Question

In Ch : Circles [Class 9], there is a theorem stated as follows :

"There is one and only one circle passing through three given non-collinear points."

So, what if the 3 given non-collinear points aren't equidistant?
How can we draw a circle through them if they are not equally distant from a centre?
Is this condition a violation of the theorem?

Explain & justify your answer

Points : 50 ☺

Answer 1

non - Colinear :- in co- ordinate geometry if any two or more points lies on line or passing through a line are called co-linear points of that line . and if these points don't lies a line are known as non- Colinear points of that line .

circle :- circle is non linear geometrical shape .e.g circle is a locus of a point which distance from a fixed point is always constant .

hence, there is not possible to draw a circle from Colinear points . only required at least three non Colinear points to draw a circle .

why we use at least three points ???

if we use at least two points then how we identified which type of shape form by two points . because we know for two points only a line can be drawn . but we choose three points we easily identified which type of shape formed .

now according to theorem , at least three non- Colinear points required to drawn a circle .
or ,

there is one and only one circle passing through three non- Colinear points .

let (x1 , y1 ) , (x2 ,y2) and (x3 ,y3) are three non -colinear points . if I ask to you a question which type of shape you by this ?

I think your first choice in answer is triangle.
this answer 100% correct because no chance, to form any other shape by these .

now you deform triangle side in such a way that all part contain like a part of circle.you see after complete work , a circle is formed .

by the way you now , ortho center, circumcentre , centroid , nine point , etc all of these relation between circle and triangle.

hence, if we draw for three non- Colinear points a triangle , then we can also draw a circle by three non- Colinear points .

there is no chance to draw more then one circle by three non- Colinear points .

now is any violation of this ?

the answer is yes ,
because circle means distance from fixed point to curve surface always constant . if three non-Colinear points don't follow this then they are not form circle .

hence, we can say that there is one and only one circle can be drawn by three non-Colinear points

Answer 2

According to geometry if any two or more points lies on line or are called co-linear points of that line . But if these points don't lies a line are known as non- Colinear points of that line .

circle is non linear geometrical shape .A circle is a simple closed shape in Euclidean geometry. It is the set of all points in a plane that are at a given distance from a given point, the centre; equivalently it is the curve traced out by a point that moves so that . Thus, it is not possible to draw a circle from Colinear points .
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We need to use three points.

if we use two points then how to identify which type of shape will be formed by two points . If we take three points we easily identified which type of shape formed . According to tge theorem , at least three non- Colinear points required to drawn a circle . Or we can say there is one and only one circle passing through three non- Colinear points .

Suppose you are asked to draw a shape from passing three collinear points at an angle. I am sure you will choose triangle. Now you are deform triangle side in such a way that all part contain like a part of circle.you see after complete work , a circle is formed .



Thus, There is no chance to draw more than one circle on three non- Colinear points .

now is any violation of this ?As circle means distance from fixed point to curve surface always constant . if three non-Colinear points don't follow this then they do not form circle .