Question

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

Answer 1

Answer: The coordinates of the circumcenter is $(\frac{9}{2}, -1)$.

Explanation:

The coordinates of triangle DEF are D(1,3) E (8,3) and F(1,-5).

Distance formula,

$d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$

$DE=\sqrt{(8-1)^2+(3-3)^2}=7$

$FE=\sqrt{(1-8)^2+(-5-3)^2}=\sqrt{7^2+8^2}$

$DF=\sqrt{(1-1)^2+(-5-3)^2}=8$

Since triangle follows pythagoras theorem,

$(DF)^2+(DE)^2=(FE)^2$

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,

$Midpoint =(\frac{8+1}{2}, \frac{3-5}{2} )=(\frac{9}{2}, -1)$

Therefore, the coordinates of the circumcenter is $(\frac{9}{2}, -1)$.