Draw and label on a circle:

Centre

Radius

Diameter

Circumference

Chord

Tangent

Arc

Sector (major/minor)

Segment (major/minor)

a

a

180 - 2a

r

r

Circle Fact 1. Isosceles Triangle

Any triangle AOB with A & B on the circumference and O at the centre of a circle is isosceles.

Circle Fact 2. Tangent and Radius

The tangent to a circle is perpendicular to the radius at the point of contact.

Circle Fact 3. Two Tangents

The triangle produced by two crossing tangents is isosceles.

Circle Fact 4. Chords

If a radius bisects a chord, it does so at right angles, and if a radius cuts a chord at right angles, it bisects it.

Circle Theorem 1: Double Angle

The angle subtended by an arc at the centre of a circle is twice the angle subtended at the circumference.

Circle Theorem 2: Semicircle

The angle in a semicircle is a right angle.

Circle Theorem 3: Segment Angles

Angles in the same segment are equal.

Circle Theorem 4: Cyclic Quadrilateral

The sum of the opposite angles of a cyclic quadrilateral is 180o.

Circle Theorem 5: Alternate Segment

The angle between a chord and the tangent at the point of contact is equal to the angle in the alternate segment.

Circle Theorem 1: Double Angle

The angle subtended by an arc at the centre of a circle is twice the angle subtended at the circumference.

a

ab

b

180 - 2b

180 – 2a

2a + 2b

= 2(a + b)

Circle Theorem 2: Semicircle

The angle in a semicircle is a right angle.

180 – 2a

180 – 2b

a

a

b

b

360 – 2(a + b) = 180

180 = 2(a + b)

90 = (a + b)

Circle Theorem 3: Segment Angles

Angles in the same segment are equal.

a

a

2a

Circle Theorem 4: Cyclic Quadrilateral

The sum of the opposite angles of a cyclic quadrilateral is 180o.

a

b

2a

2b

2a + 2b = 360

2(a + b) = 360

a + b = 180

Circle Theorem 5: Alternate Segment

The angle between a chord and the tangent at the point of contact is equal to the angle in the alternate segment.

a

90 - a

90 - a

180 – 2(90 – a)

180 – 180 + 2a 2a

a

Double Angle Semicircle

Segment Angles Cyclic Quadrilateral Alternate Segment