Question

find the coordinates of the circumcentre of the triangle whose vertices are (8,6)(8.-2) and (2,-2). also find its circum- radius

Answer 1

Sol:
Let the coordinates of the circumcentre of the triangle be (x, y).
Circumcentre of a triangle is equidistant from each of the vertices.

Distance between (8, 6) and (x, y) = Distance between (8, -2) and (x, y)
√[(x - 8)2 + (y - 6)2] = √[(x - 8)2 + (y + 2)2]
[(x - 8)2 + (y - 6)2] = [(x - 8)2 + (y + 2)2]
(y - 6)2 = (y + 2)2
y2 + 36 - 12y = y2 + 4y + 4
36 - 12y = 4y + 4
16y = 32
y = 2

Distance between (2, -2) and (x, y) = Distance between (8, -2) and (x, y)
√[(x - 2)2 + (y + 2)2] = √[(x - 8)2 + (y + 2)2]
[(x - 2)2 + (y + 2)2] = [(x - 8)2 + (y + 2)2]
(x - 2)2 = (x - 8)2
x2 + 4 - 4x = x2 - 16x + 64
4 - 4x = -16x + 64
12x = 60
x = 5.

Hence, the coordiantes of the circumcentre of the triangle are (5, 2).

Circumradius = √[(5 - 8)2 + (2 - 6)2]
= √(9 + 16)
= √25
= 5 units.

Hope This Helps :)