Circle Theorem

Remember to look for “basics”•Angles in a triangle sum to 1800

•Angles on a line sum to 1800

•Isosceles triangles (radius)•Angles about a point sum to

3600

Name parts of a circle

Diameter

radius

chord

tangent

400

800

THEOREM 1: ANGLE at the CENTRE of the CIRCLE is twice the angle at the circumference subtended by the same arc

MUST BE THE

CENTER

This rule can be hard to spot…..

THIS IS THE ONE MOST PEOPLE DON’T SEE......

1150

2300

MUST BE THE

CENTER

400800

LOOKS DIFFERENT BUT STILL

THE CENTRE

SPECIAL CASE OF THE SAME RULE……… BUT MAKES A RULE IN ITS OWN RIGHT!!

900

1800

THEOREM 2: Every angle at the circumference of a SEMICIRCLE, that is subtended by the diameter of the semi-circle is a right angle.

900

THEOREM 3: Opposite angles sum to 180 in a cyclic quadrilateral

CYCLIC QUADRILATEARAL MUST touch the circumference at all four vertices

910

890

700

1100

Now have a go at Worksheet 1

RULE 4: Angles at the circumference in the same SEGMENT of a circle are equal

PALE BLUE AREA IS THE SEGMENT

PALE BLUE AREA IS THE SEGMENT

RULE 4: Angles at the circumference in the same SEGMENT of a circle are equal

THEOREM 4: Angles at the circumference in the same SEGMENT of a circle are equal

NOTE: Will lead you to SIMILAR triangles (one is an enlargement of the other….)

Enter the world of tangents and chords…..• A tangent is a line that touches a circle at one point only. This point is called the point of contact.

• A chord is a line that joins two points on the circumference.

chord

tangent

Theorem 5 – A tangent is perpendicular to a radius

radius

tangent

900

Theorem 6 – Tangents to a circle from the same point are equal in length

Theorem 7 – The line joining an external point to the centre of a circle bisects the angle between the tangents

700350

350

Theorem 5&7 – combined can help you find the missing angles…..

700350

350

900

900

xy

Theorem 8 – A radius bisects a chord at 900

radiuscho

rd

900

And the chord will be cut perfectly in half!!!

MIDPOINT OF THE CHORD

Have a go at worksheet 2

Theorem 9 – Alternate angle theoremNeed a tangent

And a triangle that joins the tangent and two other points on the circumference of the circle

Theorem 9 – Alternate angle theorem

Theorem 9 – Alternate angle theoremThe angle between a tangent and a chord

Is equal to the angle in the alternate segment

Theorem 9 – Alternate angle theoremThe angle between a tangent and a chord

Is equal to the angle in the alternate segment

COMMON EXAM ERROR!IS TO THINK

THIS IS A DIAMETER –

SO..

THIS MUST BE 900 –

“TANGENT MEETS

RADIUS”

IT IS ONLY A DIAMETER IF YOU ARE TOLD SO…

READ QUSETIONS CAREFULLY..