Find the coordinates of the circumcenter for ∆DEF with coordinates D(1,3) E (8,3) and F(1,-5). Show your work.
Accepted Solution
A:
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].