Geometry
Circles
Use the chord, arc and cyclic-quadrilateral facts to find angles and lengths in a circle.
Every point of a circle is the same distance from its centre, and almost everything else about circles is squeezed out of that one fact. It is why the perpendicular from the centre cuts a chord exactly in half, why equal chords sit equally far from the centre, and why an angle at the centre is always double the angle standing on the same arc at the edge.
Double at the centre
Pick an arc. The angle it makes at the centre is twice the angle it makes at any point on the rest of the circle. Two consequences follow immediately: every angle standing on the same arc is the same size, and an angle in a semicircle is 90°, because the angle at the centre there is a straight 180°.
Worked example
A chord is 24 units long in a circle of radius 13 units. How far is it from the centre?
Drop a perpendicular from the centre to the chord. It cuts the chord in half, into two pieces of 12.
That is the chord theorem, and it is what turns this into a right-angled triangle.
Try it together
In cyclic quadrilateral ABCD, angle A is 85° and angle B is 70°. Find angles C and D.
The corners are named in order round the circle, so A is opposite C and B is opposite D.
1.Opposite angles of a cyclic quadrilateral add up to how many degrees?
Watch out
The doubling rule needs both angles standing on the same arc. An angle at the circumference on the minor arc and one on the major arc are not equal — they add up to 180°, which is exactly the cyclic quadrilateral theorem wearing a different hat.
Have a go
Have a go on your own: an angle at the centre is 140°. What is the angle at the circumference on the same arc, in degrees?
Hint: Halve it.
Ready to practise?
Eight questions on what you have just read. Nothing is timed, and you can play as many times as you like.