Chords

Chord theorems connect linear distances, arcs, and angles and support circle constructions, engineering cross-sections, and proofs.

Intuition and core definition

A chord is a segment whose endpoints lie on a circle. A diameter is the longest chord because it passes through the centre. In the same circle, equal chords subtend equal central angles and arcs, and chords equidistant from the centre are congruent.

Notation, language, and conditions

If radius $r$, perpendicular centre-to-chord distance $d$, and chord length $c$ are related, the perpendicular from the centre bisects the chord. A right triangle gives $(c/2)^2+d^2=r^2$, requiring $0\le d\le r$.

Why this idea matters

A radius perpendicular to a chord bisects it, turning chord-length questions into right-triangle calculations.

A dependable method

1. Draw the centre and chord and mark known lengths.
2. Drop a perpendicular from the centre to the chord.
3. Use the theorem that this perpendicular bisects the chord.
4. Apply the Pythagorean theorem to one of the congruent right triangles.
5. Double a half-chord result and compare with diameter $2r$.

Worked example

Representations and interpretation

The centre-to-chord perpendicular creates mirror-image right triangles. Sliding a chord toward the centre lengthens it; at distance zero it becomes a diameter, and near the circle it shrinks toward zero.

Reasoning about variations

A radius to an arbitrary chord endpoint does not generally bisect the chord. The bisection theorem requires a perpendicular from the centre, so the right-angle marking is essential.

Common mistakes

How to check your work

• Confirm $c\le2r$.
• Substitute half-chord, distance, and radius into the Pythagorean relation.
• Use symmetry to verify the perpendicular meets the midpoint.

Practice

1. A radius-$10$ circle has a chord $6$ units from its centre. Find the chord length.
2. What is the longest chord of a circle?
3. If two chords in one circle are equally distant from the centre, what follows?

Answers and brief solutions

Synthesis and transfer

For equal chords in one circle, equal distances from the centre provide a symmetry check; moving a chord closer to the centre must make it longer.

For a circle of radius $13$ and a chord $10$ units from the centre, the perpendicular radius bisects the chord. Half its length is $\sqrt{13^2-10^2}=\sqrt{69}$, so the complete chord is $2\sqrt{69}$. Moving the chord toward the centre decreases the perpendicular leg and increases the half-chord; at distance zero, the chord becomes a diameter. At distance equal to the radius, the chord degenerates to one tangent point. These limiting cases check whether a formula or diagram assigns the right segment to each side of the right triangle.

Related topics

• Circle Terminology
• Pythagorean Theorem
• Inscribed Angles

Teaching and accessibility note