circle theorem powerpoint updated

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

Diameterradius

chord

tangentCircumference

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

400

800

LOOKS DIFFEREN

T 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 QUADRILATEARA

L MUST touch the circumference at all four vertices

910

890

700

1100

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

The two angles marked are the same

= the same angle

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

• 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

TANGENTS AND CHORDS

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….)

Theorem 5 – A tangent is perpendicular to a radius

radius

tangent900

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

radius chord900

And the chord will be cut perfectly in half.

MIDPOINT OF THE CHORD

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

Opposite angles are the same

THEOREM 9 – ALTERNATE ANGLE THEOREM

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