Circle Theorem Remember to look for “basics” Angles in a triangle sum to Angles on a line sum to Isosceles triangles (radius) Angles about a point sum to 360 0

Name parts of a circle Diameter radius chord tangent

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 MUST BE THE CENTER

LOOKS DIFFERENT BUT STILL THE CENTRE

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

THEOREM 2: Every angle at the circumference of a SEMICIRCLE, that is subtended by the diameter of the semi-circle is a right angle. 90 0

THEOREM 3: Opposite angles sum to 180 in a cyclic quadrilateral CYCLIC QUADRILATEARAL MUST touch the circumference at all four vertices

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

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 90 0

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

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

Theorem 8 – A radius bisects a chord at 90 0 radius chord 90 0 And the chord will be cut perfectly in half!!! MIDPOINT OF THE CHORD

Have a go at worksheet 2

Theorem 9 – Alternate angle theorem Need a tangent And a triangle that joins the tangent and two other points on the circumference of the circle

Theorem 9 – Alternate angle theorem

The angle between a tangent and a chord Is equal to the angle in the alternate segment

Theorem 9 – Alternate angle theorem The 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 90 0 – “TANGENT MEETS RADIUS” IT IS ONLY A DIAMETER IF YOU ARE TOLD SO… READ QUSETIONS CAREFULLY..