Q. 14.0( 5 Votes )

I draw an acute angled triangle PQR and let us prove that the perpendicular bisectors PQ, QR and RP are concurrent. Hence let us write where the circumcentre lies (inside/outside/on the side) of the acute angle triangle.

Answer :

Let us consider an acute angle triangle PQR.


OA, OB and OC perpendicularly bisects the sides.


Joined OP, OQ and OR!


1.JPG


In Δ AOQ and Δ AOR


AO = AO (Common side)


AQ = AR(A is the midpoint of QR)


OAQ = OAR (OA is the perpendicular bisector)


Hence Δ QOA and Δ ROA are congruent to each other by S.A.S. axiom of congruency


Hence OQ = OR (Corresponding parts of Congruent Triangles)


In Δ QOC and Δ POC


CO = CO (Common side)


CQ = CP (C is the midpoint of QP)


OCQ = OCP (OC is the perpendicular bisector)


Hence Δ QOC and Δ POC are congruent to each other by S.A.S. axiom of congruency


Hence OQ = OP (Corresponding parts of Congruent Triangles)


Hence OP = OQ = OR


So it is proved that the perpendicular bisectors are concurrent


The circumcentre lies inside in an acute angled triangle


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