Question

It says find the circumcenter of Triangle EFG with E(4,4) F(4,2) and G(8,2)

Answer 1

The circumcenter is the point where all the perpendicular bisectors intersect.

The circumcenter is taken as P. This means that the distance from P to each vertex is equal.

EP = FP = GP

For EP and FP,

EP^2 = FP^2

The coordinates of vertex E is (4, 4)

The coordinate of vertex F is (4, 2)

It means that

(x - 4)^2 + (y - 4)^2 = (x - 4)^2 + (y - 2)^2

(x - 4)(x - 4) + (y - 4)(y - 4) = (x - 4)(x - 4) + (y - 2)(y - 2)

x^2 - 4x - 4x + 16 + y^2 - 4y - 4y + 16 = x^2 - 4x - 4x + 16 + y^2 - 2y - 2y + 4

x^2 - 8x + 16 + y^2 - 8y + 16 = x^2 - 8x + 16 + y^2 - 4y + 4

x^2 - x^2 + y^2 - y^2 - 8x + 8x - 8y + 4y = 4 + 16 - 16 - 16

- 4y = - 12

y = - 12/4

y = 3

Also,

FP^2 = GP^2

(x - 4)^2 + (y - 2)^2 = (x - 8)^2 + (y - 2)^2

(x - 4)(x - 4) + (y - 2)(y - 2) = (x - 8)(x - 8) + (y - 2)(y - 2)

x^2 - 4x - 4x + 16 + y^2 - 2y - 2y + 4 = x^2 - 8x - 8x + 64 + y^2 - 2y - 2y + 4

x^2 - 8x + 16 + y^2 - 4y + 4 = x^2 - 16x + 64 + y^2 - 4y + 4

x^2 - x^2 + y^2 - y^2 - 8x + 16x - 4y + 4y = 64 + 4 - 16 - 4

8x = 48

x = 48/8

x = 6

Therefore, the circumcenter, P of the triangle has the coordinates, (6, 3)