagonals AC and BD

In an trapezium ABCD, it is given that AB ∥ CD and AB = 2CD. Its diagonals AC and BD intersect at the point O such that ar(∆AOB) = 84 cm2. Find ar(∆COD)

Find the length of the altitude of an equilateral triangle of side 2a cm.

Find the length of the altitude of an equilateral triangle of side 2a cm.

A ladder 10 m long reaches the window of a house 8 m above the ground.

A ladder 10 m long reaches the window of a house 8 m above the ground. Find the distance of the foot of the ladder from the base of the wall.