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Chord and Its Properties

A chord is a line segment joining two points on a circle. The perpendicular from the centre of a circle to a chord bisects the chord, and equal chords are equidistant from the centre.

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Student-friendly explanation

The centre gives strong information about chords. If a radius from the centre meets a chord at 90 degrees, it cuts that chord into two equal parts. Also, if two chords are equal in length, their perpendicular distances from the centre are equal.

How to write this in exams

  1. 1

    Start with the exact idea

    A chord is a line segment joining two points on a circle. The perpendicular from the centre of a circle to a chord bisects the chord, and equal chords are equidistant from the centre.

  2. 2

    Then show how to use it

    Mark the centre. Draw or identify the perpendicular to the chord. Use bisection to split the chord equally. For equal chords, compare their perpendicular distances from the centre.

  3. 3

    Add one concrete example

    If OM is perpendicular to chord AB and M lies on AB, then AM = MB. If chords AB and CD are equal, then their distances from O are equal.

  4. 4

    Avoid this incomplete answer

    Writing AM = 12 cm when AB = 12 cm is wrong because AM is only half the chord after perpendicular bisection.

Definition

A chord is a line segment joining two points on a circle. The perpendicular from the centre of a circle to a chord bisects the chord, and equal chords are equidistant from the centre.

Example

If OM is perpendicular to chord AB and M lies on AB, then AM = MB. If chords AB and CD are equal, then their distances from O are equal.

Rule to remember

If OM perpendicular AB, then AM = MB. If AB = CD, then distance of AB from centre = distance of CD from centre.

Memory hook

Perpendicular from centre cuts chord into two equal pieces.

Examples and method

Worked example

Given AB = 16 cm and OM perpendicular AB at M. Since perpendicular from centre bisects chord, AM = MB = 8 cm.

Method to apply

Mark the centre. Draw or identify the perpendicular to the chord. Use bisection to split the chord equally. For equal chords, compare their perpendicular distances from the centre.

Diagram support

Draw circle with centre O, chord AB, perpendicular OM meeting AB at M, and mark AM = MB.

How CBSE asks it

Questions usually give a chord length and perpendicular distance, then ask for half-chord, full chord, or proof of equality of two chords.

Avoid common mistakes

Common confusion

Students often assume any line from the centre to a chord bisects it. The bisecting result needs the line to be perpendicular to the chord.

Common wrong answer

Writing AM = 12 cm when AB = 12 cm is wrong because AM is only half the chord after perpendicular bisection.

Exam tip

In proof questions, write both facts: perpendicular from centre to chord and hence it bisects the chord. Do not skip the 90 degree condition.

Quick check

If OM is perpendicular to chord AB at M and AB = 12 cm, what is AM?

AM is 6 cm because the perpendicular from the centre to a chord bisects the chord into two equal parts.

Study the chord and its properties diagram carefully

Use the labelled diagram to keep chord and its properties clear in short answers and revision.

What this diagram makes clear

This diagram keeps the labels and direction of chord and its properties in the right order.

Where this helps in exams

Use this for labelled diagram work and short exam answers on chord and its properties.

Revision cue

Revise chord and its properties through the labels before writing the answer.

Answer writing and exam use

1-mark answer

A chord is a line segment joining two points on a circle. The perpendicular from the centre of a circle to a chord bisects the chord, and equal chords are equidistant from the centre.

2-mark answer

A chord is a line segment joining two points on a circle. The perpendicular from the centre of a circle to a chord bisects the chord, and equal chords are equidistant from the centre. If OM perpendicular AB, then AM = MB. If AB = CD, then distance of AB from centre = distance of CD from centre. If OM is perpendicular to chord AB and M lies on AB, then AM = MB. If chords AB and CD are equal, then their distances from O are equal.

3-mark answer

The centre gives strong information about chords. If a radius from the centre meets a chord at 90 degrees, it cuts that chord into two equal parts. Also, if two chords are equal in length, their perpendicular distances from the centre are equal. If OM perpendicular AB, then AM = MB. If AB = CD, then distance of AB from centre = distance of CD from centre. Given AB = 16 cm and OM perpendicular AB at M. Since perpendicular from centre bisects chord, AM = MB = 8 cm. Questions usually give a chord length and perpendicular distance, then ask for half-chord, full chord, or proof of equality of two chords. Writing AM = 12 cm when AB = 12 cm is wrong because AM is only half the chord after perpendicular bisection.
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