Equal Chords and Equal Arcs
Simple Explanation
In the same circle (or in circles of equal radius), equal chords are always the same distance from the centre, and they always cut off equal arcs. Any one of "equal chords," "equal distance from centre," or "equal arcs" implies the other two.
Why Do We Need It?
This three-way link between chord length, distance from centre, and arc size is a recurring tool for proving other circle facts and for solving chord-length problems.
See It
A circle with two chords of equal length, one horizontal and one vertical, both the same distance from the centre
Formula
Equal Chords Theorem
Chord AB = Chord CD ⇔ AB and CD are equidistant from the centre ⇔ arc AB = arc CD
In the same circle (or equal circles), equal chords are always the same distance from the centre and always cut off equal arcs — and each of these conditions implies the other two.
- chord
- — a straight segment connecting two points on the circle
- distance from the centre
- — the length of the perpendicular segment from the centre to the chord
When to use it: Whenever you know one of "equal chords," "equal distance from centre," or "equal arcs," and need to conclude the other two.
Worked Example
Use equal chords to find an arc
In a circle, chord AB = chord CD, and arc AB = 74°. Find arc CD.
Why Does This Work?
Stated simply: draw radii to the four chord endpoints. The two triangles formed (each with two radii and a chord) are congruent by SSS (equal radii, equal chords) — so their central angles are equal, which means the corresponding arcs are equal, and the perpendicular distances from the centre (heights of these congruent triangles) are equal too.
Real-Life Example
Designing a circular gear with evenly matched teeth
An engineer designs slots around a circular gear so that certain groups of slots are the same size and placed the same distance from the centre.
Confirming the chords (slot openings) are equal automatically confirms the arcs (spacing) and centre-distances match too — the same three-way equivalence used in circle geometry.
Practice
Chord PQ is 8 cm from the centre, and chord RS is also 8 cm from the centre of the same circle. What can you conclude?
MediumCommon mistake
Assuming this three-way rule applies across circles of different sizes — it only holds within the same circle, or between circles that have exactly the same radius.
Quick Review
- Equal chords ⇔ equal distance from centre ⇔ equal arcs (same circle, or equal circles).
- Proof idea: congruent triangles (SSS, using two radii and the chord) formed for each chord.
- Any one of the three conditions guarantees the other two.