Question

Find the circumcenter of triangle ABC. A is at (2,5), B is at (2,3) and C is located at (6,3).(1 point) Responses (4,4) (4,4) (5,3) (5,3) (3,5) (3,5) (4,2)

Answer 1

To find the circumcenter of triangle ABC, we need to find the intersection point of the perpendicular bisectors of two of the sides of the triangle.

First, let's find the midpoint of side AB, which is the line segment connecting points A and B. The midpoint of AB is ((2+2)/2, (5+3)/2) = (2, 4).

The equation of the perpendicular bisector of side AB is the line passing through the midpoint (2, 4) with a slope that is the negative reciprocal of the slope of side AB.

The slope of side AB is (3-5)/(2-2) = -2/0, which is undefined. Therefore, the perpendicular bisector of AB is a vertical line passing through the midpoint (2, 4), which has the equation x = 2.

Next, let's find the midpoint of side AC, which is the line segment connecting points A and C. The midpoint of AC is ((2+6)/2, (5+3)/2) = (4, 4).

The equation of the perpendicular bisector of side AC is the line passing through the midpoint (4, 4) with a slope that is the negative reciprocal of the slope of side AC.

The slope of side AC is (3-5)/(6-2) = -2/4 = -1/2. The negative reciprocal of -1/2 is 2, so the equation of the perpendicular bisector of AC is y = 2x + b.

To find the value of b, we can plug in the coordinates of the midpoint (4, 4) into the equation y = 2x + b:

4 = 2(4) + b
4 = 8 + b
b = -4

Therefore, the equation of the perpendicular bisector of AC is y = 2x - 4.

To find the circumcenter, we need to find the intersection of the perpendicular bisectors. Since one of the perpendicular bisectors is a vertical line with the equation x = 2 and the other is the line y = 2x - 4, we can find the intersection point by plugging in x = 2 into the equation y = 2x - 4:

y = 2(2) - 4
y = 4 - 4
y = 0

Therefore, the circumcenter of triangle ABC is (2, 0).