The coordinates of the vertices of a triangle are A(3,2),B(9,2),C(6,5).
a)
The midpoint of AB is M1(23+9, 22+2)=M1(6, 2)
The slope of AB is 9−32−2=60=0. It means that AB is a horizontal line and perpendicular bisector is a vertical line. The equation of the perpendicular bisector is x=6 .
The midpoint of BC is M2(29+6, 22+5)=M2(7.5, 3.5)
The slope of BC is 9−62−5=3−3=−1. The slope of the perpendicular bisector is m2=−−11=1 .
The equation of the perpendicular bisector is y−3.5=1×(x−7.5) or y=x−4 .
The midpoint of AC is M3(23+6, 22+5)=M3(4,5, 3.5)
The slope of AC is 3−62−5=−3−3=1. The slope of the perpendicular bisector is m3=−11=−1 .The equation of the perpendicular bisector is y−3.5=−1×(x−4.5) or y=−x+8 .
b)
The circumcenter is the point of concurrency of perpendicular bisectors of a triangle. We need to solve system of any two bisector equations to find the coordinates of the circumcenter:
{x=6y=x−4 {x=6y=2
The circumcenter is O(6, 2) .
Answers: a) x=6 , y=x−4 , y=−x+8 ; b) (6, 2) .
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