Circle :

• A circle is set of points equidistant from a fixed point in a plane.
• The fixed point is called the centre of the circle and the fixed distance is called as the radius.
• The radius is also the segment joining the centre and any point on the circle.
• A chord is a segment which joins any two points on the circle.
• A diameter is the chord which passes through the centre.
• An arc is a part of the circle. The measure of an arc is equal to the measure of its corresponding central angle.

There are three types of arc : Minor arc, Major arc and Semicircular arc or Semicircle.

• Measure of a circle is 360°.
• Measure of a semicircle is 180°.
• Measure of a minor arc is less than 180°.
• Measure of major arc is more than 180°.

Example :

In the figure O is the centre of the circle.

Segment Types:

• Radius: A segment from the centre to the edge (e.g., Seg OD).
• Chord: A segment connecting two points on the circle (e.g., Seg PQ).
• Diameter: A chord that passes through the centre of the circle (e.g., Seg AB).

Arc Classifications:

• Minor Arc: An arc smaller than a semicircle (e.g., arc AXD, arc BD).
• Major Arc: An arc larger than a semicircle (e.g., arc PAB, arc PDQ).
• Semicircular Arc: An arc representing half the circle (e.g., arc ADB).

Measurement Rules:

The measure of a minor arc is equal to the measure of its central angle.

The measure of a major arc is calculated as: 360° - measure of the corresponding minor arc.

Properties of chord of a circle :

Property 1 : The perpendicular drawn from the centre of a circle to its chord bisects the chord.

In the figure, O is the centre of the circle, seg AB is the chord and seg OP ⊥ chord AB then

l(AP) = l(PB) =  l(AB).

Property 2 : The segment joining the centre of a circle and  midpoint of its chord is perpendicular to the chord.

In the figure, O is the centre of the circle, seg AB is the chord and P is the midpoint of chord AB then seg OM ⊥ chord AB.

i.e. m∠ OPA = m∠ OPB = 90°.

Arcs corresponding to the chord of a circle :

In the figure, seg AB is a chord of a circle with centre O. Arc AXB is minor arc and arc AYB is a major arc. These two arcs are called corresponding arcs of chord AB. Moreover chord AB is called corresponding chord of arc AXB and arc AYB.

Congruent arcs :

If the measures of two arcs of circle are same then two arcs are congruent.

In the circle with centre O

∴ m∠ AOB = m∠ COD

∴ m(arc AXB) = m(arc CYD)

∴ arc AXB ≅ arc CYD

Property 1: The chords corresponding to congruent arcs are themselves congruent.

Property 2: If two chords in a circle are congruent, then their corresponding minor and major arcs are also congruent.