Same length, same distance
Equal chords are equidistant from the centre
Distance from the centre
The distance of a chord from the centre is the length of the perpendicular from the centre to the chord. Spin the wheel and a chord turns with it. The perpendicular to it turns too.
Two chords of a circle are equal in length. Are they the same distance from the centre?
What this lesson covers
The idea
Chords of a circle of equal length are at the same distance from the centre, the distance of a chord being the length of the perpendicular from the centre to it.
Distance from the centre
The distance of a chord from the centre is the length of the perpendicular from the centre to the chord. Spin the wheel and a chord turns with it. The perpendicular to it turns too.
Two chords of a circle are equal in length. Are they the same distance from the centre?
- Yes, the same distance
- No, one is nearer
- Only if they are parallel
Match the chords
Chord AB is fixed. Drag D and E until the chord DE is as long as AB. Then read the distances from O.
OM = ON
Chords of a circle of equal length are at the same distance from the centre. The distance of a chord is the length of the perpendicular from the centre to it.
The perpendiculars OM and ON bisect the chords, so AM = DN (halves of equal chords). With OA = OD and right angles at M and N, triangles OMA and OND are congruent (RHS). So OM = ON.
Notes
Chords of equal length are at the same distance from the centre of the circle.
Check yourself
Chords AB and PQ of a circle are each 8 cm long. AB is 3 cm from the centre. How far is PQ from the centre, in cm?
Answer: 3 cm
Equal chords are at the same distance from the centre, so PQ is 3 cm from the centre too.
What does "the distance of a chord from the centre" mean?
A circle has radius 5 cm. Two equal chords are each 3 cm from the centre. How long is each chord, in cm?
Answer: 8 cm
Half the chord is √(5² − 3²) = √16 = 4 cm. The chord is 2 × 4 = 8 cm.
A wheel with a chord drawn on it is turned about its centre. The chord moves to a new position. What happens to its distance from the centre?
- The length of the line from the centre to one end of the chord. That is a radius.
- The length of the perpendicular from the centre to the chord — correct. Yes. It is the shortest way from the centre to the chord.
- Half the length of the chord. Half the chord is AM. The distance is the perpendicular OM.
- It becomes larger. Turning the wheel does not change the length of the chord, so the distance stays the same.
- It becomes smaller. Turning the wheel does not change the length of the chord, so the distance stays the same.
- It does not change — correct. Yes. The new chord is the same length, so it is the same distance from the centre.