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.
To prove: PQ and OT are the right bisectors.
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
∠OPT = ∠OQT = 90º ( Tangent to a circle at a point is perpendicular to the radius through the point of contact)
∴ 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
Thus it is a square,
⇒ The diagnols bisect at 90º.
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