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
(4,2)
(5,3)
(3,5)
(4,4)
To find the circumcenter of a triangle, we can first find the equations of the perpendicular bisectors of two sides of the triangle. These perpendicular bisectors will intersect at the circumcenter.
Let's start with the midpoint of the line segment AB. The coordinates of the midpoint M of AB can be found by averaging the x-coordinates and the y-coordinates of A and B respectively:
M(x, y) = ((2+2)/2, (5+3)/2) = (2, 4)
The slope of the line passing through A and B can be found using the formula: slope = (change in y) / (change in x)
slope_AB = (5-3) / (2-2) = 2/0, which is undefined as the change in x is zero. This indicates that the line AB is a vertical line.
Since the line is vertical, the equation of its perpendicular bisector will be a horizontal line passing through the midpoint M(2, 4).
Therefore, the equation of the perpendicular bisector of AB is y = 4.
Next, we'll find the midpoint N of line segment BC:
N(x, y) = ((2+6)/2, (3+3)/2) = (4, 3)
The slope of the line passing through B and C can be found using the formula: slope = (change in y) / (change in x)
slope_BC = (3-3) / (2-6) = 0/(-4) = 0.
Since the slope is zero, the line BC is a horizontal line. The equation of its perpendicular bisector will be a vertical line passing through midpoint N(4, 3).
Therefore, the equation of the perpendicular bisector of BC is x = 4.
Now, to find the circumcenter, we need to find the point of intersection of the two perpendicular bisectors. Since one of the equations is y = 4 and the other is x = 4, the coordinates of the circumcenter are (4, 4).
Therefore, the circumcenter of triangle ABC is (4, 4).