How to find the center and radius of a circle using 3 points...
My 3 points are (2,8) (-4,6) (1,-1)
I found the Equation of the perpendicular bisector of the line between the first and second point
which was y=(3/2)x + 5
And the Equation of the perpendicular bisector of the line between the second and third point
which was y=(-1/3)x+(23/3)
Finding the center:
(3/2)x + 5 = (-1/3)x+(23/3)
(11/6)x+5=(23/3)
x=(16/11)
then i plugged X into my Y equations
y=(3/2)(16/11) + 5 =(79/11)
y=(-1/3)(16/11)+(23/3) =(79/11)
Center=(16/11, 79/11) or (1.45,7.18)
Is my Center correct and how do i find Radius?
Thanks
Yes, your calculation for the center is correct. The center of the circle is (16/11, 79/11) or approximately (1.45, 7.18).
To find the radius of the circle, you can use the distance formula between the center of the circle and one of the given points. Let's use the point (2,8):
Distance = sqrt((x2 - x1)^2 + (y2 - y1)^2)
Distance = sqrt((2 - 16/11)^2 + (8 - 79/11)^2)
Distance = sqrt((-4/11)^2 + (3/11)^2)
Distance = sqrt(16/121 + 9/121)
Distance = sqrt(25/121)
Distance = 5/11
Therefore, the radius of the circle is 5/11.
To find the center of a circle given three points, you correctly found the equations of the perpendicular bisectors of the lines between the three points.
For the center, you set the two equations equal to each other and solved for x:
(3/2)x + 5 = (-1/3)x + (23/3)
(11/6)x = 18/3 - 23/3
(11/6)x = -5/3
x = -30/33
Substituting this value of x into either of the equations for the perpendicular bisectors, you found the corresponding value of y:
y = (3/2)(-30/33) + 5
y = -45/22 + 110/22
y = 65/22
So the center of the circle is (-30/33, 65/22) or approximately (-0.91, 2.95).
To find the radius, you need to find the distance between any of the three given points and the center. For example, you can choose the distance between the first point (2, 8) and the center (-0.91, 2.95):
The distance formula is √((x2 - x1)^2 + (y2 - y1)^2).
distance = √((-0.91 - 2)^2 + (2.95 - 8)^2)
distance = √((-2.91)^2 + (-5.05)^2)
distance = √(8.5081 + 25.5025)
distance = √33.0106
distance ≈ 5.75
So the radius of the circle is approximately 5.75.