How do i find the center of a circle using 3 points.
Also need to find the radius and the equation of the circle....
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)
I need to know if i found the right center too?
Thanks
To find the center of a circle using three points, you can follow these steps:
1. Find the perpendicular bisectors of two of the three lines formed by the three points.
- To find the perpendicular bisector of a line, you need to determine the slope of the line and the midpoint of the line segment.
2. Set the equations of the perpendicular bisectors equal to each other and solve for the coordinates (x, y) that satisfy the equation. This point is the center of the circle.
- In your case, you found the equations of the two perpendicular bisectors to be:
- Line 1: y = (3/2)x + 5
- Line 2: y = (-1/3)x + (23/3)
- By setting these two equations equal to each other, you get: (3/2)x + 5 = (-1/3)x + (23/3).
3. Solve the equation for x. In your calculation, you correctly solved for x = (16/11).
4. Substitute the value of x into one of the perpendicular bisector equations and solve for y. In your calculation, you substituted x = (16/11) into Line 1's equation and Line 2's equation, both resulting in y = (79/11).
5. Therefore, the center of the circle is (16/11, 79/11) or approximately (1.45, 7.18), as you correctly calculated.
To find the radius and the equation of the circle, you can use the distance formula between the center and one of the given points. Let's use the point (2, 8) as an example:
6. Use the distance formula to find the distance between the center and the given point:
- Distance = √[(x2 - x1)^2 + (y2 - y1)^2]
- Substitute the coordinates of the center (16/11, 79/11) and the point (2, 8) into the formula:
- Distance = √[(2 - 16/11)^2 + (8 - 79/11)^2]
7. Calculate the distance to find the radius of the circle.
8. Once you have the center and the radius, you can write the equation of the circle in the standard form (x - h)^2 + (y - k)^2 = r^2, where (h, k) represents the coordinates of the center and r represents the radius.
I hope this explanation helps you to understand the steps you took and confirms that you found the correct center coordinates.