The Perpendicular from the Centre to a Chord
Simple Explanation
If a perpendicular line is drawn from a circle's centre to a chord, it always bisects that chord — cutting it into two exactly equal halves.
Why Do We Need It?
This theorem is the standard tool for finding a chord's length (or the distance from the centre) using the Pythagorean theorem, by turning the situation into a right triangle with known radius.
See It
A circle with centre O, a chord AB, and a perpendicular segment from O meeting the chord at its midpoint M
Formula
The Perpendicular from the Centre to a Chord
If OM ⊥ AB (M on chord AB, O the centre), then AM = MB
A perpendicular line drawn from a circle's centre to any chord always bisects that chord exactly in half.
- O
- — the centre of the circle
- M
- — the point where the perpendicular from O meets the chord AB
- AM, MB
- — the two halves of the chord created by M
When to use it: Whenever a perpendicular is drawn from a circle's centre to a chord, letting you conclude the chord is bisected — often the key step in finding a chord's length from the radius and its distance from the centre.
Worked Example
Find a chord's length using the perpendicular distance
A circle has radius 13. A chord is 5 units from the centre. Find the chord's length.
Why Does This Work?
Stated simply: OA and OB are both radii, so OA = OB. Triangles OMA and OMB share the side OM and have equal hypotenuses (OA = OB) and both contain a right angle at M — this makes them congruent (RHS: right angle, hypotenuse, shared side), so AM = MB, meaning M truly bisects the chord.
Real-Life Example
Finding the width of a circular tunnel opening
An engineer measures the distance from a circular tunnel's centre to a straight support beam (a chord of the circular cross-section) and needs the beam's full length.
Using the known radius and the perpendicular distance, the same right-triangle method as this theorem instantly gives the beam's exact length.
Practice
A circle has radius 10. A chord is 6 units from the centre. Find the chord's length.
MediumCommon mistake
Forgetting to double the half-chord length found from the right triangle — the Pythagorean theorem step only finds HALF the chord (from the centre-foot M to one endpoint); the full chord is twice that.
Quick Review
- A perpendicular from the centre to a chord always bisects the chord.
- Proof idea: congruent right triangles (RHS), using two equal radii.
- Sets up a right triangle: radius² = (distance from centre)² + (half-chord)².