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Question

# Construct a triangle ABC such that: BC = 6 cm, ∠B=60∘ and ∠C=45∘ In the same figure, find a point which is equidistant from the vertices of the triangle.Name this point. Draw the Circumcircle of the triangle.

A
Circumcenter
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B
Incenter
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C
Midpoint
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D
Data insufficient
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Solution

## The correct option is A CircumcenterStep (I) A line BC=6cm is drawn.Step (II) ∠ABC=60o&∠ACB=45o are drawn.Step (III) OD, OE & OF are drawn as right bisectors of AB, BC & CA respectively. The right bisector intersect at O.So O is the circumcentre of ΔABC.A circle is drawn with O as centre and either of OA, OB or OC as radius,This is the circumcircle ofΔABC.Justification-Between the triangles OAF & OCF,AF=FC (F is the mid point of AC),OF is the common side,∠OFA=∠OFC(OF&isrightbisectorofAC)∴ΔOAF≡ΔOCF.∴OA=OC..........(i)Again, between ΔAOD&ΔBODAD=BD(DisthemidpointofAB),ODisthecommonside.∠ODA=∠ODB=90o∴ΔAOD≡ΔBOD.∴OA=OB........(ii).Comparing (i) & (ii),OA=OB=OC.So we can draw a circle with o as centre and OA or OB or OC as radius.So O is the circumcentre of ΔABC.

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