Answer: The coordinates of the circumcenter is [tex](\frac{9}{2}, -1)[/tex].
Explanation:
The coordinates of triangle DEF are D(1,3) E (8,3) and F(1,-5).
Distance formula,
[tex]d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}[/tex]
[tex]DE=\sqrt{(8-1)^2+(3-3)^2}=7[/tex]
[tex]FE=\sqrt{(1-8)^2+(-5-3)^2}=\sqrt{7^2+8^2}[/tex]
[tex]DF=\sqrt{(1-1)^2+(-5-3)^2}=8[/tex]
Since triangle follows pythagoras theorem,
[tex](DF)^2+(DE)^2=(FE)^2[/tex]
Therefore the given triangle is a right angle triangle.
Or plot these points on a coordinate plane. From the figure we can say that the triangle DEF is a right angle triangle.
The circumcenter of a right angle triangle is the midpoint of the hypotenuse.
The hypotenuse is EF. The midpoint of EF is,
[tex]Midpoint =(\frac{8+1}{2}, \frac{3-5}{2} )=(\frac{9}{2}, -1)[/tex]
Therefore, the coordinates of the circumcenter is [tex](\frac{9}{2}, -1)[/tex].
