Circles

Radius Diameter

Tangent (major) Segment Chord (minor) Segment Arc

Circumference of a circle = or r D p 2

Area of a circle = r2

Angles held up by the diameter are called “Angles in the semi-circle” and are all 900

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

Angles in the same segment are equal x

x · 2x Angles from chord to centre are twice the size of angles from chord to circumference

The angle at the centre is 1800 , so the angle at the circumference is half that - 900

Opposite angles in cyclic quadrilateral add up to 1800 (supplementary) 680 . c a b = 1120 opposite angle of a cyclic quadrilateral

Opposite angles in cyclic quadrilateral add up to 1800 (supplementary) 680 . c a b = 1120 opposite angle of a cyclic quadrilateral Adjacent angles in cyclic trapezium are equal - angles subtended by an arc.

a 420 b . d c Find the missing angles a, b, c and d

. d = 840 angle at the centre is twice the angle at the circumference = 420 angle in the same segment 420 b = 420 angle in the same segment . d c

770 . c a b Find the missing angles a, b, and c

. c a = 770 adjacent angle of a cyclic trapezium b = 1030 adjacent angle of a cyclic trapezium, or opposite angle of a cyclic quadrilateral b = 1030 opposite angle of a cyclic quadrilateral

For the following circles, where O is the centre of the circle, find the missing angles 870 1350 c . . . e o o d o a b 470 480 i k 920 . . f 580 h o l o j 390 g m 1100

For the following circles, where O is the centre of the circle, find the missing angles 870 1350 . . . d = 900 e = 960 d o o o e a a =930 b =450 b 470 480 k=320 f h=1220 i=900 920 . f = 390 . 580 i k h l o j o l=460 j=320 390 g = 310 m g m=460 310 1100

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

Reminder of “segment” (major) Segment (minor) Segment

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

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

Angle between tangent and radius is a right angle 900 B Angle at the centre is twice the angle at the circumference 600 A 850 1700 100 O 250 The angle between a chord and a tangent = the angle in the opposite segment 950 C E 250 In kite BEDO, BED = 360-known angles 900 + 1700 + 900 =100 D Opposite angles of a cyclic quad are supplementary Find each of the following angles OBE BOD BED BCD CAB