Centre to the middle
Centre to the midpoint of a chord
A line to the middle
Draw a chord AB of a circle and find its midpoint M. Now draw the line from the centre O to M.
How does the line OM meet the chord AB?
What this lesson covers
The idea
The line joining the centre of a circle to the midpoint of a chord is perpendicular to the chord.
A line to the middle
Draw a chord AB of a circle and find its midpoint M. Now draw the line from the centre O to M.
How does the line OM meet the chord AB?
- At a right angle
- At 60°
- At an angle that changes from chord to chord
Measure the angle
M is always the midpoint of AB. Drag A and B round the circle to make different chords, and read ∠OMB.
Always 90°
The line joining the centre of a circle and the midpoint of a chord is perpendicular to the chord.
In triangles OMA and OMB, OA = OB (radii), AM = MB (M is the midpoint) and OM is common. So they are congruent (SSS) and ∠OMA = ∠OMB. These two angles lie along the straight line AB, so they add up to 180°. Each one is 90°.
Notes
The line joining the centre of a circle to the midpoint of a chord is perpendicular to the chord.
Check yourself
M is the midpoint of chord AB of a circle with centre O. What is ∠OMA, in degrees?
Answer: 90 °
The line from the centre to the midpoint of a chord is perpendicular to the chord, so ∠OMA = 90°.
M is the midpoint of chord AB, and O is the centre. Why is ∠OMA = ∠OMB?
M is the midpoint of chord AB of a circle with centre O. AM = 4 cm and OM = 3 cm. What is the radius OA, in cm?
Answer: 5 cm
OM is perpendicular to AB, so triangle OMA is right-angled at M. OA² = 3² + 4² = 25, so OA = 5 cm.
AB is a diameter of a circle with centre O. Where is the midpoint of AB?
- Because AB is a diameter. If AB were a diameter, M would be the centre O. There is no triangle OMA then.
- Triangles OMA and OMB are congruent (SSS) — correct. Yes. OA = OB, AM = MB and OM is common to both triangles.
- Because OM = AM. Nothing says OM and AM are equal.
- Halfway between O and the circle. The midpoint of a diameter is the centre itself.
- On the circle. The ends A and B are on the circle. The midpoint is the centre.
- At the centre O — correct. Yes. A diameter passes through the centre, and the centre is its midpoint.