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 \(PO\perp OQ,\)
⇒ ∠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
Hence PT=TQ=OQ= OP
Thus it is a square,
⇒ The diagnols bisect at 90º .