1. What is a Circle?
A circle is a simple closed curve in which every point is at the same distance from a fixed point. This fixed point is called the centre of the circle.
The constant distance of any point on the circle from the centre is called the radius.
\(\text{Circle} = \{ P : CP = r \}\)
This means a circle is the set of all points \(P\) whose distance from centre \(C\) is equal to \(r\) (the radius).
2. Centre of a Circle
The centre is the fixed point from which every point on the circle is equidistant. We usually denote it by letters like \(O\) or \(C\).
The entire circle is named using the centre, for example: Circle O.
3. Radius of a Circle
A radius (plural: radii) is a line segment joining the centre of the circle to any point on the circle.
\(OA = OB = OC = r\)
All radii of a circle are equal.
4. Diameter of a Circle
A diameter is a line segment that passes through the centre of the circle and has its endpoints on the circle.
It is the longest chord of the circle.
\(\text{Diameter} = 2r\)
4.1. Relation Between Radius and Diameter
\(d = 2r\)
Every diameter divides the circle into two equal semicircles.
5. Chord of a Circle
A chord is a line segment whose endpoints lie on the circle.
Every diameter is a chord, but not every chord is a diameter.
5.1. Length of Chord (Idea)
For a fixed radius, a chord gets longer as it moves closer to the centre. The longest chord is the diameter.
6. Arc of a Circle
An arc is a portion of the circumference of a circle. It is like a 'curved line segment'.
Arc notation: Minor arc \(AB\), Major arc \(ADB\).
6.1. Minor and Major Arcs
Minor Arc: Smaller arc between points \(A\) and \(B\).
Major Arc: Larger arc between points \(A\) and \(B\).
7. Sector of a Circle
A sector is the region enclosed by two radii and the arc between them. It looks like a 'slice of pizza'.
7.1. Types of Sectors
Minor sector: Smaller region formed by the radii.
Major sector: Larger region formed by the radii.
8. Segment of a Circle
A segment is a region formed by a chord and the arc lying between the chord's endpoints.
8.1. Types of Segments
Minor segment: Smaller region cut off by the chord.
Major segment: Larger region cut off by the chord.
9. Circumference
The circumference is the total length of the boundary of a circle. It is like the perimeter of a circle.
\(C = 2\pi r\)