What is the circumcenter? triangle ABC has vertices at A (2,3) B (-4,-3) and C(2,3)

Question

a. (-1,-3)
b. (-1,0)
c. (2,-3)
d. (0,0)

Answer 1

The circumcenter is the point where the perpendicular bisectors of the sides of a triangle intersect.

To find the circumcenter of triangle ABC, we need to find the equations of the perpendicular bisectors of the sides AB, BC, and CA.

The midpoint of side AB is ((2 - 4) / 2, (3 - 3) / 2) = (-1, 0). The slope of the line perpendicular to AB is the negative reciprocal of the slope of AB. The slope of AB is (3 - (-3)) / (2 - (-4)) = 6 / 6 = 1. Therefore, the slope of the perpendicular bisector is -1, and the equation of the perpendicular bisector of side AB is y - 0 = -1(x - (-1)), which simplifies to y = -x - 1.

Similarly, the midpoint of side BC is ((2 - (-4)) / 2, (3 - (-3)) / 2) = (1, 0). The slope of BC is (3 - (-3)) / (2 - (-4)) = 6 / 6 = 1. Therefore, the equation of the perpendicular bisector of side BC is y - 0 = -1(x - 1), which simplifies to y = -x + 1.

Finally, the midpoint of side CA is ((2 - 2) / 2, (3 - 3) / 2) = (0, 0). The slope of CA is (3 - 3) / (2 - 2) = 0 / 0, which is undefined. This is because CA is a vertical line. The equation of the perpendicular bisector of side CA is therefore x = 0.

Now, we have the equations y = -x - 1, y = -x + 1, and x = 0. Solving these equations simultaneously, we find that the point of intersection is (-1, 0). Therefore, the circumcenter of triangle ABC is answer choice b. (-1, 0).