The angle in a semi-circle is 90 0 Angles at the circumference are equal. A B They must come from the same arc. Look out for a diameter. 2x Cyclic Quadrilateral Opposite angles add up to 180 0 A They must come from the same arc. B Circle Theorems equal x The angle at the centre is twice the angle at the circumference. From any point you can only draw two tangents... The angle between a tangent and a radius is 90 0 equal Look out for radii. Alternate Segment Theorem.... and they ll be equal.
180 0 straight lines triangles round a point 60 0 allied angles quadrilaterals opposite alternate parallel lines polygons angle sum = (n 2) x 180 0 corresponding Equal interior angle exterior angle add up to 60 0
rectangle a triangle is half the area of a rectangle base Area = base x base base Area = base x 2 parallelogram base Area = base x AREA Always use the perpendicular circle base trapezium Area = (a + b) x h 2 a b radius Area = πr 2
Y = 4x + Y=x+2 Y=x-4 Quadratic Graphs y = x 2 x - 2 Linear Graphs intercept y = -x + 5 4 gradient y = mx + c y = ax 2 + bx + c Parallel lines have GRAPHS Y=x+2 y = 1 the same gradient. x y = tanx Perpendicular lines have gradients with a product of -1. y = x 2 Square numbers. U shaped parabola. Reciprocal y = sinx Sine Curve Trigonometric Graphs y = cosx Cosine Curve Tangent Cubic Graphs Cube numbers. y = x
Angle Sum triangle 4 quadrilateral (n 2) x 180 0 5 pentagon 6 hexagon 4 x 180 0 = 720 0 interior angle number of 7 - heptagon triangles 8 octagon Polygons exterior angle 9 - nonagon 10 - decagon angle sum number of sides 60 0 number of sides OR exterior 180 0 angle OR interior 180 0 angle
Solving: Factorising Formula Completing the square Drawing a graph Completing the square: x 2 + 4x - = 0 (x + 2) 2 4 = 0 (x + 2) 2 7 = 0 x + 2 = ± 7 x = ± 7-2 half of 4x subtract 2 2 Factorising: easy x 2 + 7x + 12 = 0 (x + )(x + 4) = 0 x = - or x = 4 brackets more difficult! Quadratic Equations ax 2 + bx + c x x 7 Difference of Two Squares: multiply 6 1x6 x 2-5x + 2 2x x 2 - x 2x + 2 x(x 1) 2(x -1) (x 2)(x - 1) The formula: Graphs: x = -b ± b 2 4ac 2a draw lines to find solutions x 2-16 (x 4)(x + 4) x squared subtract 4 squared Parabola u shaped graph
y = fx + a y = -fx y = kfx plus a - up minus a down 0 reflection in x-axis k stretch in y-axis a a y=x 2 y=sinx scale factor k y = f(x + a) a plus a - left minus a right -a 0 opposite to what u might think! y = f(-x) reflection in y-axis y=x y = f(kx) stretch in x-axis scale factor 1/k
Translation 4 4 Describe with a vector squares right squares up Rotation To describe a rotation you need: the angle of rotation the direction the coordinates of the centre Rotation of 90 0, clockwise, about centre (2,-1) anticlockwise clockwise Reflection 2 Transformations Centre Enlargement, scale factor, centre (0,7) 1 Centre of rotation Negative enlargements HIGHER only! Enlargement Describe by naming the line of x = 2 Reflection in the line x = 2. Always use TRACING PAPER for translation, reflection & rotation. To describe an enlargement you need: the scale factor Enlargement of scale factor -2. coordinates of the centre
Prisms Crosssection Volume = length area of cross-section length Prisms have a uniform cross-section Volume V = πr 2 h Non-Prisms Cones radius Pyramids Volume = area of base x a cone is one third of a cylinder cylinders Frustrums cuboids πr 2 a frustrum is a pyramid Spheres with the top cut off. width length Volume = length x width x Volume = πr 2 h You need to find the volume of both pyramids. radius Often you need to use similar shapes in frustrum problems. V = 4πr
Square 4 equal sides opposite sides are parallel kite diagonals meet at 90 0 2 pairs of equal sides Parallelogram diagonals meet at 90 0 4 lines of Rhombus 4 equal sides opposite sides are parallel rotational of order 2 2 lines of rotational of order 4 diagonals of equal length diagonals meet at 90 0 rotational of order 1 Trapezium one pair of parallel sides rotational of order 1 1 line of Quadrilaterals an isosceles trapezium has a line of rotational of order 2 Rectangle rotational of order 2 no line opposite sides are equal & parallel 2 lines of diagonals of equal length angles in a quadrilateral add up to 60 0 opposite sides are equal & parallel
10 millimetres = 1 centimetre 100 centimetre = 1 METRE pints 1000 METRES = 1 kilometre Metric units Imperial units gallons Metric units inches feet yards MILES Imperial units 1000 millilitres = 1 litre Imperial units pounds (lbs) Units 1000 grams = 1 kilogram ounces 1 inch = 2.5 cm 1 kg = 2.2 pounds 5 miles = 8 km 1 mile = 1.6 km 4 litres = 7 pints 1 litre = 1¾ pints A litre of water is a pint and three quarters STONES 1 gallon = 4.5 litres Metric units An average man is about 1.7 or 1.8 metres tall. (6 foot) 1 foot = 12 inches That s 0cm the length of a ruler! feet = 1 yard A yard is almost 1 metre (its 90cm).
on a calculator 9% of 82 0.9 x 82 fraction to % 15 = 75 = 75% 20 100 x 5 5 OR without a calculator Change to a decimal and multiply 15 20 x 100 = 75% % 50% - half 25% - half and half 75% - 50% + 25% increasing Percentages decreasing 10% - divide by 10 5% - half 10% 20% - double 10% increase 60 by 12% 12% of 60 = 0.12 x 60 = 7.20 New amount = 60 + 7.20 = 67.20 decrease 60 by 12% ADD 12% of 60 = 0.12 x 60 = 7.20 New amount = 60-7.20 = 52.80 SUBTRACT