Question

# Prove that the tangent at the extremities of any chord makes equal angles at the chord.

Solution

## Let $$AB$$ be a chord of a circle with centre $$O$$, and let $$AP$$ and $$BP$$ be the tangents at $$A$$ and $$B$$ respectively. Suppose the tangents meet at $$P$$. Join $$OP$$. Suppose $$OP$$ and meets $$AB$$ at $$C$$. We have to prove that $$\angle PAC=\angle PBC$$.In triangle $$PCA$$ and $$PCB$$, we have $$\Rightarrow$$$$PA=PB\quad$$          [$$\because\quad$$ tangents from an external point are equal]$$\Rightarrow$$$$\angle APC=\angle BPC\quad$$  [$$\because\quad PA$$ and $$PB$$ are equal inclined to $$OP$$].$$\Rightarrow$$$$PC=PC$$So, by $$SAS------$$criterion of congruence, we have     $$\triangle PAC\cong \triangle PBC$$$$\Rightarrow \angle PAC= \angle PBC$$ Maths

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