The vertices of triangle ABC are (2,2), B(3,-1) and C(0,1). Find the coordiantes of the circumcenter.
To find the coordinates of the circumcenter of triangle ABC, you can follow these steps:
1. Find the midpoint of two sides of the triangle: AB and AC.
The midpoint, M, of two points A(x₁, y₁) and B(x₂, y₂) is given by the formulas:
Mx = (x₁ + x₂) / 2
My = (y₁ + y₂) / 2
Midpoint of AB:
MABx = (2 + 3) / 2 = 2.5
MABy = (2 - 1) / 2 = 0.5
Midpoint of AC:
MACx = (2 + 0) / 2 = 1
MACy = (2 + 1) / 2 = 1.5
2. Calculate the slopes of the perpendicular bisectors of two sides of the triangle: AB and AC.
The slope, m, of the perpendicular bisector of two points A(x₁, y₁) and B(x₂, y₂) is given by the formula:
m = -1 / ((y₂ - y₁) / (x₂ - x₁))
Slope of AB:
mAB = -1 / ((-1 - 2) / (3 - 2)) = -1 / (-3) = 1/3
Slope of AC:
mAC = -1 / ((1.5 - 2) / (1 - 1)) = -1 / (-0.5) = 2
3. Find the equations of the perpendicular bisectors using the midpoints and slopes.
Equation of the perpendicular bisector passing through midpoint MAB(2.5, 0.5) and slope mAB(1/3):
y - 0.5 = (1/3)(x - 2.5)
y - 0.5 = (1/3)x - (5/6)
y = (1/3)x - (5/6) + 0.5
y = (1/3)x - (5/6) + (3/6)
y = (1/3)x - (2/6)
y = (1/3)x - (1/3)
Equation of the perpendicular bisector passing through midpoint MAC(1, 1.5) and slope mAC(2):
y - 1.5 = 2(x - 1)
y - 1.5 = 2x - 2
y = 2x - 2 + 1.5
y = 2x - 0.5
4. Solve the system of equations formed by the perpendicular bisectors to find the coordinates of the circumcenter.
(1/3)x - (1/3) = 2x - 0.5
Simplifying the equation:
(1/3)x - 2x = -0.5 + (1/3)
(-5/3)x = -1/6
Multiplying both sides by -3/5 to isolate x:
x = -1/6 * (-3/5)
x = 1/10
Substituting the value of x back into one of the perpendicular bisector equations (e.g., y = (1/3)x - (1/3)):
y = (1/3)(1/10) - (1/3)
y = 1/30 - 10/30
y = -9/30
y = -3/10
The coordinates of the circumcenter are (1/10, -3/10).