Question

in triangle ABC, AP perpendicular BC, CQ perpendicular AB and BR perpendicular AC, prove that angle OPQ=angle OPR

Answer 1

Answer:

As the two right triangles ABR and APB are on the same side of AB and which is also the hypotenuse of both , the circle drawn taking AB as diameter will pass through R and P. So ABPR is a cyclic quadrilateral.

Hence ∠ RPC =∠BAR

Similarly ACPQ is also cyclic quadrilateral

So   ∠ BPQ =∠QAC

But ∠BAR =∠QAC

So ∠ RPC=∠ BPQ

=>90- ∠ RPC=90-∠ BPQ

=> ∠ OPQ =∠OPR

hope u will get ans. ..

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