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..