Circle Theorems Identify a tangent to a circle

Circle Theorems ¡ ¡ Identify a tangent to a circle Find angles in circles l. Tangents l. Angles in a semicircle l. Cyclic quadrilateral l. Major and minor segments l. Angles at the circumference/centre l. Alternate segments

(1) Tangents ¡ A tangent to a circle is a straight line that touches the circle at one point. Not. Drawing a radius tangents to any tangent Tangents to a circle always produces a 900 angle ¡ The point where the tangent touches the circle is called the point of contact.

The angles in circles For these triangles: One side is the radius of the circle One side is a tangent to the circle Right angle Since these two lines meet at right angles- these are right angle triangles! Right angle

Remember: Angles in a triangle add up to 1800 Finding angles 900 32º Once you can spot the right angles trianglesyou can find missing angles a d 900 d 320 + 900 + a = 1800 32 + 90 + 58 = 180 a= 580 d + 900 = 1800 45 + 90 = 180 d = 450

Angles in a semi circle Use the diameter of a circle as the base of a triangle Any triangle you make using this base will form a right angled triangle

(2) Two tangents will create Isosceles Triangles You need to identify: Equal pairs of lines (length) Equal pairs of angles

Two tangents. Use “Isosceles triangle” to show 700 angles Use the right angle (tangent to radius) to calculate the 200 700 40º 200 1400 200 700 Since angles in a triangle add up to 1800 show the 1400 Use the isosceles triangle to show 200 angle

(3) Cyclic Quadrilateral So: In any cyclic Quadrilateral opposite corners sum to 1800 B A + C = 1800 D + B = 1800 A C D E H F G So: 4 points on a Any E + Gjoined = 1800 to form a circle quadrilateral F + H = 1800

Cyclic Quadrilateral Questions What are the missing angles? 1400 900 1100 50º 40º 1300 70º

(4) Major and minor segments Any circle can be divided into two unequal parts. are called segments You can use your chord as the base of a triangle d The larger part is the major segment. The smaller part is the minor segment. c b Minor segment Major segment Or… a These The line that divides the circle is called a chord. Providing all the angles are in the same segments they will be equal. a=b=c=d

(5) Angles at the centre/circumference From a chord you can create a triangle at the centre or at the circumference of the circle. At the centre At the circumference The angle at the centre is twice the size of the angle at the circumference

Questions b = 600 300 a b 130º 2500 a = 650 1100 c C = 1250

Putting your rules together Angles at a tangent = 900 An isosceles triangle has two equal angles Angles in a triangle add up to 1800 Angle at the centre is twice that at the circumference Angles round a point add up to 3600 500 40º 500 1000 400 2600

(6) Alternate segments Angles in alternate segments are equal A B