Question

Two tangents TP and TQ are drawn to a circle with centre O from an external point T. Prove that
PTO = 2 OPQ​

Answer 1

Solution:-

We know that the lengths drawn from an external point to a Circle are equal.

• TP = TQ

Now, In ∆ TPQ

=> TP = TQ

=> Angle TPQ = TQP ____________eqn(1)

(In ∆ opposite sides have equal Angles opposite to them)

Now , AngleTPQ + TQP + PTQ = 180° ( Angle sum property of Triangle)

2TPQ + PTQ ( By eqn (1)

PTQ = 180 - 2TPQ __________eq(2)

A tangent to a circle is perpendicular to the radius through the point of contact.

=> OP | PT

OPT = 90°

=> angle TPQ+ OPQ = 90°

=> OPQ = 90 - TPQ

=> 2OPQ = 2(90 - TPQ) = 180 - 2TPQ____q(2)

From eqn(1) & (2) we get

PTQ = 2OPQ Hence Proved!

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