Question

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

Answer 1

Answer:

Given:

A circle with center O.

P

Tangents PA and PB drawn from external point P

B

To prove: Z APB + ZAOB = 180°

Proof:

Since PA is tangent,

OAL PA

(Tangent at any point of circle is perpendicular to the radius through point of contact)

:: LOAP = 90°

Since PB is tangent,

OB L PB

(Tangent at any point of circle is perpendicular to the radius through point of contact)

:. 2 OBP = 90°In quadrilateral UAPB

ZOAP + ZAPB + Z OBP + ZAOB = 360°

Putting values of angles

90° + LAPB + 90° + L AOB = 360°

180° + APB + ZAOB = 360°

ZAPB + ZAOB = 360° - 180°

ZAPB + ZAOB = 180°

(Angle sum property of quadrilateral)

Hence proved