My Math Forum  

Go Back   My Math Forum > High School Math Forum > Algebra

Algebra Pre-Algebra and Basic Algebra Math Forum


Reply
 
LinkBack Thread Tools Display Modes
March 21st, 2012, 01:56 AM   #1
Senior Member
 
Joined: Jan 2007
From: India

Posts: 161
Thanks: 0

Are 3 points required to define a circle!?

Hi All,
I read somewhere a while ago that we need 3 points to define a circle. I understand 2 points define a line. But cant a circle also be defined with 2 points, say the center point and one more point on the circumference.

Please clarify if I am missing something.

Cheers ,
Arun
arun is offline  
 
March 21st, 2012, 02:43 AM   #2
Math Team
 
agentredlum's Avatar
 
Joined: Jul 2011
From: North America, 42nd parallel

Posts: 3,372
Thanks: 233

Re: Are 3 points required to define a circle!?

The way i understand it, the center is not on the circle, when they talk about points defining a unique circle, they mean points on the circle. In this sense, you need 3 NON-COLINEAR points (all 3 not on the same line) to define a circle because if the points are on a straight line then at least 1 point will not be on the circle if you try to draw a circle through 3 colinear points.


also, extending your analogy backwards, you can define a unique line using only 1 point and the slope but if you want to get a unique line using points on the line you need 2. Similarly, if you want to get a unique circle using points on the circle you need 3 NON-COLINEAR points.
agentredlum is offline  
March 21st, 2012, 10:21 AM   #3
Senior Member
 
Joined: Mar 2012
From: Belgium

Posts: 654
Thanks: 11

Re: Are 3 points required to define a circle!?

so to make his answer easier. if you have 2 points you can take them both as the center. so thats why you need 3 then you can't chose which of them is the center
gelatine1 is offline  
March 21st, 2012, 10:58 AM   #4
Global Moderator
 
The Chaz's Avatar
 
Joined: Nov 2009
From: Northwest Arkansas

Posts: 2,766
Thanks: 4

Re: Are 3 points required to define a circle!?

Quote:
Originally Posted by gelatine1
so to make his answer easier. if you have 2 points you can take them both as the center. so thats why you need 3 then you can't chose which of them is the center
Uh... no.

x^2 + y^2 + ax + by + c = 0 is an equation of a circle.

We can solve the system of equations given by three points, to determine values of a,b, and c.
The Chaz is offline  
March 21st, 2012, 04:57 PM   #5
Senior Member
 
Joined: Jul 2011

Posts: 245
Thanks: 0

Re: Are 3 points required to define a circle!?

Quote:
Originally Posted by arun
But cant a circle also be defined with 2 points, say the center point and one more point on the circumference.
I don't fully understand the current comments. I've yet to take Geo, but. . .

It makes perfect sense that a center and a point on your circle can define the circle.

Consider,

Define O, the center of the circle, as (x,y). Consider the point on the circle (n,m). The radius of the circle, r, is given by the distance between the center and the point on the circle. This would be simply sqrt((n-x)^2+(y-m)^2). Given this amount, r, we can determine any other point on the circle simply by choosing a particular angle and calculating what point is the distance of r from the center at that angle. (This last sentence was a slight abuse of language.)

I don't feel like going into the details too much, but Arun makes a perfectly valid point to me. Does anyone disagree? If so, why?
CherryPi is offline  
March 21st, 2012, 05:48 PM   #6
Math Team
 
Joined: Nov 2010
From: Greece, Thessaloniki

Posts: 1,990
Thanks: 133

Math Focus: pre pre pre pre pre pre pre pre pre pre pre pre calculus
Re: Are 3 points required to define a circle!?

Quote:
Originally Posted by CherryPi
It makes perfect sense that a center and a point on your circle can define the circle.
[color=#000000]I think[/color] [color=#FF0000]TheChaz[/color] [color=#000000]answered it! You have to consider randomness too, what you do is to name a point (not random) "a center" pick another one (not random again) and name it "point of the circumference" .[/color]

Quote:
Originally Posted by TheChaz
Uh... no.

x^2 + y^2 + ax + by + c = 0 is an equation of a circle.

We can solve the system of equations given by three points, to determine values of a,b, and c.
ZardoZ is offline  
March 22nd, 2012, 01:12 AM   #7
Math Team
 
agentredlum's Avatar
 
Joined: Jul 2011
From: North America, 42nd parallel

Posts: 3,372
Thanks: 233

Re: Are 3 points required to define a circle!?

A circle is the set of all points equidistant from a fixed point called the center. The center is not part of the circle.


Yes, TheChaz is right about his system of equations but the points chosen must be on the circumference of the sought after circle to be determined.

Yes, if you give a center and a point on the circumference a unique circle is defined, also you can give a center and radius to define a unique circle, however if you are given only points on the circumference then you need minimum 3 NON-COLINEAR points to define a unique circle.
agentredlum is offline  
March 22nd, 2012, 03:19 PM   #8
Senior Member
 
Joined: Jul 2011

Posts: 245
Thanks: 0

Re: Are 3 points required to define a circle!?

And the goal of this website was just accomplished. Good job, everyone. Our unique perspectives are invaluable to us.
CherryPi is offline  
March 22nd, 2012, 03:30 PM   #9
Global Moderator
 
The Chaz's Avatar
 
Joined: Nov 2009
From: Northwest Arkansas

Posts: 2,766
Thanks: 4

Re: Are 3 points required to define a circle!?

Quote:
Originally Posted by agentredlum
A circle is the set of all points equidistant from a fixed point called the center. The center is not part of the circle.


Yes, TheChaz is right about his system of equations but the points chosen must be on the circumference of the sought after circle to be determined.

Yes, if you give a center and a point on the circumference a unique circle is defined, also you can give a center and radius to define a unique circle, however if you are given only points on the circumference then you need minimum 3 NON-COLINEAR points to define a unique circle.
I should have emphasized that the "center" is not part of the circle (except in the degenerate case where r = 0).

Maybe someone with a bunch of time on their hands would like to show that the system







Has solutions only when are non-colinear...


Not me!
The Chaz is offline  
Reply

  My Math Forum > High School Math Forum > Algebra

Tags
circle, define, points, required



Search tags for this page
Click on a term to search for related topics.
Thread Tools
Display Modes


Similar Threads
Thread Thread Starter Forum Replies Last Post
points on unit circle WWRtelescoping Complex Analysis 2 February 8th, 2014 01:04 AM
distinct points on a circle guru123 Algebra 2 June 24th, 2012 04:59 AM
Rational Points on a circle guru123 Algebra 18 June 22nd, 2012 01:38 PM
Points on the circle zolden Advanced Statistics 3 February 2nd, 2009 11:54 AM
Points of intersection of a circle. Oxymoron Algebra 4 July 23rd, 2008 04:11 PM





Copyright © 2019 My Math Forum. All rights reserved.