Equal Chords Theorem

The Equal Chords Theorem states that:

In the same circle or in congruent circles, equal chords subtend equal arcs and lie at the same distance from the centre.

This theorem helps link chord length, arc length, and central angle.

If two chords have the same length, then the arcs they cut off from the circle are also equal.

For chords \(AB\) and \(CD\):

AB = CD \Rightarrow \overset{\frown}{AB} = \overset{\frown}{CD}

If \(AB\) and \(CD\) are equal chords, then their perpendicular distances from the centre are the same.

If \(OM \perp AB\) and \(ON \perp CD\), then:

AB = CD \Rightarrow OM = ON

The converse of the theorem is also true. If the arcs of two chords are equal, or if the chords are equidistant from the centre, then the chords must be equal.

Imagine two different chords inside the same circle. If both chords are equally long, they will look symmetric in the diagram, at the same distance from the centre, and will subtend arcs of the same size.

In a circle of radius \(10\text{ cm}\), two chords have lengths:

• Chord \(AB = 12\text{ cm}\)
• Chord \(CD = 12\text{ cm}\)

Then:

\overset{\frown}{AB} = \overset{\frown}{CD}

OM = ON

This theorem is helpful in:

• Wheel and gear symmetry designs
• Engineering layouts involving circular parts
• Art, mandala patterns, and circle-based geometric designs