Find the circumcenter of triangle ABC. A is at (2,5), B is at (2,3) and C is located at (6,3)
To find the circumcenter of a triangle, we first need to find the midpoints of two of its sides. Then, we will find the slopes of the perpendicular bisectors of those sides and find the point where they intersect, which will be the circumcenter.
Let's find the midpoint of side AB first. The coordinates of A are (2, 5) and the coordinates of B are (2, 3). To find the midpoint, we average the x-coordinates and the y-coordinates separately:
Midpoint of AB:
x-coordinate: (2 + 2) / 2 = 4 / 2 = 2
y-coordinate: (5 + 3) / 2 = 8 / 2 = 4
So, the midpoint of AB is M(2, 4).
Now, let's find the midpoint of side BC. The coordinates of B are (2, 3) and the coordinates of C are (6, 3).
Midpoint of BC:
x-coordinate: (2 + 6) / 2 = 8 / 2 = 4
y-coordinate: (3 + 3) / 2 = 6 / 2 = 3
So, the midpoint of BC is N(4, 3).
Next, we will find the slope of the line passing through M and N, which is the negative reciprocal of the slope of the segment MN. The slope of the segment MN can be found using the formula:
slope = (y₂ - y₁) / (x₂ - x₁)
Slope of MN:
(3 - 4) / (4 - 2) = -1 / 2
So, the slope of the line passing through M and N is 2.
Since the slope of the perpendicular bisector is the negative reciprocal of the slope between the two points, the slope of the perpendicular bisector is -1/2.
Now, we can find the equation of the perpendicular bisector passing through M(2, 4) using the slope-intercept form: y = mx + b.
Using the point-slope form: y - y₁ = m(x - x₁), where (x₁, y₁) is the midpoint M(2, 4):
y - 4 = (-1/2)(x - 2)
y - 4 = (-1/2)x + 1
y = (-1/2)x + 5
Similarly, we can find the equation of the perpendicular bisector passing through N(4, 3):
y - 3 = (-1/2)(x - 4)
y - 3 = (-1/2)x + 2
y = (-1/2)x + 5
Now, we have two equations for two perpendicular bisectors:
y = (-1/2)x + 5 (Equation 1)
y = (-1/2)x + 5 (Equation 2)
Since the equations are the same, they represent the same line. Therefore, the perpendicular bisectors intersect at infinitely many points, and the triangle ABC is an isosceles triangle where AB and BC are equal in length.
In this case, we can consider any point on the perpendicular bisector as the circumcenter. Let's choose the midpoint M(2, 4) as the circumcenter.
Therefore, the circumcenter of triangle ABC is located at point M(2, 4).