PQ and RS are two parallel tangents to a circle with centre'O',another tangent AB with the point of contact 'C' - Maths - Circles

Question

PQ and RS are two parallel tangents to a circle with
centre'O',another tangent AB with the point of contact 'C'
intersecting PQ and RS at A and B respectively Prove that angle
AOB=90 degree

Gursheen kaur. · Accepted Answer

If your question is - PQ and RS are two parallel tangents to a circle with centre 'O ',another tangent XY with the point of contact 'C ' intersecting PQ and RS at A and B respectively Prove that angle AOB=90 degree Then the solution is as follows:

Construction: Join O to A, O to B and O to C Since tangent is perpendicular to the radius through the point of contact âˆ´âˆ x=âˆ y=90Â° Now, In right Î”OMA and Î”OCA Hyp OA = Hyp OA [common] OM= OC [radii of the same circle] âˆ OMA=âˆ OCA [each equal to 90Â°] â‡’Î”OMA $≅$ Î”OCA[By RHS congruence criterion] â‡’âˆ 1=âˆ 2 [By Cpct] Similarly, âˆ 3=âˆ 4 Now, âˆ 1+âˆ 2+âˆ 3+âˆ 4=180Â° [ $∵$ Linear pair] â‡’2(âˆ 2+âˆ 3)=180Â° â‡’âˆ 2+âˆ 3=90Â° âˆ´âˆ AOB=90Â° Hence Proved

PQ and RS are two parallel tangents to a circle with centre'O',another tangent AB with the point of contact 'C' intersecting PQ and RS at A and B respectively.Prove that angle AOB=90 degree.