1. Understanding the Perpendicular Bisector Theorem
The Perpendicular Bisector Theorem states that:
The perpendicular drawn from the centre of a circle to a chord bisects the chord.
This means if you draw a right angle from the centre to a chord, it cuts the chord into two equal parts.
2. Statement of the Theorem
If a radius (or a line from the centre) is perpendicular to a chord at point \(M\), then:
AM = MB
Where \(A\) and \(B\) are endpoints of the chord \(AB\).
3. Why the Theorem is True
When the centre \(O\) is joined to the endpoints of the chord \(A\) and \(B\), two triangles are formed:
- \(\triangle OAM\)
- \(\triangle OBM\)
Since:
- \(OA = OB = r\) (radii)
- \(OM = OM\) (common side)
- \(\angle OMA = \angle OMB = 90^\circ\)
The two triangles are congruent, giving:
AM = MB
4. Converse of the Theorem
The converse is also true:
If a line from the centre of the circle bisects a chord, then it is perpendicular to that chord.
So, the direction of the line doesn't matter—bisecting implies perpendicularity.
5. Application in Geometry
This theorem is used to locate the centre of a circle through chords, solve problems involving chord lengths, and prove equalities in diagrams.
5.1. Finding the Centre Using Chords
If two perpendicular bisectors of two different chords are drawn, their intersection gives the centre of the circle.
5.2. Example
In a circle, a chord \(AB\) is \(10\text{ cm}\). If a line from the centre is perpendicular to the chord, then it divides the chord into:
AM = MB = 5\text{ cm}
6. Real-Life Connections
This principle appears in practical situations such as:
- Designing wheels and gears
- Locating the centre of circular plates or disks
- Ensuring symmetry in circular art and patterns