Diameter Radius Circumference of a circle = or Area of a circle = r2r2.

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Presentation transcript:

Diameter Radius

Circumference of a circle = or

Area of a circle = r2r2

Chord Tangent (minor) Segment (major) Segment

Sector

Angles in the same segment are equal x x x

2a a

Angles held up by the diameter are called “Angles in the semi-circle” and are all 90 0

.. The angle in a semicircle is 90° Isosceles triangles are formed by two radii. Radius Tangent Tangent and Radius meet at 90° 90°. Chord Any chord bisector is a diameter

68 0. c a b = opposite angle of a cyclic quadrilateral Opposite angles in cyclic quadrilateral add up to (supplementary) Adjacent angles in cyclic trapezium are equal - angles subtended by an arc. Only true if trapezium.

77 0. O a f e c b d Find the missing angles a, b, c, d, e and f 42 0

77 0. O = 42 0 angle in the same segment f e c b d a = opposite angle of a cyclic quadrilateral = interior angle = 77 0 adjacent angle of a cyclic trapezium 42 0 f = 84 0 angle at the centre is twice the angle at the circumference

a b. For the following circles, where O is the centre of the circle, find the missing angles. e f g ik h j 92 0 l m d c o o o o o

a =93 0 b =45 0. For the following circles, where O is the centre of the circle, find the missing angles d = 90 0 c= 90 0 e = 96 0 f = g = 31 0 i=90 0 k=32 0 j=32 0 h=122 0 l=46 0 m=46 0 b c d e f g h a i j k l m o o o o o

m m The angle between chord and tangent The angle in the opposite segment The angle between a chord and a tangent = the angle in the opposite segment n n

always equal in length Two tangents drawn from an outside point are always equal in length, two congruent right-angled triangles so creating an isosceles situation with two congruent right-angled triangles

Two tangents drawn from an outside point are always equal in length, so creating an isosceles situation with two congruent right-angled triangles m m The angle between chord and tangent The angle in the opposite segment The angle between a chord and a tangent = the angle in the opposite segment

A O CE B D 85 0 Find each of the following angles OBE BOD BED BCD CAB Angle between tangent and radius is a right angle In kite BEDO, BED = 360-known angles =10 0 Opposite angles of a cyclic quad are supplementary The angle between a chord and a tangent = the angle in the opposite segment Angle at the centre is twice the angle at the circumference