Question

Find the coordinates of the circumcenter for ∆DEF with coordinates D(1,3) E (8,3) and F(1,-5).

*Please show all work!

Answer 1

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

1. Distance OD=distance OE, then:

`(x-1)^2+(y-3)^2=(x-8)^2+(y-3)^2`.

2. Distance OD=distance OE, then:

`(x-1)^2+(y-3)^2=(x-1)^2+(y+5)^2`.

Solve the system:

`\left\{\begin{array}{l} (x-1)^2+(y-3)^2=(x-8)^2+(y-3)^2 \\ (x-1)^2+(y-3)^2=(x-1)^2+(y+5)^2 \end{array}\right.`,

`\left\{\begin{array}{l} x^2-2x+1+y^2-6y+9=x^2-16x+64+y^2-6y+9 \\ x^2-2x+1+y^2-6y+9=x^2-2x+1+y^2+10y+25 \end{array}\right.`,

`\left\{\begin{array}{l} 14x=63 \\ -16y=16 \end{array}\right.`.

Then

`x=(63)/(14) =(9)/(2) =4.5,\\ \\ y=-1`.

Answer: the coordinates of the circumcenter for ∆DEF are x=4.5, y=-1.