How far is a chord from the centre?

Fold a paper circle to make a chord, then fold its two ends together. The second crease is the perpendicular from the centre, and it meets the chord at its midpoint. Turning a copy shows that equal chords seem equally far from the centre.

But examples are not a proof. Theorem 6: chords of the same length are the same distance from the centre. For AB = FG with midpoints E and H, ΔCAB ≅ ΔCFG (SSS), so their heights CE and CH are equal. (Or: ΔCEA ≅ ΔCHF by RHS.)

Theorem 7, the converse: chords that are equally far from the centre are equal.

The working formula. For any chord: half-chord² + distance² = radius². So a chord at distance d in a circle of radius r has length 2√(r² – d²).

Theorem 8: the longer chord is closer to the centre.

• the diameter, at distance 0, is the longest chord;
• push a chord away from the centre and it shrinks;
• at distance r it is just a single point.

Careful: distance and length are not in simple proportion. Doubling the distance does not halve the length; use the formula.