Question

Prove that the angle between the two tangents drawn from an external point to a circle is supplementary to the angles subtended by the line segment joining the points of contact at centre.​

Answer 1

$\huge\mathfrak{Answer:}$

Given:

• We have been given a circle with centre O and two tangents are drawn from the external point P to the circle.

To Prove:

• We need to prove that the angle between the two tangents drawn from an external point to a circle is supplementary to the angles subtended by the line segment joining the points of contact at centre.

Solution:

Let PA and PB be two tangents drawn from an external point P to a circle with centre O.

Now, in right ΔOAP and right ΔOBP, We have

PA = PB [Tangents drawn from an external point]

OA = OB [Radii of the same circle]

OP = OP [Common]

∴ ΔOAP ≅ ΔOBP by S.S.S congruency criteria.

=> ∠OPA = ∠OPB [C.P.C.T]

and ∠AOP = ∠BOP [C.P.C.T]

=> ∠APB = 2∠OPA and ∠AOB = 2∠AOP

But, ∠AOP = 90° - ∠OPA

=> 2∠AOP = 180° - 2∠OPA

=> ∠AOB = 180° - ∠APB

=> ∠AOB + ∠APB = 180°

Hence proved!!