Circle - Introduction

Center of the circle

Radius Diameter

Circumference

Arc

Tangent

Secant

Chord

Circle – Finding length of a Chord

rd

1. A perpendicular drawn from center of

the circle on the chord bisects the

chord.

Hence l(AC) = l(CB)

if mOCB = 900 and O is center of the

circle.

2. By Pythagoras theorem,

[l(OA)]2 = [l(OC)]2 + [l(AC)]2

r2 = d2 + [l(AC)]2

[l(AC)]2 = r2 - d2 l(AC) = Sqrt(r2 - d2)

3. Chord Length = 2 × l(AC)

= 2 × Sqrt(r2 - d2)

A BC

O

dr22

2Length Chord

Circle – Congruent Chords

rd

A BP

O rd C

D If two chords of a circle are of equal length, then they are at equal distance from the center of the circle.i.e. If l(AB) = l(CD) then l(OP) = l(OQ)

Conversely if two chords of a circle are at equal distance from the center, they are of equal length.i.e. If l(OP) = l(OQ) then l(AB) = l(CD)

Q

Circle – Angles subtended at Center byCongruent Chords

r

A BP

Or

C

D If chords AB and CD are of equal length, then angles subtended by them at the center viz. DOC and AOB are congruent.

Conversely if DOC and AOB are congruent, then chords AB and CD are of equal length.

Q

Circle – Central Angle Theorem

The central angle subtended by two points on a circle is twice the inscribed angle subtended by those points.

i.e. mAOB = 2 × mAPB

Circle – Thale’s Theorem

The diameter of a circle always subtends a right angle to any point on the circle

Circle – Cyclic Quadrilateral

A cyclic quadrilateral is a quadrilateral whose vertices all lie on a single circle

In a cyclic simple quadrilateral, opposite angles are supplementary.