question_answer

Prove that tangents drawn at the ends of a diameter of a circle are parallel to each other.

Answer:

Given, PQ is a diameter of a circle with centre O. The lines AB and CD are tangents at P and Q respectively.

To Prove: \[AB\parallel CD\]

Proof: AB is a tangent to the circle at P and OP is the radius through the point of contact

\[\therefore \]                              \[\angle OPA=90{}^\circ \]

Similarly CD is a tangent to circle at Q and OQ is radius through the point of contact

\[\therefore \]                              \[\angle OQD=90{}^\circ \]

\[\Rightarrow \]                           \[\angle OPA=\angle OQD\]

But both form pair of alternate angles

\[\therefore \]                              \[AB\parallel CD\]                       Hence Proved.