Find the coordinates of the circumcentre of the triangle whose vertices are (8, 6), (8, – 2) and (2, – 2). Also, find its circumradius. [3 MARKS]
Concept : 1 Mark
Application : 1 Mark
Calculation : 1 Mark
Recall that the circumcentre of a traingle is equidistatnt from the verices of a traingle.
Let A (8 , 6 ), B ( 8, -2 ) and C (2 , -2) be the vertices of the given traingle.
and let P ( x, y) be the circumcentre of this traingle . Then,
PA=PB=PC⇒PA2=PB2=PC2
Now ,
PA2=PB2
⇒(x−8)2+(y−6)2=(x−8)2+(y+2)2
⇒x2+y2−16x−12y+100=x2+y2−16x+4y+68
⇒16y=32
⇒y=2
and
PB2=PC2
⇒(x−8)2+(y+2)2=(x−2)2+(y+2)2
⇒x2+y2−16x+4y+68=x2+y2−4x+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)
=√9+16
=5