In figure 4, PQ is a chord of length 16cm, of a circle of radius 10 cm. The tangents at P and Q intersect at a point T. Find the length of TP.

In the given figure, PQ = 16 cm

PO = 10 cm (radius)

⸫ TP = TQ [Tangents drawn from an external point to the same circle are equal]

ΔTPQ is an isosceles triangle

⸫ OT is the perpendicular bisector of PQ

⸫ PR = 8 cm

In ΔTPR,

TP² = TR² + PR² [Pythagoras theorem]

TP² = TR² + 8² …. (i)

∠TPO = 90° [A tangent is perpendicular to the radius at the point of tangency]

In ΔTPO,

TO² = TP² + PO² [Pythagoras theorem]

TO² = TP² + 10²

TP² = TO² – 100 …. (ii)

In ΔOPR,

OR² = OP² – PR² [Pythagoras theorem]

OR² = 10² – 8²

⸫ OR = √36 = 6 cm

Equation eq (i) and eq (ii),

TR² + 64 = TO² – 100

TR² + 64 = (TR + OR)² – 100 [From the figure, (TR + OR) = TO]

TR² + 64 = TR² + 36 + 12TR – 100

12TR = 128

⸫ TR =

Substituting the value of TR in eq (i), we get

TP² =

TP² =

⸫ TP =

⸫ TP = 13.33 cm