Question

what is the circumcenter of the triangle whose vertices are (3,4) (8,4) and (3,16) ?​

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 =PC2

Now,PA2 =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 =PC2

⇒(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