Question

The circumcentre of the triangle formed by (-2,-3),(2,-1),(4,0) is...

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Answer 1

Given points are,

A=(−2,3),B=(2,−1),C=(4,0)A=(−2,3),B=(2,−1),C=(4,0)

To find out the circumcenter we have to solve any two bisector equations and find out the intersection points.

So, mid point of AB=(−2+22,3−12)=(0,1)AB=(−2+22,3−12)=(0,1)

Slope of AB=(−1−32−(−2))=−1AB=(−1−32−(−2))=−1

Slope of the bisector is the negative reciprocal of the given slope.

So, the slope of the perpendicular bisector =−(1−1)=1=−(1−1)=1

Equation of ABAB with slope 11 and the coordinates (0,1)(0,1) is,

(y−1)=1(x−0)(y−1)=1(x−0)

x−y=−1x−y=−1 ……………… (1)(1)

Similarly, for ACAC

Mid point of AC=(−2+42,3+02)=(1,32)AC=(−2+42,3+02)=(1,32)

Slope of AC=(0−34−(−2))=−12AC=(0−34−(−2))=−12

Slope of the bisector is the negative reciprocal of the given slope.

So, the slope of the perpendicular bisector =−1−12=2=−1−12=2

Equation of ACAC with slope 22 and the coordinates (1,32)(1,32) is,

(y−32)=2(x−1)(y−32)=2(x−1)

2y−3=4x−42y−3=4x−4

4x−2y=14x−2y=1 ……………… (2)(2)

Equation (1)(1) and (2)(2) ⟹x−y=−(4x−2y)⟹x−y=−(4x−2y)

or y=53xy=53x

putting this value back in (1)(1) ⟹x−53x=−1⟹x−53x=−1 or x=32x=32

y=53x=53×32=52y=53x=53×32=52

So the circumcenter is (32,52)(32,52)

Answer 2

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