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QExample 3 : PQ is a chord of length 8 cm of a circle of radius 5 cm. The tangents at P and Q intersect at a point T (see Fig. 10.10). Find the length TP.

  Solution : Join OT. Let it intersect PQ at the point R. Then $∆$ TPQ is isosceles and TO is the angle bisector of $\angle$PTQ . So, OT $\perp$PQ and therefore, OT bisects PQ which gives PR = RQ = 4 cm. 

  Also, PR = $\sqrt{{\mathrm{OP}}^2 -{\mathrm{PR}}^2} =\sqrt{5^2 -4^2} \mathrm{cm} =3\mathrm{cm}$