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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
answered-by-expert 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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