Question

Find the co-ordinates of the circumcenter of ∆ if A(2,3),B(4,-1) and C(5,2).​

Answer 1

Answer:

Recall that the circumcentre of a triangle is equidistant from the vertices of a triangle. Let A(8,6), B(8,-2) and C(2,-2) be the vertices of the given triangle and let P(x,y) be the circumcentre of this triangle. Then

PA=PB=PC

⇒PA

2

=PB

2

=PC

2

Now,PA

2

=PB

2

⇒(x−8)

2

+(y−6)

2

=(x−8)

2

+(y+2)

2

⇒x

2

+y

2

−16x−12y+100=x

2

+y

2

−16x+4y+68

⇒16y=32

⇒y=2

and,PB

2

=PC

2

⇒(x−8)

2

+(y+2)

2

=(x−2)

2

+(y+2)

2

⇒x

2

+y

2

−16x+4y+68=x

2

+y

2

−4x+4y+8

⇒12x=60

⇒x=5

So, the coordinates of the circumcentre P are (5,2)

Also, Circum-radius=PA=PB=PC=

(5−8)

2

+(2−6)

2

=5