Circle Theorems. Angles at the circumference are equal. The angle in a semi-circle is x The angle at the centre. Cyclic Quadrilateral

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1 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 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.

2 180 0 straight lines triangles round a point 60 0 allied angles quadrilaterals opposite alternate parallel lines polygons angle sum = (n 2) x corresponding Equal interior angle exterior angle add up to 60 0

3 rectangle a triangle is half the area of a rectangle base Area = base x base base Area = base x 2 parallelogram AREA Always use the base Area = base x perpendicular circle base trapezium Area = (a + b) x h 2 a b radius Area = πr 2

4 Cubic Graphs Reciprocal Linear Graphs Y = 4x + Y=x+2 Y=x-4 Quadratic Graphs y = x 2 x - 2 intercept 4 y = x 2 Square numbers. gradient U shaped parabola. y = mx + c y = ax 2 + bx + c y = -x + 5 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 = sinx Trigonometric Graphs y = cosx Tangent Cube numbers. Sine Curve Cosine Curve y = x

5 Angle Sum triangle 4 quadrilateral (n 2) x pentagon 6 hexagon 4 x = 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 angle OR interior angle

6 Solving: Factorising Formula Completing the square Drawing a graph Factorising: x 2 + 9x + 20 = 0 x 4 x x 2 4x 5 5x 20 (x + 5)(x + 4) = 0 x = -5 x = -4 fill in the blue squares first work out the factors (red numbers) x Completing the square: x x 2 2 2x 2 2x 4 Quadratic Equations ax 2 + bx + c x 2-10x + 8 = 0 x the white squares make the x term x -4 x 2 x 2-4x + 4x = (x + 2) 2-4 complete the square -2-6x 8 x 2 + 4x - = 0 half of 4x (x + 2) 2 4 = 0 (x + 2) 2 7 = 0 subtract 2 2 x + 2 = ± 7 x = ± 7-2 The formula: Graphs: x = -b ± b 2 4ac 2a draw lines to find solutions Difference of Two Squares: x 2-16 (x 4)(x + 4) x squared subtract 4 squared (x - 4)(x - 2) = 0 x = x = 2 Parabola u shaped graph

7 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

8 x = 2 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 symmetry 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

9 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

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11 Square 4 equal sides opposite sides are parallel kite diagonals meet at pairs of equal sides Parallelogram diagonals meet at lines of symmetry Rhombus 4 equal sides opposite sides are parallel rotational symmetry of order 2 2 lines of symmetry rotational symmetry of order 4 diagonals of equal length diagonals meet at 90 0 rotational symmetry of order 1 Trapezium one pair of parallel sides rotational symmetry of order 1 1 line of symmetry Quadrilaterals an isosceles trapezium has a line of symmetry rotational symmetry of order 2 Rectangle rotational symmetry of order 2 no line symmetry opposite sides are equal & parallel 2 lines of symmetry diagonals of equal length angles in a quadrilateral add up to 60 0 opposite sides are equal & parallel

12 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).

13 Brackets The Power Zero division, so subtract the powers Multiplying... but any number divided by itself equals 1... add the powers Indices Negative Indices Any number to the power zero equals 1. Fractional Indices Dividing = 5 = 2 must be equal 25 x 25 = x 5 = 25 add powers subtract the powers 2 5 = - equal

14 on a calculator 9% of x 82 fraction to % 15 = 75 = 75% x 5 5 OR without a calculator Change to a decimal and multiply 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 = = decrease 60 by 12% ADD 12% of 60 = 0.12 x 60 = 7.20 New amount = = SUBTRACT