# What are the coordinates of the center of the circle that passes through the points (1, 1), (1, 5), and (5, 5)?

Jan 30, 2016

$\left(3 , 3\right)$

#### Explanation:

Along with the point $\left(5 , 1\right)$ these points are the vertices of a square, so the centre of the circle will be at the midpoint of the diagonal between $\left(1 , 1\right)$ and $\left(5 , 5\right)$, that is:

$\left(\frac{1 + 5}{2} , \frac{1 + 5}{2}\right) = \left(3 , 3\right)$

The radius is the distance between $\left(1 , 1\right)$ and $\left(3 , 3\right)$, that is:

$\sqrt{{\left(3 - 1\right)}^{2} + {\left(3 - 1\right)}^{2}} = \sqrt{8}$

So the equation of the circle may be written:

${\left(x - 3\right)}^{2} + {\left(y - 3\right)}^{2} = 8$

graph{((x-3)^2+(y-3)^2-8)((x-3)^2+(y-3)^2-0.01)((x-1)^2+(y-1)^2-0.01)((x-5)^2+(y-1)^2-0.01)((x-1)^2+(y-5)^2-0.01)((x-5)^2+(y-5)^2-0.01)((x-3)^100+(y-3)^100-2^100)(x-y)(sqrt(17-(x+y-6)^2)/sqrt(17-(x+y-6)^2)) = 0 [-5.89, 9.916, -0.82, 7.08]}