Finding the Center of a Circle Using Perpendicular Bisectors

So you can find the intersection of any two perpendicular bisectors, and that will be the center of the circle. Then you can use the distance formula to find the radius of the circle. In summary, to find the coordinates of point P, the center of the circle containing points F, G, and H, you can either solve a system of simultaneous equations using the general equation of a circle, or find the intersection of any two perpendicular bisectors of the sides of the triangle formed by F, G, and H.
  • #1
L²Cc
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Homework Statement


In the coordinate plane, the points F(-2, 1), G(1, 4), and H(4, 1) lie on a circle with center P. What are the coordinate of point P?

Homework Equations





The Attempt at a Solution



I tried subtracting the x-values, but my answer would be always wrong. I think I'm approaching the question incorrectly.
 
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  • #2
General equation of a circle is [tex]x^2+y^2+2fx+2gy+c=0[/tex] where [tex](-f,-g)[/tex] is the centre of the circle(i.e. P) and radius,[tex]r= \sqrt{f^2+g^2-c}[/tex]sub the points F,G and H into this equation
and you'll get 3 equations with 3 unknowns
more precisely you should get to solve these equations

[tex]8f+2g+c=-17[/tex]
[tex]2f+8g+c=-17[/tex]
[tex]-4f+2g+c=-5[/tex]
 
Last edited:
  • #3
The three points F, G, H, form a triangle; and the segment bisectors of each side will intersect at the center of the circle which contains F, G, and H. Find the lines for the segment bisectors (you only need two of them) and find their point of intersection. That is the center point of the circle.

How do you find each line? It contains the midpoint of a side and has slope which is negative reciprocal of the side; substitute into formula for equation of a line.
 
  • #4
Another way is, let the coordinates of point p be (x,y). Find the point of intersection of the lines PF and GH, and let it be M. Then you can use the property PM*FM=GM*HM to solve for x and y.
 
  • #5
This will be the circumcentre right? Then we can find the equations of any 2 perpendicular side bisectors and equate them.

Am i right?
 
  • #6
Messages #3 and #5 express the same concept. Message #2 is nice because it immediately puts the information into simultaneous equations which are fairly easy to solve.

About #3 and #5:
The perpendicular bisectors of the sides of a triangle are concurrent (meaning they intersect) at a point equidistant from the vertices. This also means that the vertices lie on a circle whose center is that point of concurrency.
 

1. What are coordinates?

Coordinates are a set of numbers that represent the location of a point on a map or graph. They consist of an x-coordinate (horizontal) and a y-coordinate (vertical).

2. What is the point P?

Point P is a specific location on a map or graph that is defined by its coordinates. It is represented by a dot or symbol and can be used to identify a specific location or to plot data.

3. How do you find the coordinates of point P?

The coordinates of point P can be found by identifying its position on the x-axis and y-axis. The x-coordinate is the horizontal distance from the origin (0,0) to the point, and the y-coordinate is the vertical distance from the origin to the point.

4. What is the importance of coordinates in science?

Coordinates are important in science as they allow for precise and accurate measurement and location of points on maps and graphs. They also enable scientists to analyze and interpret data in a quantitative manner.

5. Can coordinates be negative?

Yes, coordinates can be negative. This indicates that the point is located to the left of the origin on the x-axis or below the origin on the y-axis. Negative coordinates are important in representing locations in the southern and western hemispheres, as well as negative values in scientific data.

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