What are the coordinates of the circumcenter of a triangle with vertices a(−3, 3) , b(−1, 3) , and c(−1, −1) ? enter your answer in the box. ( , )

ANSWER

The circumcenter of ∆ABC has coordinates


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



This gives





The equation of the perpendicular bisector is


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,



This implies that,



This gives,



The equation of the perpendicular bisector is


These two bisectors will meet at,



Therefore the circumcenter is

The answer is all about the number 4