Long and close

The idea

Of two unequal chords of a circle, the longer chord is at a smaller perpendicular distance from the centre.

Which is closer?

Two chords of one circle are not equal: one is longer. Drop the perpendicular from the centre to each chord. Its length tells you how far the chord is from the centre.

Which of the two chords is closer to the centre?

• The longer chord
• The shorter chord
• Both are the same distance

Slide the chord

AB stays where it is. Touch the circle to slide the chord DE. It starts as long as AB. Slide it farther from O, then nearer. Read the table.

Longer, closer

Of two unequal chords of a circle, the longer chord is at the smaller perpendicular distance from the centre.

Let AB > DE, with perpendiculars OF and OG from the centre. The perpendiculars bisect the chords, so AF > GD. The radii are equal, so OF² + AF² = OG² + GD². AF² is bigger than GD², so OF² must be smaller than OG². That is, OF < OG.

Notes

Of two unequal chords of a circle, the longer chord is closer to the centre.

Check yourself

AB and DE are chords of one circle, and AB is longer. Which chord is nearer to the centre?

A circle has radius 5 cm. A chord is 4 cm from the centre. How long is the chord, in cm?

Answer: 6 cm

Half the chord is √(5² − 4²) = √9 = 3 cm. The chord is 2 × 3 = 6 cm.

A chord of a circle is moved farther from the centre. What happens to its length?

Chord AB is 2 cm from the centre of a circle. Chord DE of the same circle is 5 cm from the centre. Which statement is true?

• AB — correct. Yes. The longer chord is at the smaller distance from the centre.
• DE. It is the other way round. The longer chord is nearer to the centre.
• They are the same distance. Equal chords are at the same distance. These chords are unequal, so their distances differ.
• It becomes longer. It is the other way round. Chords nearer the centre are longer.
• It becomes shorter — correct. Yes. The farther a chord is from the centre, the shorter it is.
• It stays the same. Only chords at the same distance from the centre are equal. A different distance means a different length.
• DE is longer than AB. DE is farther from the centre, so it is the shorter chord.
• AB and DE are equal. Equal chords are at the same distance from the centre. These two are at different distances.
• AB is longer than DE — correct. Yes. AB is nearer to the centre, so it is longer.