Circle
A Circle is a closed two-dimensional figure with no corners and edges consisting of all the points at an equidistance from a fixed point in a plane.
Parts of a Circle:
| Parts | Definition |
| Centre | The fixed point in which the circle is described is called the centre of the circle. |
| Radius | The equidistant from the centre to any point on the circle is its radius. |
| Chord | The intersecting line having its endpoints on the circle is called a chord of the circle. |
| Diameter | The chord of a circle passing through the centre is called the diameter of the circle. |
| Circumference | The boundary line of the circle is called its circumference. |
Position of a point with respect to a circle:
Consider any point on the circle in a plane, then:
-
If the distance between the centre and the point is equal to the radius of the circle, then the point lies on the circle.
-
If the distance between the centre and the point is less than the radius of the circle, then the point lies inside the circle.
-
If the distance between the centre and the point is greater than the radius of the circle, then the point lies outside the circle.
Therefore, the circle divides the plane into three distinct region in which it lies.
Symmetries of a Circle:
A circle has the maximum number of symmetries among all plane figures.
The circle exhibits both the rotaional symmetry and reflection symmetry.
Rotational symmetry: A circle looks exactly the same when rotated through any angle about its centre. Hence, it has infinite rotational symmetry.
Reflection symmetry: A circle can be divided into two identical halves by any line passing through its centre. Each half is the mirror image of the other.
Lines of symmetry: A circle has infinitely many lines of symmetry because every diameter of the circle acts as a line of reflection symmetry.
Circle through a point
Given a point \(O\), it is always possible to draw infinitely many circles through the given point.
In other words, an infinite number of circles can be drawn through a given point.
Circle through two points
Given any two points, it is always possible to draw infinitely many circles through the given points.
In other words, an infinite number of circles can be drawn, passing through a pair of points.
Circle through three points
Case 1: Collinear points
If the three points are collinear, then it is impossible to draw a circle using all three points.
Case 2: Non-collinear points
If the three points are non-collinear, then it is possible to draw only one circle using all the three points.
Theorem on circle passing through three points:
There is a unique circle passing through three non-collinear points.
Angle Subtended by Chord at the Centre:
Theorem: Equal chords of a circle subtend equal angles at the centre of the circle.
Explanation:
The theorem states that if the chords \(PQ\) and \(RS\) are equal, then the angle subtended by the chords respectively at the centre \(O\) are equal (i.e.) \(\angle POQ = \angle ROS\).
Converse of the theorem: Chords of a circle that subtend equal angles at the centre
are equal.
are equal.
Explanation:
The theorem states that, if \(PQ\) and \(RS\) are two chords subtending equal angle at the centre \(O\) then the chords \(PQ\) and \(RS\) equal (i.e.) \(PQ = RS\).
Theorems on Midpoints and Perpendicular Bisectors of Chords:
Theorem: The line joining the centre of a circle and the midpoint of a chord of the circle is perpendicular to the chord.
Explanation:
The theorem states that if \(O\) is the centre and \(R\) is the mid-point of the chord \(PQ\), the line joining the centre \(O\) and the mid-point \(R\) is perpendicular to the chord \(PQ\).
Converse of the theorem: The perpendicular from the centre of a circle to a chord of a circle bisects the chord.
Explanation:
The theorem states that if \(O\) is the centre and \(AB\) is the chord, then the perpendicular \(OC\) from the centre \(O\) to the chord \(AB\) bisects the chord \(AB\) (i.e.) \(CA = CB\).
Distance of Chords from the Centre:
Theorem: Chords of a circle having the same length are all at the same distance from the centre of the circle.
Explanation:
The theorem states that if the chords \(PQ\) and \(RS\) are equal, then the distance between the chords \(PQ\) and \(RS\) from the centre \(O\) are equal.
\(\text{Distance from }O\text{ to chord }PQ\) \(=\) \(\text {Distance from }O\text{ to chord}\) \(RS\)
Converse of the theorem: Chords of a circle that are equidistant from the centre have equal length.
Explanation:
The theorem states that, if the distance between the chords \(PQ\) and \(RS\) from the centre \(C\) are equal, then the chords \(PQ\) and \(RS\) are equal (i.e.) \(PQ = RS\).