Take a paper circle. Fold the circle from the boundary, inwards. Open the fold. The crease is now a chord (see the figure B). Now fold the paper again, so that the end points of the chord meet. Open the fold (see the figure C).
Measure the lengths of the parts into which the chord is divided. The chord gets bisected where the folds intersect. Measure the angle between the creases. The crease of the second fold is along the perpendicular from the centre to the chord. Measure the distance from the centre to the midpoint of the chord. It is the distance from the centre to the chord.
Now draw another chord of the same length. How will you do this? We will let you figure this out yourself. Join the centre to the midpoint of the new chord and measure its length. Is it the same as distance from the centre to the first chord?

Step-by-step solution

Idea: The second fold puts one end of the chord on the other, so the crease is the perpendicular bisector of the chord, and it passes through the centre. A chord of the same length is just the first chord turned about the centre, so its perpendicular from the centre has the same length too.

1. Measuring the parts: for example, in a circle of radius 5 cm with a crease-chord of 8 cm, the two parts measure 4 cm and 4 cm: the chord is bisected.½ mark
2. Angle between the creases: it measures 90°. The second crease is the perpendicular bisector of the chord (folding one end onto the other), so it passes through the centre.½ mark
3. Distance of the chord: measure from the centre to the midpoint. With radius 5 cm and chord 8 cm, this is 3 cm (since 3² + 4² = 5²).½ mark
4. Drawing another chord of the same length: open the compasses to the length of the first chord (8 cm). Put the compass point at any point R on the circle and draw an arc cutting the circle at S. Then RS = 8 cm is a chord of the same length.1 mark
5. Comparing: join the centre to the midpoint of RS and measure it: again 3 cm. Yes, it is the same. Equal chords are at equal distances from the centre (the circle can be turned about its centre to carry one chord onto the other).½ mark

Answer to write in the exam

Common mistakes that cost marks

• Measuring the distance from the centre to an end of the chord. That is the radius; the distance to a chord is measured along the perpendicular, to its midpoint.
• Drawing the “same length” chord by eye. Use compasses opened to the exact chord length.
• Making the first fold through the centre. Then the crease is a diameter and the distance is 0; fold off-centre to get a general chord.

How this can come in the exam

Try one yourself

A circle has radius 14.5 cm. A chord of 21 cm is drawn. How far is it from the centre? How far is any other 21 cm chord?

More questions like this

• Chords of a circle having the same length are all at the same distance from the centre of the circle.
• Use the Baudhāyana–Pythagoras theorem to show why chords of a circle having the same length are all at the same distance from the centre of the circle.
• Consider the figure. If CE is perpendicular to AB, CH is perpendicular to GH, and CE = CH, show that AB = GF.
• If CE is perpendicular to AB, CH is perpendicular to GH, and CE = CH, show that AB = GF. Solve this using the Baudhāyana–Pythagoras theorem.
• You have two chords on a circle. One is longer than the other. Which chord is closer to the centre? Can you guess?
  Activity: Draw a circle. Draw chords of various lengths. Drop a perpendicular to each chord from the centre. Record the length of the chord and its distance from the centre in a table.
  What do you observe?

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