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.
Let us consider an acute angle triangle PQR.
OA, OB and OC perpendicularly bisects the sides.
Joined OP, OQ and OR!
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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