In the given figure, XY and X'Y' are two parallel tangents to a circle with centre O and another tangent AB with point of contact C intersects XY at A and X'Y' at B. Then ∠ AOB is _____.
In quadrilateral APQB, we have
∠APO=90∘ and ∠BQO=90∘
[∵ Tangent is perpendicular to the radius at the point of contact- By Theorem].
Now, ∠APO+∠BQO+∠QBC+∠PAC=360∘
⇒∠PAC+∠QBC=360∘−(∠APO+∠BQO)=180∘ . . . (i)
We have
∠CAO=12∠PAC and ∠CBO=12∠QBC [∵ Tangents from an external point are equally inclined to the line segment joining the centre to that point- By Theorem].
On adding ∠CAO and ∠CBO and using (i) we get,
∠CAO+∠CBO=12(∠PAC+∠QBC)=12×180∘=90∘ . . .(ii)
In Δ AOB, we have
∠CAO+∠AOB+∠CBO=180∘
⇒ ∠AOB=180∘−(∠CAO+∠CBO)=90∘ [using (ii)].
Hence, ∠AOB=90∘