CBSE Class 10 Answered
Prove that the angle between two tangents drawn from an external point to a circle is supplementary to the angle subtended by the line segments joining the points of contact at the centre.
Asked by Topperlearning User | 08 Dec, 2013, 04:36: AM
Expert Answer
Let PA and PB be two tangents drawn from an external point P to a circle with centre O. We have to prove that angles AOB and APB are supplementary i.e. AOB+APB=1800.
In right triangles OAP and OBP, we have
PA=PB (tangents drawn from an external point to a circle are equal in length)
OA=OB (radii of same circle)
OP=OP (common)
∆OAP ∆OBP (by SSS)
OPA=OPB and AOP=BOP (by cpct)
APB = 2OPA and AOB =2AOP
But, AOP=90-OPA
2AOP =180-2OPA
AOB =180-APB
AOB+APB=180
Answered by | 08 Dec, 2013, 06:36: AM
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