Circle Theorems

Circle TheoremRemember the 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

Parts of a Circle

Diame

ter

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.

THIS IS THE ONE MOST PEOPLE DONT SEE

1150

2300

400800

LOOKS DIFFERENT BUT STILL

THE CENTRE

SPECIAL CASE OF THE SAME RULE BUT MAKES A RULE BY ITSELF

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

RULE 3: 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 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)

THEOREM 4: Opposite angles sum to 180 in a cyclic quadrilateral

CYCLIC QUADRILATERAL MUST touch the circumference at all four vertices

910

890

700

1100

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

radius cho

rd

900

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

MIDPOINT OF THE CHORD

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 ERRORS

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..