Lesson Basic properties of a circle

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In this lesson, we will look at the properties of the Circle.

Before we start looking that the properties of the circle. we need to know basic 
terminology used to define the properties of circle.

<A HREF=Chord_%28geometry%29.wikipedia>Chord</A>
Chord is a line segment joining two points on the circumference of the circle.

{{{drawing(160,160, 0,6,0,6,circle(3,3,2), line(4,1.35,4,4.65),circle(3,3,.15), line(1,3,3,3),locate(3,2.7,C),locate(2,3,r),locate(4,1.35,A),locate(4,5.3,B)
)}}}

In the above figure, AB is called the chord of the given circle of radius 'r' and center 'C'.

Properties of the chord
1. Diameter is the longest chord of the circle. {{{d=2*r}}}
2. Perpendicular line to the chord passing through center 'bisects' the Chord.
3. Chords equidistant from the center of the circle are of equal in length. 
4. An angle inscribed (by diameter)in a Semi-circle is a right angle. <A HREF=http://www.algebra.com/algebra/homework/Circles/Theorems-on-Triangle-and-Circle.lesson>Theorem</A>

<A HREF=Tangent.wikipedia>Tangent</A>

Tangent is a line which touches the circle at only one point(It doest not cross the circle).

{{{drawing(160,160, 0,6,0,6,circle(3,3,2), line(5.05,0,5.05,6),circle(3,3,.1), locate(5.3,1,P), locate(5.3,5.5,Q), locate(5.3,3,H))}}}

In the above figure, Line PQ is the tangent at the point H on circle.

Properties of the Tangent
1. The line drawn perpendicular to the end point of a radius is a tangent to 
the circle.
{{{drawing(160,160, 0,6,0,6,circle(3,3,2), line(5.05,0,5.05,6), line(3,3,5,3),circle(3,3,.1), locate(5.3,1,P), locate(5.3,5.5,Q), locate(4,3,r), locate(5.3,3.2,H))}}}

2. Two tangents can always be drawn from a point outside of the circle.
{{{drawing(160,160, 0,6,0,6,circle(3,3,2), line(5,5,5,3), line(3,5,5,5),circle(3,3,.1), locate(3,4.7,P), locate(5.3,5.5,Q), locate(5.3,3.2,H))}}}

3. Tangents drawn from a point outside to the circle are equal in length.

Length of the tangent is the distance between the point of origination 
   of tangent(outside the circle) and the point of contact to the circle. 
   i.e in the above diagram the length of the tangents are PQ and QH.

<A HREF=Locus_%28mathematics%29.wikipedia>Locus</A>
In general, locus is set of all the points satisfying the certain given conditions.

"Locus of a circle" is the set of all the points which satisfies the definition 
of the circle. i.e. All the points have a same distance(radius)from a fixed
point(center)(conditions).

Locus of all the points have a same distance('r')from a fixed point(a,b)is given 
by the following equation:

{{{(x-a)^2+(y-b)^2=r^2}}}
Which is nothing but the equation of a "Circle" in "<A HREF=Cartesian_coordinate.wikipedia>Cartesian Coordinate System</A>".

Similarly, the locus of the circle in the "<A HREF=Polar_coordinate.wikipedia>Polar Coordinate System</A>" with center 
at ({{{r[o]}}}, {{{rho}}}) and radius "R" is:

{{{r^2-2*r*ro*Cos(theta - rho)+(ro)^2=R^2}}}

For more information on Circles refer to lesson on <A HREF=http://www.algebra.com/algebra/homework/Circles/Basic-formulas-in-a-circle.lesson>basic formulas in circle</A> and <A HREF=Circle.wikipedia>wikipedia</A>.