Equations of Circles

[ThisIsMeFAA]

Okay, I'm in ninth grade Pre-AP Geometry, and I'm completely lost on anything to do with the equations of circles! I do know that if the center is (h,k), then the equation of a circle is (x-h)^2+(y-k)^2=r^2. The instructions are:Write an equation of a circle that contains each set of points.The problem is:A(1,6), B(5,6), C(5,0)Can you please explain how to do this??

 

[pka]

Okay, I'm in ninth grade Pre-AP Geometry, and I'm completely lost on anything to do with the equations of circles! I do know that if the center is (h,k), then the equation of a circle is (x-h)^2+(y-k)^2=r^2. The instructions are:Write an equation of a circle that contains each set of points.The problem is:A(1,6), B(5,6), C(5,0)Can you please explain how to do this??

Find the intersection of the two perpendicular bisectors of \(\displaystyle \overline{AB}~\&\overline{BC}~\).
That point is the center of the circle.

The radius is the distance from that center to point \(\displaystyle A\).

 

[ThisIsMeFAA]

Find the intersection of the two perpendicular bisectors of \(\displaystyle \overline{AB}~\&\overline{BC}~\).
That point is the center of the circle.

The radius is the distance from that center to point \(\displaystyle A\).

Umm...I'm lost. I don't really understand perpendicualar bisectors and such, so can you please explain a little more detailed?

 

[ThisIsMeFAA]

Nevermind! I had a slow moment lol. But don't I find the circumcenter? Won't that be the intersection of the perpendicular bisectors?

 

[pka]

Umm...I'm lost. I don't really understand perpendicualar bisectors and such, so can you please explain a little more detailed?

If that is true, then you have no hope of understanding the problem.
You may well simply be over your head in this AP course.
You ought to back up about four weeks and learn the basics.
Or change to a different course.

 

[ThisIsMeFAA]

If that is true, then you have no hope of understanding the problem.
You may well simply be over your head in this AP course.
You ought to back up about four weeks and learn the basics.
Or change to a different course.

I just had to think a minute. I haven't done anything with perpendicular bisectors in several weeks, because we've been doing Trig lately..But now I understand it. Thank you. And I do wish I could get out of Pre-AP Geometry, but it's too late in the year to switch classes, so I have to just try to understand it.

 

[Oaky]

A more intuitive method, though perhaps slower, would be to set up three simultaneous equations.

Utilising \(\displaystyle (x-h)^2 + (y-k)^2 = r^2\) with the points (1,6), (5,6) and (5,0)...

\(\displaystyle [1]\) \(\displaystyle (1-h)^2 + (6-k)^2 = r^2\)
\(\displaystyle [2]\) \(\displaystyle (5-h)^2 + (6-k)^2 = r^2\)
\(\displaystyle [3]\) \(\displaystyle (5-h)^2 + k^2 = r^2\)

and simply solve for h, k and r.

It should be incredibly simple given that they all have the same radius, as well as the fact that you have two points where x=5 and two where y=6.