Circle Geometry

There are more rules about a circle than you would expect. Learn them all though and you’ll be able to outfox the examiner.

A Tangent And A Radius Meet At 90°

tangent and radius form a right angle

A tangent is a line that touches one point on the circumference of the circle. The radius is just a straight line from the centre of the circle. When the two meet they always form a right angle.

Two Radii Form An Isosceles Triangle

Two radii form an isosceles triangleBe careful though, they might not have the tick marks on the side like here. The examiner will expect you to know it’s an isosceles triangle, and that the two red angles marked are the same, just because two radii are involved.

The Perpendicular Bisector Of A Chord Passes Through The Centre Of A Circle

The perpendicular bisector of a chord
A chord is just a line drawn across a circle. Wherever it is though if you cut it in half with a perpendicular line, that line will go through the centre.

Angle At The Centre Is Twice The Angle At Circumference

The angle at the centre of a circle is double that at the circumference

From the same two points on a circle, the angle made at the centre will be twice that made at the circumference.

Angle In A Semicircle is Always 90°

Angle in a semicircle is 90

Angles In The Same Segment Are Equal

Angles in the same segment are equalAny triangles drawn from a chord will have the same angle where they touch the circumference.

Opposite Angles in A Cyclic Quadratic Add Up To 180°

Opposite angles in a cyclic quad add up to 180In a four sided shape where every corner touches the circumference the opposite angles will add up to 180°.

Tangents From The Same Point Are The Same Length

Tangents from the same point are the same lengthAlternate Segments Are Equal

Alternate segment theoremThe angle between a tangent and a chord is equal to the angle in the opposite segment (that means the angle made at the circumference by two lines drawn from the chord).

Phew! OK this is pretty hard A* stuff. You will probably have to use a number of these rules to figure out some angles in a question. Ask your teacher for lots of examples.