Circle

Circle is defined as the set of points that is at an equal distant from the centre of the circle.

There are a number of terminologies involved in a Circle. Some of them are as follows:

Centre: The predetermined point from which the surface of the circle is at an equidistant is called the centre of a circle.

Radius: The constant distance from the centre to a point on the surface of the circle is called its radius .

Circumference: The boundary of a circle is called its circumference.

Chord: A line segment whose end points is present on the circumference of a circle is called a chord .

Diameter: A chord crossing through the midpoint of a circle is called its diameter.

Diameter of a Circle: Diameter = 2 X Radius

Radius of a Circle: Radius(R) = Diameter / 2

Area of a Circle: Area = pi X R²

Circumference of a Circle: Circumference = 2 X pi X R

Theorem 1: A perpendicular from the centre of a circle to a chord bisects the chord.

Given : AB is a chord in a circle with centre O. OC ⊥ AB.

To prove: The point C bisects the chord AB.

Construction: Join OA and OB

Proof: In triangles OAC and OBC,

m∠OCA = m∠OCB = 90 (Given)

OA = OB (Radii)

OC = OC (common side)

∠OAC = ∠OBC (RHS)

CA = CB (corresponding sides)

The point C bisects the chord AB.

Hence the theorem is proved.

Theorem 2: AB and CD are equal chords of a circle whose centre is O. OM ⊥ AB and ON ⊥ CD. Prove that m∠OMN = m∠ONM.

Given : In a circle with centre O chords AB and CD are equal

OM ⊥ AB, ON ⊥ CD (Fig.6.11).

To prove : ∠OMN = ∠ONM

Proof : AB = CD (given)

OM ⊥ AB (given); ON ⊥ CD (given)

OM = ON (equal chords equidistant from the centre)

In triangle OMN,

m∠OMN = m∠ONM ( Δ OMN is isosceles)

Hence Proved.