Q. 244.4( 9 Votes )

In Fig. 10.62, . The tangents to the circle at P and Q intersect at a point T. Prove that PQ and OT are right bisectors of each other.

Answer :

To prove: PQ and OT are the right bisectors.

Proof:

To prove PQ and OT are the right bisectors,

We need to prove ∠PRT= ∠TRQ=∠QRO=∠ORP = 90º

As it is given that ,

⇒ ∠POQ = 90º

In Δ POT and  Δ OQT

OP=OQ (Radius)

∠OPT = ∠OQT = 90º ( Tangent to a circle at a point is perpendicular to the radius through the point of contact)

OT=OT (common)

∴ Δ POT ≅ Δ OQT

Thus PT=OQ ( BY C.P.C.T)  ..... (1)

Now in Δ PRT and Δ ORQ

∠TPR = ∠OQR ( alternate angles)

∠PTO = ∠TOQ (alternate angles)

PT=OQ ( from (1) )

∴ Δ PRT ≅ Δ ORQ

Thus TQ = OP ( By C.P.C.T 0

Hence PT=TQ=OQ=OP

Thus it is a square,

⇒ The diagnols bisect at  90º.

Hence proved





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