Circle theorems
Move a point around the circle and watch the angle at the circumference refuse to change while the centre angle stays exactly double.
- Angle at the centre
- Angles in the same segment
- Angle in a semicircle
Angle at centre
110°
Angle at circumference
55°
Ratio
2.0 : 1
Predict
Angle AOB at the centre is 140°. What is angle APB at the circumference?
Experiment
- 1Set the arc to 120° and move P from one end of the major arc to the other. Watch the angle at P stay fixed.
- 2Now change the arc and watch both angles change together, always in the ratio 2 : 1.
- 3Set the arc to exactly 180° and read the angle at P — you have just derived the angle in a semicircle.
- 4Write down, in the wording an examiner would accept, the reason the ratio is 2 : 1.
Explain
The angle at the centre is twice the angle at the circumference when both stand on the same arc. That is the theorem, and it is the reason a great many circle problems collapse to a single division by two.
Because the angle at P does not depend on where P is, every point on the major arc gives the same angle. That is the second theorem — angles in the same segment are equal.
Set the arc to 180° and AB becomes a diameter, so the angle at the centre is 180° and the angle at P is 90°. The angle in a semicircle is not a separate fact to memorise; it is this theorem with the arc set to a straight line.
In the exam, write the reason next to every angle you find. Most of the marks in a circle-theorem question are for the reasoning, not the number.