Circle Theorem Remember to look for basics Angles

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 m a i D radius d r o h c tangent r e et

400 MUST BE THE CENTER 800 THEOREM 1: ANGLE at the CENTRE of the CIRCLE is twice the angle at the circumference subtended by the same arc

This rule can be hard to spot…. .

1150 2300 THIS IS THE ONE MOST PEOPLE DON’T SEE. . . MUST BE THE CENTER

800 400 LOOKS DIFFERENT BUT STILL THE CENTRE

900 1800 SPECIAL CASE OF THE SAME RULE……… BUT MAKES A RULE IN ITS OWN RIGHT!!

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

PALE BLUE AREA IS THE SEGMENT RULE 3: Angles at the circumference in the same SEGMENT of a circle are equal

THEOREM 3: 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…. )

CYCLIC QUADRILATEARAL MUST touch the circumference at all four vertices 890 1100 700 910 THEOREM 4: Opposite angles sum to 180 in a cyclic quadrilateral

Enter the world tangents and • A tangent is a line thatof touches a circle at one point only. This chords…. . point is called the point of contact. tangent d r o ch • A chord is a line that joins two points on the circumference.

Theorem 5 – A tangent is perpendicular to a 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 350 0 70 350

Theorem 5&7 – combined can help you find the missing angles…. . 900 x 350 0 70 350 y 900

Theorem 8 – A radius bisects a chord. MIDPOINT at 900 chord OF THE CHORD 900 radius And the chord will be cut perfectly in half!!!

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

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! IT A IS IS TOONLY THINK DIAMETER THIS IS A IF YOU ARE– DIAMETER TOLD SO… SO. . THIS MUST BEREAD 900 – QUSETIONS “TANGENT CAREFULLY. . MEETS RADIUS”

Hope this has been useful – if you are still stuck, double check your “basics” from slide 2.