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**Coursework Notes - Shape & Space**

**Circles**

parts |

__The parts of a circle__

**centre** - the point within the circle where the distance to points on the circumference is the same.

**radius** - the distance from the centre to any point on the circle. The diameter is twice the radius.

**circumference**(perimeter) - the distance around a circle.

**chord** is a straight line joining two points on the circumference.

**diameter **- a chord(of max. length) passing through the centre

**sector** - a region enclosed by two radii and an arc.

**segment** - the region enclosed by a chord and an arc of the circle.

**tangent** - a straight line making contact at one point on the circumference, such that the radius from the centre is at right angles to the line.

__Subtended angles __

When a chord subtends an angle on the circumference of a circle, the angle subtended at the centre of the circle is twice the angle.

A diameter subtends a right-angle at the circumference.

**angle XPZ = 90 deg. **

Angles subtended by a chord onto the circumference of a circle are equal.

**angle ADB = angle ACB **

__Chords__

The line joining the centre of a circle and the mid-point of a chord is perpendicular to the chord.

The chord is bisected into two equal halves.

**XP = PY **

__Tangents__

The tangents to a circle from a point are equal in length.

**AP = BP **

also,

the tangents subtend equal angles at the centre of the circle

**angle POA = angle POB **

and,

the angles between the tangents and the line joining the centre of the circle and the point are equal.

**angle APO = angle BPO **

**note** : Triangle APO and triangle BPO are congruent.

__The angle between a tangent and a chord__

The angle between a tangent and a chord is equal to the angle subtended by the chord in the opposite segment.

**angle ZPY = angle PXY **

__Cyclic quadrilaterals__

**Opposite** angles in a **cyclic** quadrilateral add up to **180 deg**.

As with all quadrilaterals, the **sum of the interior angles = 360 deg**.

Any **exterior angle** of a cyclic quadrilateral **equals the interior opposite angle**.

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