circle theorem powerpoint updated
TRANSCRIPT
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