The Chords Of A Circle Which Are Equidistant From The Centre Are Equal

In geometry, circles are one of the most fundamental and fascinating shapes, and their properties often appear in various mathematical problems and proofs. Among these properties, the relationship between chords and their distance from the center of the circle is particularly elegant. It is known that the chords of a circle which are equidistant from the centre are equal in length. This statement is not just a theoretical observation it can be proved mathematically and applied in problem solving, construction, and reasoning about symmetrical figures. Understanding why this property is true gives deeper insight into the symmetrical nature of circles and the role of radii, perpendicular bisectors, and distances in geometric relationships.

Understanding the key terms

Before exploring the proof and reasoning, it is important to clarify the meaning of the terms involved

• CircleA set of all points in a plane that are at the same distance from a fixed point called the centre.
• ChordA straight line segment whose endpoints both lie on the circle.
• CentreThe fixed point from which all points on the circle are equidistant.
• Equidistant from the centreTwo or more objects are said to be equidistant from the centre if the perpendicular distances from the centre to each object are the same.

Restating the property

The statement The chords of a circle which are equidistant from the centre are equal means that if two chords have the same perpendicular distance from the centre of the circle, then their lengths will be the same. This is an important result in Euclidean geometry and is often used in proofs involving circles and symmetry.

Visualizing the concept

Imagine a circle with centre O. Draw two chords AB and CD in such a way that the perpendicular distance from O to AB is the same as the perpendicular distance from O to CD. Intuitively, because the circle is perfectly symmetrical in all directions, these two chords must have the same length. The formal proof confirms this intuition through logical reasoning.

Mathematical proof

Let us prove the statement step-by-step.

Given

• Two chords AB and CD of a circle with centre O.
• The perpendicular distances from O to AB and from O to CD are equal.

To prove

AB = CD

Construction

Draw perpendiculars OM and ON from O to AB and CD respectively. Let M and N be the midpoints of AB and CD, as the perpendicular from the centre of a circle to a chord bisects the chord.

Proof

• In right triangle OMB

  OM is given (distance from O to AB), OB is the radius of the circle.

  MB = √(OB² − OM²)

• In right triangle OND

  ON is given (distance from O to CD, equal to OM), OD is the radius of the circle.

  ND = √(OD² − ON²)

• Since OB = OD (both are radii) and OM = ON (given),

  MB = ND

• Therefore, AB = 2 Ã MB and CD = 2 Ã ND, so AB = CD.

Key ideas behind the proof

The proof relies on two important facts about circles

• The perpendicular from the centre to a chord bisects the chord.
• All radii of a circle are equal in length.

Using these facts, the congruence of segments in right triangles leads directly to the conclusion that the chords must be equal.

Applications in problem solving

This property is useful in many geometric problems. Here are a few examples

• Finding unknown chord lengthsIf the distance from the centre is given, the chord length can be calculated directly.
• Determining equalityIf two chords are equidistant from the centre, you can immediately state that their lengths are equal without further measurement.
• Circle constructionsIn compass and straightedge constructions, this property helps in creating symmetrical designs and equal segments.

Example problem

In a circle of radius 10 cm, two chords are each 6 cm away from the centre. What is the length of each chord?

Solution

• Radius r = 10 cm
• Perpendicular distance from centre to chord d = 6 cm
• Half chord length = √(r² − d²) = √(10² − 6²) = √(100 − 36) = √64 = 8 cm
• Full chord length = 2 Ã 8 = 16 cm

Therefore, both chords have a length of 16 cm, confirming the property.

Relation to symmetry of the circle

This property is a direct consequence of the symmetry of a circle. A circle is symmetrical about any diameter, meaning that if two chords are at the same perpendicular distance from the centre, they are mirror images of each other with respect to the horizontal or vertical line passing through the centre at that distance.

Reverse statement

Interestingly, the converse of this property is also true if two chords in a circle are equal in length, they are equidistant from the centre. This can be proved using the same perpendicular bisector logic in reverse.

Common misconceptions

• Equidistant does not mean equal radiusThe distance from the centre to the chord is different from the radius unless the chord is a diameter.
• Chords not passing through the centreOnly perpendicular distances are considered; slanted distances do not determine equality.
• Applies only in the same circleThis property works within a single circle, not across different circles of varying radii.

Real-world analogy

Imagine a circular pizza. If you cut two straight slices (chords) at the same distance from the centre, they will have the same length of crust along the edge. This is because the geometry of the circle ensures uniformity.

The property that the chords of a circle which are equidistant from the centre are equal beautifully illustrates the harmony and balance of circular geometry. It stems from fundamental definitions, right triangle relationships, and the constant nature of a circle’s radius. This principle is not only a key part of theoretical geometry but also has practical applications in design, engineering, and construction where precision and symmetry matter. Understanding and applying this property equips learners with a deeper appreciation for the logical structure of mathematics and the elegance of geometric forms.