Question

Intercept of a tangent between two parallel tangents to a circle subtends a right angle at the centre.

• A. True
• B. False

Answer 1

The correct option is A True

Given:$XYand{X}^{\prime }{Y}^{\prime }$ are two parallel tangents to the circle wth centre O and AB is the tangent at the point C

which intersects $XY$ at $A$ and $X^{\prime }{Y}^{\prime }$ at $B$.

In $△OPA$ and $△OCA$

$OP=OC$ (radii)

$AP=AC$ (Tangents from point $A$)

$AO=AO$ (Common )

$△OPA\cong △OCA$ (By $SSS$ criterion)

$\therefore \angle POA=\angle COA-\left(1\right)$ (By C.P.C.T)

$OQ=OC$ (radii)

$BQ=BC$ (Tangents from point A)

$BO=BO$ (Common )

$△OQB\cong △OCB$

$\angle QOB=\angle COB-\left(2\right)$

$POQ$ is a diameter of the circle.

Hence, it is a straight line.

$\angle POA+\angle COA+\angle COB+\angle QOB=180$

From equations (1) and (2), it can be observed that

$2\angle COA+2\angle COB=180$

$⟹\angle COA+\angle COB=90$

$⟹\angle AOB={90}^{\circ }$