Question

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

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

$x=\dfrac{63}{14} =\dfrac{9}{2} =4.5,\\ \\ y=-1$.

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