C
CraftExam
medium difficulty6 min

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.

In one line: Perpendicular from centre cuts chord into two equal pieces.

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Understand it

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.

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.

Formula and rule

If OM perpendicular AB, then AM = MB.

If AB = CD, then distance of AB from centre = distance of CD from centre.

See it worked out

Follow the example line by line, then use the same method on new questions.

Worked example

  1. 1Given AB = 16 cm and OM perpendicular AB at M.
  2. 2Since perpendicular from centre bisects chord, AM = MB = 8 cm.

Method you can reuse

  1. 1Mark the centre.
  2. 2Draw or identify the perpendicular to the chord.
  3. 3Use bisection to split the chord equally.
  4. 4For equal chords, compare their perpendicular distances from the centre.

Sketch it in your answer

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

Check yourself

Answer the question in your head before opening the answer.

Try it yourself

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

Think first, then show answer
AM is 6 cm because the perpendicular from the centre to a chord bisects the chord into two equal parts.

Common wrong answer

Writing AM = 12 cm when AB = 12 cm.

Why it loses marks: AM is only half the chord after perpendicular bisection.

Don't mix this up

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.

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.

Chord and Its Properties diagram

Open full screen to read every label clearly.

Write it in the exam

Each point is worth about one mark. Write point 1 for a 1-mark question and keep adding points as the marks go up.

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.
  1. 1

    Enough for 1 mark

    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

    Add for 2 marks

    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.

  3. 3

    Add for 3 marks

    The centre gives strong information about chords.

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