Circle geometry (or circle theorems) involves looking at angle properties when lines and shapes are drawn in circles. Once you can identify and label parts of a circle, it is possible to add lines, triangles and quadrilaterals inside a circle. Circle theorems can help find missing angles.

Circle properties

You should be able to label parts of a circle.

NAME

DESCRIPTION

ILLUSTRATION

Radius

A line from the centre to the edge of the circle.

Diameter

A line across the circle, going through the centre.

Circumference

The perimeter of the circle.

Arc

A section of the circumference.

Chord

A line segment joining two points on the circumference.

Sector

Part of a circle bound by an arc and two radii.

Segment

Part of a circle bound by a chord and an arc.

Circle theorems

There are eight circle theorems that you should learn.

THEOREM

DESCRIPTION

ILLUSTRATION

Angles in the same segment are equal.

$\angle CAD = \angle CBD$

The angle at the centre is twice the angle at the circumference.

$\angle COB = 2 \times \angle CAB$

The angle in a semicircle is $90\degree$.

$\angle ACB = 90\degree$

The radius and tangent are perpendicular.

$\angle CAP = 90\degree$

The perpendicular bisector of a chord goes through the centre.

$AB \ \perp \ OC$

Opposite angles in a cyclic quadrilateral add up to $180\degree$.

$\angle a + \angle b = 180\degree$

$\angle c+ \angle d = 180\degree$

Two tangents from the same point are the same length.

$AC = BC$

The alternate segment theorem.

The angle at the tangent is equal to the angle in the alternate segment.

Example 1

In the circle shown, $\angle AOD=120\degree$. Find $\angle ABD$, $\angle DBC$ and $\angle DOC$.

$\angle AOD$is twice the angle at the circumference $\angle ABD$.

$\begin{aligned}\angle AOD &= 2 \times \angle ABD \\\angle ABD &= \dfrac {120}{2}\\\angle ABD &= \underline{60\degree}\end{aligned}$

$\angle ABC = 90\degree$as it is the angle in a semicircle. Therefore $\angle ABD$ and $\angle DBC$ add up to $90\degree$.

The alternate segment theorum states that the angle at the tangent is the same as the angle in the alternate segment.

What is an arc?

An arc is a section of the circumference.

What is a cyclic quadrilateral?

A cyclic quadrilateral is a quadrilateral that is inside a circle, where the vertices of the quadrilateral touch the circumference of the circle. Opposite angles in a cyclic quadrilateral add up to 180 degrees.

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