The vertices A,B,C of a triangle ABC have coordinates (4,4),(5,3),(6,0) respectively. Find the equations of the perpendicular bisectors of AB and BC ,the coordinates of the circumvented and the equation of the circumcircle of triangle ABC

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We have:A4,4, B5,3 and C6,0Slope of AB=3-45-4=-1Slope of perpendicular bisector of AB=m1=-1Slope of AB=-1-1=1Perpendicular bisector of AB will pass through mid point of AB.Mid point of AB x1,y1=5+42,4+32=92,72Hence equation of perpendicular bisector of AB will be:y-y1=m1x-x1y-72=x-922y-72=2x-922y-7=2x-92x-2y=2x-y=1 ;iSlope of BC=0-36-5=-3Slope of perpendicular bisector of BC=m2=-1-3=13Perpendicular bisector of BC will pass through mid point of AB.Mid point ofBC x2,y2=5+62,3+02=112,32Hence equation of perpendicular bisector of BC will be:y-y2=m2x-x2y-32=13x-1122y-32=2x-1162y-3=2x-1136y-9=2x-112x-6y=2x-3y=1 ;iiPoint of intersection of i and ii will give circumcentre.i-iix-y-x+3y=1-1y=0Put in ix=1Hence circumcentre is O1,0Radius of circumcentre=OA=OB=OCC6,0r=OC=6-12+0-02=5 unitsHence equation of circumcircle will be:x-12+y2=52x-12+y2=25

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