Same distance, same length
Equidistant chords are equal
The other way round
Now turn it around. Take two chords that are the same distance from the centre.
Two chords of a circle are the same distance from the centre. Are they equal in length?
What this lesson covers
The idea
Chords of a circle that are equidistant from the centre have equal length.
The other way round
Now turn it around. Take two chords that are the same distance from the centre.
Two chords of a circle are the same distance from the centre. Are they equal in length?
- Yes, equal
- No, they can differ
- Only if they are parallel
Match the distances
Chord AB is fixed. Drag D and E until DE is as far from O as AB. Then read the lengths.
AB = DE
Chords of a circle that are equidistant from the centre have equal length.
OM = ON, OA = OD, and there are right angles at M and N. So triangles OMA and OND are congruent (RHS), and AM = DN. M and N are midpoints, so AB = 2 × AM = 2 × DN = DE.
Notes
Chords that are equidistant from the centre of a circle have equal length.
Check yourself
Two chords of a circle are each 4 cm from the centre. One chord is 6 cm long. How long is the other, in cm?
Answer: 6 cm
Chords at the same distance from the centre are equal, so the other chord is 6 cm long too.
Two chords of one circle are both 2 cm from the centre. Which statement is true?
A circle has radius 13 cm. Two chords are each 5 cm from the centre. How long is each chord, in cm?
Answer: 24 cm
Half the chord is √(13² − 5²) = √144 = 12 cm. Each chord is 2 × 12 = 24 cm.
Chord AB is longer than chord DE in the same circle. Can they be the same distance from the centre?
- The one on the left is longer. Their positions do not matter. Equal distances from the centre mean equal lengths.
- They have equal length — correct. Yes. Chords that are equidistant from the centre are equal.
- We cannot tell without measuring them. We can tell: chords at the same distance from the centre are equal.
- Yes, if they are parallel. Being parallel does not matter. Chords at the same distance from the centre are always equal.
- Yes, if both are inside the circle. Every chord is inside the circle. Chords at the same distance from the centre are always equal.
- No, chords at the same distance are equal — correct. Yes. If they were the same distance from the centre, they would be equal.