Question

What are the coordinates of the circumcenter of a triangle with vertices A(−3, 3) , B(−1, 3) , and C(−1, −1) ?

Answer 1

The answer to this is (-2, 1).

Answer 2

ANSWER

The circumcenter of ∆ABC has coordinates
$(-2,1)$

EXPLANATION

The circumcenter is the point of intersection of the perpendicular bisectors of any two sides of triangle ABC.

Considering side AB with coordinates (-3,3) and (-1,3) respectively, we can see that this is a horizontal line. The perpendicular bisector of this line is a vertical line that has equation x equals the x-coordinate of the midpoint of AB.

Midpoint of AB has coordinates

$( \frac{ - 3 + - 1}{2} , \frac{3 + 3}{2} )$

This gives

$( \frac{ - 4}{2} , \frac{6}{2} )$

$( - 2, 3)$

The equation of the perpendicular bisector is
$x = - 2$

Similarly, the coordinates of B(-1,3) and C(-1,-1) tells us that, line BC is a horizontal line since the x-values are constant. Therefore the perpendicular bisector is a horizontal line that has equation , y equals the y-value of the midpoint of BC.

The midpoint of BC has coordinates,

$( \frac{ - 1 + - 1}{2} , \frac{3 + - 1}{2} )$

This implies that,

$( \frac{ - 2}{2} , \frac{2}{2} )$

This gives,

$( - 1 , 1)$

The equation of the perpendicular bisector is
$y = 1$

These two bisectors will meet at,

$(-2,1)$

Therefore the circumcenter is
$(-2,1).$