Shape, Space & Measures

Transcription

1 Section 3

2 Shape, Space & Measures Page Topic Title ngles D and 3D shapes Measures Length, area and volume Symmetry Transformations Loci Pythagoras Theorem and Trigonometry Vectors ircle theorems Revision Websites dd your favourite websites and school software here. This section of the Salford GSE Maths Revision Package deals with Shape, Space and Measures. This is how to get the most out of it: 1 Start with any topic within the section for example, if you feel comfortable with Symmetry, start with Topic 28 on page Next, choose a grade that you are confident working at. 3 omplete each question at this grade and write your in the answer column on the right-hand side of the page. 4 Mark your using the page of at the end of the topic. 5 If you answered all the questions correctly, go to the topic s smiley face on pages 4/5 and colour it in to show your progress. Well done! Now you are ready to move onto a higher grade, or your next topic. 6 If you answered any questions incorrectly, visit one of the websites listed left and revise the topic(s) you are stuck on. When you feel confident, answer these questions again. When you answer all the questions correctly, go to the topic s smiley face on pages 4/5 and colour it in to show your progress. Well done! Now you are ready to move onto a higher grade, or your next topic. Lnet GSE Revision 2006/7 - Mathematics 105

3 24. ngles Grade Learning Objective Grade achieved G Recognise right angles Know and use names of types of angle (acute, obtuse and reflex) F Know the sum of the angles in a triangle and use this fact to find missing angles Use notation of angle Know the sum of the angles on a straight line and the sum of the angles round a point E Know and use the fact that the base angles in an isosceles triangle are equal Know and use the fact that angles in an equilateral triangle are equal Know and use the fact that vertically opposite sides are equal D Know and use the fact that corresponding and alternate angles are equal Find interior and exterior angles of regular shapes * Know that the sum of exterior angles for a convex shape is 360 degrees alculate three-figure bearings Make sure you are able to meet LL the objectives at lower grades Make sure you are able to meet LL the objectives at lower grades 106 GSE Revision 2006/7 - Mathematics Lnet

4 24. ngles Grade G Recognise right angles Know and use names of types of angle (acute, obtuse and reflex) On this diagram mark a right angle with a letter R an acute angle with a letter (c) an obtuse angle with a letter O (d) a reflex angle with a letter F Grade G See Diagram (c) (d) Grade F Know the sum of the angles in a triangle and use this fact to find missing angles. Use notation of angle In the diagram below, work out the size of angle angle D 81º Grade F 37º D Know the sum of the angles on a straight line and the sum of the angles round a point 2. (i) Work out the size of the angle marked x (ii) Give a reason for your answer 2. (i) (ii) 75º x Work out the size of the angle marked y 68º 94º 112º y Lnet GSE Revision 2006/7 - Mathematics 107

5 24. ngles Shape, Space and Measures Grade E Know and use the fact that the base angles in an isosceles triangle are equal. Know and use the fact that angles in an equilateral triangle are equal. Know and use the fact that vertically opposite sides are equal. What is the special name given to this type of triangle? Work out the size of the angles marked (i) a (ii) b X 50º (3 marks) Grade E (i) (ii) XY = XZ Y a b Z What is the special name given to this type of triangle? What is the size of each angle? Know and use the fact that vertically opposite angles are equal. 3. In the diagram QR and ST are straight lines S 74º (i) Work out the value of a (ii) Give a reason for your answer (i) Work out the value of b (ii) Give a reason for your answer (3 marks) (c) (i) Work out the value of c (ii) Give a reason for your answer 3. (i) (ii) (i) (ii) Q b 43º a c T R (c) (i) (ii) 108 GSE Revision 2006/7 - Mathematics Lnet

6 24. ngles Grade D Find interior and exterior angles of regular shapes. a 110º 73º 95º b Diagrams NOT Grade D Diagram Diagram c Diagram shows a quadrilateral Work out the size of the angle marked a Diagram shows a regular hexagon Work out the size of the angle marked b Diagram d (c) Diagram shows a regular octagon (i) Work out the size of the angle marked c (ii) angle d is an exterior angle. Work out its size. (c) (i) (ii) Know and use the fact that corresponding and alternate angles are equal. 2. The diagram shows a quadrilateral D and a straight line E. 2. is parallel to E. D x 106º y E 75º 83º Work out the size of the angle marked x (i) Write down the size of the angle marked y (ii) Give a reason for your answer (i) (ii) º 70º 3. z (i) Write down the size of the angle marked z (ii) Give a reason for your answer (i) (ii) Lnet GSE Revision 2006/7 - Mathematics 109

7 24. ngles Shape, Space and Measures Grade Know that the sum of the exterior angles for a convex shape is 360º. The diagram shows a regular hexagon. alculate the size of the angle marked x x Work out the size of an exterior angle Grade Grade Grade alculate 3 figure bearings. 2. The diagram shows the positions of three schools, and. School is 9 kilometres due West of school. School is 5 kilometres due North of school. N 2. N 5km x 9km alculate the size of the angle marked x Give your answer correct to one decimal place. (3 marks) Jeremy s house is 9 kilometres due East of school. alculate the bearing of Jeremy s house from school 110 GSE Revision 2006/7 - Mathematics Lnet

8 24. ngles - nswers Grade G Examples R F Grade F O O 180º - (81º + 37º) = 62º 180º - 62º = 118º 2. (i) 180º - 75º = 105º (ii) Sum of angles on a straight line = 180º 360º - (68º + 112º + 94º) = 86º Grade E Isosceles (i) 180º - 50º = 130º 130º 2 = 65º a = 65º (ii) 180º (straight line) 180º - 65º = 115º b = 115º 2. Equilateral 180º 3 = 60º 3. (i) 137º (ii) ngles on a straight line = 180º 180º - 43º = 137º (i) 63º (ii) ngles of a triangle = 180º 180º - (74º + 43º) = 63º (c) (i) 43º (ii) Vertically opposite angles are equal. R Grade D The sum of the interior angles of a quadrilateral = 360º (2 180º) 360º - (110º + 95º + 73º) = 82º a = 82º The sum of the interior angles of a hexagon = 720º (4 180º) 720º 6 sides = 120º b = 120º The sum of the interior angles of an octagon = 1 080º (6 180º) (c) (i) 1 080º 8 sides = 135º c = 135º (ii) Sum of exterior angles of a polygon = 360º 360º 8 sides = 45º d = 45º º - (106º + 83º + 75º) = 96º x = 96º (i) y = 83º (ii) lternate angles are equal 3. (i) z = 110º (ii) orresponding angles are equal Grade 360º 6 = 60º 360º 6 = 60º Grade Tan 5/9 = 29.1º 29.1º 9km 119.1º N 5km Exterior angle equals sum of opposite interior angles 90º º = 119.1º bearing = 119º Lnet GSE Revision 2006/7 - Mathematics 111

9 25. 2D & 3D shapes Grade Learning Objective Grade achieved G Measure lengths and angles Recognise notation (symbols) for parallel, equal length and right angle Know names of triangles (including scalene, isosceles, equilateral) Know the names of 2D shapes (including trapezium, parallelogram, square, rectangle, kite) Know the names of 3D shapes (including cylinder, cuboid, cube, cone, prism) Know and use terms horizontal and vertical Recognise nets of solids F Draw triangles given Side, ngle and Side Use notation (symbol) for parallel Use terms face, edge, vertex and vertices E Know the names of 3D shapes (including sphere, square based pyramid and triangular based pyramid) Sketch 3D shapes from their nets Understand what is meant by perpendicular Make isometric drawings Draw triangles given Side, Side and Side D * Visualise spatial relationships to find touching vertices or edges Understand how a 3D shape can be represented using 2D drawings of a plan (top) view, side and front elevations Make sure you are able to meet LL the objectives at lower grades Make sure you are able to meet LL the objectives at lower grades Make sure you are able to meet LL the objectives at lower grades Make sure you are able to meet LL the objectives at lower grades 112 GSE Revision 2006/7 - Mathematics Lnet

10 25. 2D & 3D shapes Grade G Measure lengths and angles Here is an accurately drawn triangle. x Giving your in centimetres and millimetres Measure side Measure side (c) Measure side (d) Using an angle measurer, measure the size of angle x Grade G (c) (d) Measure lengths and angles Know names of triangles and angles Know and use the terms horizontal and vertical 2. The diagram shows a triangle on a centimetre grid 2. y x O x Write down the co-ordinates of the point (i) (ii) Write down the special name for triangle (c) Measure the length of the line Give your answer in millimetres (d) (i) Measure the size of the angle x (ii) Write down the special name given to this type of angle (e) (i) Draw a horizontal line on the grid and label it H (ii) Label the vertical line on the grid V (i) (ii) (c) (d) (i) (ii) (e) (i) See Diagram (ii) See Diagram Lnet GSE Revision 2006/7 - Mathematics 113

11 25. 2D & 3D shapes Shape, Space and Measures Grade G Know the names of 2D shapes Recognise notation (symbols) for parallel, equal length and right angle 3. Write down the mathematical name for each of the following 2D shapes. (Total 6 marks) (i) (ii) (iii) Grade G 3. (i) (ii) (iii) (iv) (v) (vi) (iv) (v) (vi) Look at the shapes above and label (i) right angle with an R (ii) Parallel lines with a P (iii) Equal length marks with an E (3 marks) See Diagram (i) (ii) (iii) Know the names of 3D shapes 4. Write down the mathematical name for each of the following 3D shapes. (Total 5 marks) 4. (i) (ii) (iii) (iv) (v) (i) (ii) (iii) (iv) (v) 114 GSE Revision 2006/7 - Mathematics Lnet

12 25. 2D & 3D shapes Grade G Recognise nets of solids 5. The diagrams below show some solid, 3D shapes and their nets. n arrow has been drawn from one 3D shape to its net. Draw an arrow from each of the other solid shapes to its net. (Total 5 marks) Grade G 5. See Diagram a (i) (ii) b (iii) c (iv) d e (v) Lnet GSE Revision 2006/7 - Mathematics 115

13 25. 2D & 3D shapes Shape, Space and Measures Grade F Draw triangles given Side, ngle, Side This diagram shows a sketch (not accurately drawn) of a triangle. 5.8cm 6.7cm Make an accurate drawing of the triangle x Diagram not Grade F See Drawing (i) On your drawing, measure the size of the angle marked x (ii) Write down the special mathematical name of the angle marked x (i) See Drawing (ii) Use notation (symbol) for parallel Use terms face, edge, vertex and vertices 2. This diagram shows a sketch of a solid, 3D shape. 2. Write down the name of the solid Label two pairs of the parallel lines using the correct markings (c) For this solid, write down (i) The number of faces (ii) the number of edges (iii) the number of vertices (3 marks) See Diagram (c) (i) (ii) (iii) 116 GSE Revision 2006/7 - Mathematics Lnet

14 25. 2D & 3D shapes Grade E Know the names of 3D shapes Write down the mathematical name for each of these 3D shapes. (3 marks) a b c Grade E (c) Sketch 3D shapes from their nets 2. Sketch the 3D shapes belonging to the nets below. (Total 10 marks) 2. See Drawing See Drawing (c) (c) See Drawing (d) (d) See Drawing (e) (e) See Drawing Lnet GSE Revision 2006/7 - Mathematics 117

15 25. 2D & 3D shapes Shape, Space and Measures Grade E Make isometric drawings Understand what is meant by perpendicular 3. Here is a net of a prism. 6cm 3cm 60º 60º Grade E 3. 3cm Mark with a P, a line that is parallel to the line Mark with an X, a line that is perpendicular to the line (c) Make an accurate drawing of the net. See Diagram See Diagram (c) See Drawing (d) Sketch the prism (d) See Drawing 118 GSE Revision 2006/7 - Mathematics Lnet

16 25. 2D & 3D shapes Grade E Draw triangles given Side, Side, Side 4. Here is a sketch of a triangle. 5.6cm 4.3cm Grade E 4. See Drawing 6.2cm Use a ruler and compasses to construct this triangle accurately in the space below. You must show all your construction lines. (3 marks) Grade D Grade D Visualise spatial relationships to find touching vertices or edges Here is a net of a cube. The net is folded to make a cube. Two other vertices meet at. 3cm Mark each of them with the letter. The length of each edge is 3cm. Work out the volume of the cube. See Diagram Lnet GSE Revision 2006/7 - Mathematics 119

17 25. 2D & 3D shapes Shape, Space and Measures Grade D Understand how a 3D shape can be represented using 2D drawings of plan (top) view, side and front elevations 2. elow are a plan view and a front elevation of a prism. The front elevation shows a cross section of the prism. Grade D 2. Plan View Front Elevation On the grid below, draw a side elevation of the prism (3 marks) See Grid Draw a 3D sketch of the prism See Drawing 120 GSE Revision 2006/7 - Mathematics Lnet

18 25. 2D & 3D shapes - nswers Grade G (i) = 5cm 4mm (ii) = 6cm 9mm (iii) = 2cm 6mm 20º 2. (i) (8,2) (ii) (0,4) Isosceles (c) 78mm (d) (i) 27º (ii) cute (e) (i) ny horizontal line (ii) should be labelled V 3. (i) Right-angled triangle (ii) Equilateral triangle (iii) Scalene triangle (iv) Parallelogram (v) Trapezium (vi) Kite (i) ottom right corner on right angled triangle (ii) < and << on parallelogram and trapezium (iii) \ and \\ on equilateral triangle and kite 4. (i) uboid (ii) ylinder (iii) one (iv) ube (v) Triangular prism 5. = (v) = (iii) (c) = (i) (d) = (ii) (e) = (iv) Grade E Square-based pyramid Triangular-based pyramid (c) Sphere 2. (c) (d) (e) 3. ny horizontal line ny vertical line (c) ccurate drawing (d) 4. orrectly constructed triangle and arcs (3 marks) orrect triangle and incorrect arcs orrect arcs and two correct sides Two correct sides Grade D = 27cm 3 2. Grade F ccurately drawn triangle (i) 40º (ii) cute 2. uboid < and << on parallel edges (c) (i) 6 (ii) 12 (iii) 8 Lnet GSE Revision 2006/7 - Mathematics 121

19 26. Measures Grade Learning Objective Grade achieved G hoose appropriate units with which to measure weights, lengths, areas and volumes hange between units for weight, length, volume and time F Make estimates of weights, lengths and volumes in real-life situations onvert metric units to imperial units of weight, length and volume E Make sure you are able to meet LL the objectives at lower grades D hange between units for area, eg. m 2 to cm 2 Make sure you are able to meet LL the objectives at lower grades Make sure you are able to meet LL the objectives at lower grades Make sure you are able to meet LL the objectives at lower grades * Make sure you are able to meet LL the objectives at lower grades 122 GSE Revision 2006/7 - Mathematics Lnet

20 26. Measures Grade G hoose appropriate units with which to measure weights, lengths, areas and volumes. elow is a table of measurements. omplete the table by writing a sensible metric unit on each dotted line. The first one has been done for you. The weight of a small bag of crisps 25 grams (3 marks) Grade G See Table. The distance from Manchester to London The height of a man The volume of petrol in a car s petrol tank hange between units for weight, length, volume and time. 2. hange 250 millimetres to centimetres hange 3.7 litres to millilitres (c) hange 400 seconds to minutes and seconds 2. (c) Lnet GSE Revision 2006/7 - Mathematics 123

21 26. Measures Shape, Space and Measures Grade F Make estimates of weights, lengths and volumes in real-life situations. Here is a picture of a man standing near a giraffe. Grade F oth the man and the giraffe are drawn to the same scale. Estimate the height of the man, in metres. Estimate the height of the giraffe, in metres. (3 marks) onvert metric units to imperial units of weight, length and volume. 2. hange 10 kilograms into pounds. hange 5 litres into pints. (c) hange 5 miles into kilometres. 2. (c) Grade D Grade D hange between units for area, eg. m 2 into cm 2. hange 2.8m 2 to cm GSE Revision 2006/7 - Mathematics Lnet

22 26. Measures - nswers Grade G Kilometres entimetres (c) Litres mm = 1cm 250 divided by 10 = litre = ml = (c) 6 minutes, 40 seconds 60 seconds = 1 minute 400 divided by 60 = 6 remainder 40 Grade F 5-2 metres man s height pounds 1 kilogram = 2.2 pounds = litre = pproximately 75 pints 5 75 = 8.75 (c) 8 kilometres 1 mile = pproximately 6 kilometres 5 6 = 8 Grade D cm (or ) Lnet GSE Revision 2006/7 - Mathematics 125

23 27. Length, rea and Volume Grade Learning Objective Grade achieved G ount squares to find areas Measure perimeters Find volume by counting cubes F alculate the area of a triangle alculate the area of a square alculate the perimeter of a compound shape Understand and use the words length and width E Estimate areas for shapes without straight lines alculate volumes alculate area of a rectangle alculate areas and perimeters of compound shapes onvert between metric units for length, area and volume D alculate the circumference and area of a circle alculate the diameter and radius given the circumference of a circle alculate missing dimensions of a cuboid given its volume alculate the area of a trapezium alculate missing dimensions of a prism given its volume alculate the volume of a prism * Recognise algebraic expressions for Length, rea and Volume alculate the length of an arc alculate the area of a sector Make sure you are able to meet LL the objectives at lower grades Make sure you are able to meet LL the objectives at lower grades 126 GSE Revision 2006/7 - Mathematics Lnet

24 27. Length, rea and Volume Grade G ount squares to find areas Measure perimeters Grade G If each square on the grid is 1cm 2 Find the area, in cm 2, of the shaded shape. Find the perimeter, in cm, of the shaded shape. Find volume by counting cubes This solid shape is made up from cubes of side 1cm Find the volume, in cm 3, of the shape. Grade F alculate the perimeter of a compound shape alculate the area of a square alculate the area of a triangle Use the words length and width Work out the perimeter of the 60m Grade F whole shape D. In part you must write down the units with your answer. 80m E Work out the area of (i) the square ED. (ii) the triangle E. 50m (i) (ii) (c) Label the length with the letter L (c) See Diagram (d) Label the width with the letter W D 50m (d) See Diagram Lnet GSE Revision 2006/7 - Mathematics 127

25 27. Length, rea and Volume Shape, Space and Measures Grade E alculate areas for shapes without straight lines The shaded area on the grid represents the surface of a lake in winter. Estimate the area, in cm 2, of the diagram that is shaded. If each square on the grid represents an area with sides of length 120m: Work out the area, in m 2, represented by one square on the grid Grade E (c) Estimate the area, in m 2, of the lake (c) In summer the area of the lake decreases by 15% (d) Work out the area, in m 2, of the lake in summer (d) alculate volumes onvert between metric units for Length, rea and Volume 2. In this question you must write down the units of your answer cm 10cm 25cm Work out the area of the base of the solid shape. (i) Work out the volume of the solid shape (ii) Write this volume in litres (i) (ii) alculate the area of a rectangle alculate the area and perimeter of a compound shape 3. This diagram shows the plan of a floor m 6m 9m 5m Work out the perimeter of the floor. Work out the area of the floor. (3 marks) 128 GSE Revision 2006/7 - Mathematics Lnet

26 27. Length, rea and Volume Grade D alculate the circumference and area of a circle. Some oil is spilt. The spilt oil is in the shape of a circle. The circle has a diameter of 15 centimetres. Work out the circumference, in cm, of the spilt oil. Give your answer correct to one decimal place. Work out the area, in cm 2, of the spilt oil. Give your answer correct to 2 decimal places. (3 marks) 15cm Grade D alculate the diameter and radius given the circumference of a circle. 2. udrey has a circular dining table. The perimeter of the circular tablecloth is 6.5m 2. Work out the diameter of the tablecloth. Give your answer correct to 3 significant figures. Work out the radius of the tablecloth. Give your answer correct to 3 significant figures. alculate missing dimensions of a cuboid given its volume. 3. cuboid has a volume of 72cm 3 a length of 4cm a width of 3cm Work out the height of the cuboid Grade alculate the area of a trapezium. The diagram (not accurately drawn) shows a trapezium D. Grade is parallel to D. = 4.2m D = 5.8m D = 2.6m ngle D = 90º ngle D = 90º alculate the area of trapezium D. D Lnet GSE Revision 2006/7 - Mathematics 129

27 27. Length, rea and Volume Shape, Space and Measures Grade alculate missing dimensions of a prism given its volume 2. The diagram shows a triangular prism. = 3cm, F = 9cm and angle = 90º The volume of the triangular prism is 54cm 3. Work out the height of the prism. (4 marks) E D F Grade 2. alculate the volume of a prism 3. The cylinder has a height of 25cm. It has a base radius of 8cm. The cube has side of edges 15cm. alculate the total volume, in cm 3, of the cylinder. Give your answer to the nearest cm 3. (3 marks) alculate the total volume, in cm 3, of the container. Give your answer to the nearest cm 3. (3 marks) 3. Grade Grade Recognise algebraic expressions for Length, rea and Volume Here are some expressions. See Table (a+b)ch 2πa 3 2ab ab h 2πb 2 2(a 2 +b 2 ) πa 2 b The letters a, b, c and h represent lengths. π and 2 are numbers that have no dimensions. Tick the boxes underneath the three expressions which could represent areas. (3 marks) alculate the length of an arc alulate the area of a sector 2. This is the sector of a circle, radius = 10cm. alculate the length of the arc. 32º centre 10cm Give your answer correct to 3 significant figures. (4 marks) alculate the area of the sector. Give your answer to 3 significant figures. (4 marks) GSE Revision 2006/7 - Mathematics Lnet

28 27. Length, rea and Volume - nswers Grade G rea = 19cm 2 Perimeter = 24cm 2. 44cm 3 Grade F = 240cm (i) = 2 500m 2 (ii) (50 30) 2 = 750m 2 (c) Length = side D (d) Width = side D Grade E 10cm = m 2 (1 square) (c) = m 2 (d) /100 = m 2 (100% - 15% = 85%) = 250cm 2 (i) = 5 000cm 3 (ii) = 5 litres (1litre = 1 000cm 3 ) Perimeter = 40m (9 5 = 45m 2 ) + (6 6 = 36m 2 ) = 81m 2 area = 81m 2 Grade D ircumference = πd π 15 = = 47.1cm rea = πr 2 π (7.5) 2 = π = = cm π = = 2.07m = 034 = 03m 3. 4(L) 3(W) = 12 72/12 = 6 height = 6cm Grade rea of trapezium = average of parallel sides height = ( ) = = = 13m 2 2. Volume of a prism = rea of base Length rea of base 9 = 54 rea of base = 54 9 = 6 = ½ 3 height = 6 height = 4cm 3. πr 2 h π (8) 2 25 = = 5027cm (15 3 ) = 3 375cm cm 3 = 8 402cm 3 Grade 3rd: 2ab 5th: 2πb 2 6th: 2(a 2 + b 2 ) cm (to 3 significant figures) = π d rc = Oº /360 circle s circumference = 32/360 π 20 = or 5.59 to 3 significant figures 27.9cm 2 to 3 significant figures = πr 2 Sector = Oº /360 circle s area = 32/360 π 100 = or 27.9 (to 3 significant figures) Lnet GSE Revision 2006/7 - Mathematics 131

29 28. Symmetry Grade Learning Objective Grade achieved G Draw lines of symmetry in shapes and recognise shapes having a line of symmetry Recognise shapes having rotational symmetry F Recognise and draw planes of symmetry in 3D shapes Find the order of rotational symmetry for a shape E Find the centre of rotation given an object and its image Draw shapes with a given line of symmetry and / or order of rotational symmetry D Make sure you are able to meet LL the objectives at lower grades Make sure you are able to meet LL the objectives at lower grades Make sure you are able to meet LL the objectives at lower grades Make sure you are able to meet LL the objectives at lower grades * Make sure you are able to meet LL the objectives at lower grades 132 GSE Revision 2006/7 - Mathematics Lnet

30 28. Symmetry Grade G Recognise shapes having a line of symmetry and draw lines of symmetry in shapes Draw in all the lines of symmetry on each of the following shapes. (4 marks) (c) (d) Grade G See Shapes Recognise shapes having rotational symmetry 2. Draw a circle around each of the shapes below that have rotational symmetry. 2. See Shapes (c) (d) (e) Grade F Recognise and draw planes of symmetry in 3D shapes The diagram represents a prism. Draw in one plane of symmetry. Grade F See Diagram Find the order of rotational symmetry 2. Write down the order of rotational symmetry for each of the shapes below. (3 marks) 2. (c) (c) Lnet GSE Revision 2006/7 - Mathematics 133

31 28. Symmetry Shape, Space and Measures Grade E Find the centre of rotation given an object and its image Here is a triangle and its image, after being rotated 90º clockwise Find the centre of rotation y Grade E See Grid x Draw shapes with a given line of symmetry and/or order of rotational symmetry 2. On these shapes draw in all lines of symmetry. 2. See Shapes Write down the order of rotational symmetry for these shapes. (c) On the grid below draw a shape with 4 lines of symmetry and rotational symmetry of order 4. (c) See Grid 134 GSE Revision 2006/7 - Mathematics Lnet

32 28. Symmetry - nswers Grade G 2. 2 lines 4 lines (c) 1 line (d) 1 line 2. Draw a circle around, (c) and (e) Grade F (i) 8 (ii) 4 (c) Pupils own, eg square or (c) 2 Grade E 7 y x entre of rotation is where the perpendicular bisectors cross (2, 1) Lnet GSE Revision 2006/7 - Mathematics 135

33 29. Transformations Grade Learning Objective Grade achieved G F Reflect a shape in a mirror line Show how a shape can tesselate E Enlarge a shape by a positive integer scale factor Recognise congruent shapes Find a scale factor from a drawing Find distances on a map for a given scale factor Rotate shapes given a centre of rotation and angle of rotation D Plot points given a three-figure bearing Understand the effect of enlargement on the area of a shape Describe rotations and reflections, giving angles and equations of mirror lines Produce enlargements by a fractional positive scale factor and a given centre of enlargement Translate simple 2D shapes using vectors Understand that enlargements produce mathematically similar shapes preserving angles within the shapes Find side length for similar shapes * Enlarge shapes by negative scale factors Make sure you are able to meet LL the objectives at lower grades 136 GSE Revision 2006/7 - Mathematics Lnet

34 29. Transformations Grade G Reflect a shape in a mirror line shaded shape is shown in the grid of centimetre squares. Grade G Mirror Line Work out the perimeter of the shaded shape Work out the area of the shaded shape (c) Reflect the shaded shape in the mirror line (1 mark (c) Grade F Show how a shape can tesselate Show how the shape in the grid will tesselate. You should draw at least 6 shapes. See Grid Lnet GSE Revision 2006/7 - Mathematics 137

35 29. Transformations Shape, Space and Measures Grade E Enlarge a shape by a positive integer scale factor shaded shape is shown on grid. On grid draw an enlargement, scale factor 2, of the shaded shape. Grade E See Drawing Grid Grid Recognise congruent shapes Find a scale factor from a drawing 2. Here is a triangle J. Here are nine more triangles. J Grade E 2. D E F G H I Write down the letters of the triangles that are congruent to triangle J. (i) Write down the letter of a triangle that is an enlargement of triangle J. (ii) Find the scale factor of the enlargement. (i) (ii) 138 GSE Revision 2006/7 - Mathematics Lnet

36 29. Transformations Grade E Find distances on maps for a given scale factor 3. Isobel uses a map with a scale of 1 to 50,000. She measures the distance between two towns on the map. The distance Isobel measures is 7.3cm Give the actual distance between the two towns - in kilometres. Rotate shapes given a centre and angle of rotation 4. Rotate triangle J 90º clockwise about the the point (1,1) Grade E See Grid y J x Lnet GSE Revision 2006/7 - Mathematics 139

37 29. Transformations Shape, Space and Measures Grade D Plot points given a three-figure bearing The scale drawing below shows the positions of a lighthouse, L, and a ship, S. 1 cm on the diagram represents 20 km. N S Grade D L (i) Measure, in centimetres, the distance LS. (ii) Work out the distance, in kilometres, of the ship from the lighthouse. (i) Measure and write down the bearing of the ship from the lighthouse. (ii) Write down the bearing of the lighthouse from the ship. (c) tug boat is 70 km from the lighthouse on a bearing of 300 degrees. Plot the position of the tug boat, using a scale of 1 cm to 20 km on the scale diagram above. (3 marks) (i) (ii) (i) (ii) (c) See Diagram Understand the effect of enlargement on the area of a shape 2. The diagram represents two photographs. 2. Diagram not 3 cm 5 cm Work out the area of the small photograph. State the units of your answer. The photograph is to be enlarged by scale factor 4. Write down the measurements of the enlarged photograph. (c) How many times bigger is the area of the enlarged photograph than the area of the small photograph? (c) 140 GSE Revision 2006/7 - Mathematics Lnet

38 29. Transformations Grade D Describe rotations and reflections giving angles and equations of mirror lines y x Grade D Describe fully the single transformation which takes shape onto shape. Describe fully the single transformation which takes shape onto shape. (3 marks) Lnet GSE Revision 2006/7 - Mathematics 141

39 29. Transformations Shape, Space and Measures Grade Produce enlargements by a fractional positive scale factor and a given centre of enlargement Shape P is shown on the grid. Shape P is enlarged, centre (0,0), to obtain shape Q. One side of shape Q has been drawn for you. Write down the scale factor of the enlargement. On the grid, complete shape Q. (c) The shape Q is enlarged by scale factor 1/2, centre (5,12) to give shape R. On the grid, draw shape R. (3 marks) Grade See Grid (c) See Grid y Q 4 3 P x (d) d) Shapes P, Q and R are mathematically similar. What does this mean? Translate simple 2D shapes using vectors (3) 2. On the grid, translate triangle by the vector See Grid Label the new triangle y x 142 GSE Revision 2006/7 - Mathematics Lnet

40 29. Transformations Grade Understand that enlargements produce mathematically similar shapes preserving angles within the shapes Find the side length for similar shapes Triangle is similar to triangle PQR. ngle = angle PQR ngle = angle PRQ. P 13 cm Grade 6 cm Q R 8 cm 10 cm alculate the length of PQ. alculate the length of. Grade Enlarge shapes by negative scale factors Grade Enlarge triangle T by scale factor -1½, centre O. (3 marks) See Grid y T x Lnet GSE Revision 2006/7 - Mathematics 143

41 29. Transformations - nswers Shape, Space and Measures Grade G Perimeter = 12cm rea = 5cm 2 (c) Grade E 2., E, H (i) F or I (ii) = cm = 3.65km 4. o-ordinates (1,1), (1,0), (3,1) Grade D Mirror Line (i) 5.7cm (ii) = 114 km (i) 068º (ii) 248º (360º - 068º = 248º) (c) N Grade F T 3.5cm S 300º L Grade E = 15cm 2 Height 4 3 = 12cm Length 4 5 = 20cm (c) 16 rea of small photo = 15cm 2 rea of large photo = = 240cm = Reflection in the y axis Rotation 90º clockwise about the origin (0,0) nywhere on the grid. 144 GSE Revision 2006/7 - Mathematics Lnet

42 29. Transformations - nswers Grade 2 See diagram (c) See diagram (d) They are the same shape with the same angles, but a different size. y O R Q 5 4 P x Grade 6 (10/8) = 7.5cm 13 10/8 = 10.4cm or 13 8/10 = 10.4cm Grade Vectors at (-5,-5), (-3,-5), (-5,-4.5) y T x y x Lnet GSE Revision 2006/7 - Mathematics 145

43 30. Loci Grade Learning Objective Grade achieved G No objectives at this grade F No objectives at this grade E onstruct shapes from given information using only compasses and a ruler D Locate the position of an object given information about its bearing and distance onstruct perpendicular bisectors and angle bisectors using only compasses and a ruler onstruct loci in terms of distance from a point, equidistance from two points and distance from a line Shade regions using loci to solve problems, eg vicinity to lighthouse/port onstruct loci in terms of equidistance from two lines Make sure you are able to meet LL the objectives at lower grades * Make sure you are able to meet LL the objectives at lower grades 146 GSE Revision 2006/7 - Mathematics Lnet

44 30. Loci Grade E onstruct shapes from given information using only compasses and a ruler Here is a sketch of a triangle, not drawn to scale. In the space below, use ruler and compasses to construct this triangle accurately. 6.7cm You must show all construction lines. (Total 3 marks) 5.2cm Grade E See Drawing 7.3cm Grade D Grade D Locate the position of an object given information about its bearing and distance 1 The scale drawing below shows the positions of two ships, P and Q. 1 cm on the diagram represents 20 km. See Diagram N N Q P ship R is 100 km away from ship P, on a bearing of 058. Ship R is also on a bearing of 279 from ship Q. In the space above, draw an accurate diagram to show the position of ship R. Mark the position of ship R with a cross. Label it R. (Total 4 marks) Lnet GSE Revision 2006/7 - Mathematics 147

45 30. Loci Shape, Space and Measures Grade onstruct perpendicular bisectors and angle bisectors using only compasses and a ruler Use ruler and compasses to construct the perpendicular bisector of the line segment YZ. You must show all construction lines. (Total 2 marks) Grade See Drawing Y Z onstruct loci in terms of distance from a point, equidistance from two points and distance from a line Q 2. Triangle PQR is shown on the right. 2. On the diagram, draw accurately the locus of the points which are 4cm from Q. See Diagram On the diagram, draw accurately the locus of the points which are the same distance from QP as they are from QR. See Diagram J is a point inside triangle PQR P J is 4cm from Q J is the same distance from QP as it is from QR (c) On the diagram, mark the point J clearly with a cross. Label it with the letter J. R (c) See Diagram 148 GSE Revision 2006/7 - Mathematics Lnet

46 30. Loci Grade Shade regions using loci to solve problems 3. The diagram represents a triangular pool. The scale of the diagram is 1cm represents 2m. fountain is to be built so that it is nearer to than to, within 7m of point. On the diagram, shade the region where the fountain may be built. (Total 3 marks) Grade 3. See Diagram Grade Grade onstruct loci in terms of equidistance from 2 lines The diagram shows three points, and on a centimetre grid. On the grid, draw the locus of points which are equidistant from and D. On the grid, draw the locus of points which are 3.5 cm from E. (c) On the grid, shade the region in which points are nearer to than D and also less than 3.5cm from E. y See Diagram See Diagram (c) See Diagram E x D -2 Lnet GSE Revision 2006/7 - Mathematics 149

47 30. Loci - nswers Shape, Space and Measures Grade E Grade 2. Q 4cm 6.7cm 5.2cm J P R 7.3cm Grade D N N R 279º cm 5cm Q 58º P 58º angle (1mark) 279º angle (1mark) 5cm line (1mark) Letter R (1mark) Grade y Grade 6 4 Y Z 2 E y = x D -2 Horizontal line equidistant from and D ircle radius 3.5cm from E 150 GSE Revision 2006/7 - Mathematics Lnet

48 3 Pythagoras Theorem & Trigonometry Grade Learning Objective Grade achieved G F E D No objectives at this grade No objectives at this grade No objectives at this grade No objectives at this grade Recall Pythagoras Theorem and use it to find the length of any side of a right-angled triangle Use Pythagoras theorem to solve problems such as bearings, areas of triangles, diagonals of rectangles, etc Use sine, cosine and tangent ratios to calculate angles and sides in right-angled triangles pply sine, cosine and tangent ratios to solve problems involving right-angled triangles, including bearings and angles of depression and elevation Use Pythagoras Theorem and trigonometry in 3-dimensional problems Use the sine rule to find the size of an angle or side in a non-right-angled triangle Use the cosine rule to find the size of an angle or side in a non-right-angled triangle * Solve more complex sine and cosine rule problems, when the quadratic formula is required Understand the ambiguous case for the sine rule Lnet GSE Revision 2006/7 - Mathematics 151

49 3 Pythagoras Theorem & Trigonometry Shape, Space and Measures Grade Recall Pythagoras Theorem and use it to find the length of any side of a right-angled triangle 13cm D 19cm D is a rectangle. = 19 cm and D = 13 cm alculate the length of the side D. Give your answer correct to one decimal place. (3 marks) Grade Use Pythagoras Theorem to solve problems such as bearings, areas of triangles and diagonals of rectangles 2. paint can is a cylinder of radius 11cm and height 21cm. Vincent, the painter, drops his stirring stick into the tin and it disappears. Work out the maximum length of the stick. 21cm Give your answer correct to two decimal places. (3 marks) Diagram not 11cm 2. Grade Use sine, cosine and tangent ratios to calculate angles and sides in right-angled triangles The diagram shows a right-angled triangle. = 15cm 15cm ngle = 39 ngle = 90 Find the length of the side. 39º Give your answer correct to 3 significant figures. (3 marks) Grade pply sine, cosine and tangent ratios to solve problems involving right-angled triangles including bearings and angles of depression and elevation. 2. D represents a vertical cliff 16m high. boat,, is 25 m due east of D. alculate the size of the angle of elevation of from. Give your answer correct to 3 significant figures. What is the angle of depression of from? Give a mathematical reason for this. (3 marks) GSE Revision 2006/7 - Mathematics Lnet

50 3 Pythagoras Theorem & Trigonometry Grade Use Pythagoras Theorem and trigonometry in 3-dimensional problems The diagram (not accurately drawn) represents a cuboid DEFGH. = 7 cm, = 9 cm E = 5 cm. 5cm E F H D 9cm G Grade alculate the length of G. 7cm Give your answer correct to 3 significant figures. alculate the size of the angle between G and the face D. Give your answer correct to 1 decimal place. Use the sine rule to find the size of a side in a non-right-angled triangle 8cm 2. In triangle (not accurately drawn), 70º 60º 6cm = 8 cm, = 6 cm ngle = 60 and ngle = 70 alculate the length of. Give your answer correct to 3 significant figures. (3 marks) 2. Use the sine rule to find the size of an angle in a non-right-angled triangle 3. In triangle 3. = 5 cm = 9 cm 100º 5cm ngle = 100 alculate the size of angle. 9cm Give your answer correct to 1 decimal place. Use the cosine rule to find the size of a side or angle in a non-right-angled triangle 4. In triangle (not accurately drawn) 8cm = 8 cm, = 14 cm and ngle = 69. alculate the length of. (3 marks) Give your answer correct to 3 significant figures cm 69º alculate the size of angle. Give your answer correct to 1 decimal place. Lnet GSE Revision 2006/7 - Mathematics 153

51 3 Pythagoras Theorem & Trigonometry Shape, Space and Measures Grade * Solve more complex sine and cosine rule problems, when the quadratic formula is required (2x+1)m (x+4)m 30º Grade * In triangle (not accurately drawn) = (2x + 1) metres. = (x + 4) metres. ngle = 30. The area of the triangle is 4m 2. alculate the value of x. Give your answer correct to 3 significant figures. (Total 5 marks) Understand the ambiguous case for the sine rule 2. Triangle (not accurately drawn) is obtuse. alculate the size of angle giving your answer to 3 significant figures. (3 marks) 2. 15cm 30º 20cm 154 GSE Revision 2006/7 - Mathematics Lnet

52 3 Pythagoras Theorem & Trigonometry - nswers Grade = = Length D = 13.9 cm = = = 30.41cm Grade = os39 15 = = 8.94m Grade 4. a 2 = b 2 + c 2-2bcos ( os69) = = = 13.4 cm (3 sf) 4. os = b2 + c 2 - a 2 2bc os = = 77.18º = 77.2º (1 dp) tan -1 = ngle of elevation = 32.6º 32.6º ngle of depression is equal to the angle of elevation because they are alternate angles. Grade G 2 = G = = = 130 G 2 = G 2 = = 155 G 2 = 155 = G 2 = 12.4 cm (3 sf) Find angle G Sin -1 ( ) = 23.8º (1 dp) 2. a Sin70 = 8 Sin60 a = Sin70 8 Sin 60 a = 8.68 cm (3 sf) 3. Sin = Sin 9 5 Sin = Sin 5 9 = Sin = = 33.2º (1 dp) Grade * 4 = ½ (x + 4) (2x + 1) Sin 30 4 = ¼ (x + 4) (2x + 1) 16 = 2x² + 9x + 4 2x² + 9x 12 = 0 a = 2, b = 9, c = -12 x = -9 ± (4 2-12) 4 x = -9 ± x = 076 (Reject negative value from ( ) 4 as length can t be negative). 2. Sin 20cm = Sin 30 15cm Sin = 20 Sin 30 15cm Sin = Sin = 0.6 recurring ngle = inverse Sin (0.6 recurring) ngle = 481º. However, remember that the sine curve has symmetry. n angle of 180º - 481º will also give the same sine. So could be either 481º or º. To decide which is right we must remember that the largest angle is always opposite the largest side. If were 481º then would be 180º - 30º - 481º which gives º Therefore must be º. This is an acute angle so satisfies the constraint in the question. = 138º to 3 significant figures. Lnet GSE Revision 2006/7 - Mathematics 155

53 32. Vectors Grade Learning Objective Grade achieved G No objectives at this grade F No objectives at this grade E No objectives at this grade D No objectives at this grade No objectives at this grade Understand and use vector notation alculate the sum, difference, scalar multiple and resultant of 2 vectors * Solve geometrical problems in 2D using vector methods 156 GSE Revision 2006/7 - Mathematics Lnet

54 32. Vectors Grade Understand and use vector notation is the point (3,2) and is the point (-1,0) Find as a column vector. ( 9 ) is a point such that = 4 Write down the co-ordinates of the point. (c) X is the midpoint of. O is the origin. Grade (c) Find OX as a column vector. Grade alculate the sum, difference, scalar multiple and resultant of 2 vectors ( 1 ) ( 4 ) ( 1 ) Given that a = 4 b = 1 c = -3 Grade Work out the following: 2a a + 2b (c) a b + c (d) 2a + b c (e) ½ a (c) (d) (e) 2. In the triangle, = j and = k and D is the midpoint of. 2. Work out the vectors: j D D (c) D k (c) Lnet GSE Revision 2006/7 - Mathematics 157

55 32. Vectors Shape, Space and Measures Grade * Solve geometrical problems in 2D using vector methods The diagram shows two triangles O and OD. O and OD are straight lines. is parallel to D. O = a and O = b The point cuts the line O in the ratio O:O = 2:3 Express D in terms of a and b Grade * O a b D 158 GSE Revision 2006/7 - Mathematics Lnet

56 32. Vectors - nswers Grade ( -2 ) -4 (7, 11) (c) = (3, 2) = (-1, 0) X = (2, 1) ( 1 ) OX = 2 Grade ( 1 ) ( 2 ) 2a = 2 4 = 8 a + 2b = = ( 6 1 ) ( 4 ) ( 9 ) (c) a - b + c = = ( 0 1 ) ( 4 ) ( 1 ) ( -2 ) (d) 2a + b - c = = ( 12 1 ) ( 4 ) ( 1 ) ( 5 ) (e) ½a = ½ 4 = 2 ( 1 ) ( 0.5 ) 2. = + = -j + k = k-j D = ½ = ½(k-j) (c) D = + D = j + ½(k-j) = j + ½k - ½j = ½j + ½k = ½(j-k) Grade * 3/2 (b - a) = (-a + b) = (b - a) D = 3/2 = 3/2 (b - a) Lnet GSE Revision 2006/7 - Mathematics 159

57 33. ircle Theorems Grade Learning Objective Grade achieved G No objectives at this grade F No objectives at this grade E No objectives at this grade D No objectives at this grade No objectives at this grade Solve problems by understanding and applying circle theorems Solve more complex problems by understanding and applying circle theorems * Make sure you are able to meet LL the objectives at lower grades 160 GSE Revision 2006/7 - Mathematics Lnet

58 33. ircle Theorems Grade Solve problems by understanding and applying circle theorems - ngle at the centre of a circle is twice as big as the angle at the circumference - ngle in a semi-circle is a right angle - ngles in the same segment are equal,, and D are points on the circumference of a circle. O is the centre of the circle. ngle = 58º Work out the size of angle O. Give a reason for your answer. Work out the size of angle. D O 58º Grade Give a reason for your answer. (c) Work out the size of angle D. Give a reason for your answer. (c) - Know the sum of the opposite angles in a cyclic quadrilateral - Know the sum of the angles on a straight line - Know the sum of the angles in a triangle - Know the angles in the same segment are equal 2. Work out the size of these angles. Give a reason for each answer. Diagrams NOT r p a 110º 120º 125º b c q 33º (i) ngle a (ii) ngle b (iii) ngle c Work out the size of these angles. Give a reason for each answer. (i) ngle p (ii) ngle q (iii) ngle r (6 marks) (6 marks) 2. (i) (ii) (iii) (i) (ii) (iii) - Two tangents drawn to a circle from outside it are of equal length 3. X, Y and Z are points on the circumference of a circle. O is the centre of the circle. ngle XZY = 65º Find the size of angle XOY. Give a reason for your answer. Z 65º O Y 3. Find the size of angle XTY. T Give a reason for your answer. (3 marks) X Lnet GSE Revision 2006/7 - Mathematics 161

59 33. ircle Theorems Shape, Space and Measures Grade Solve problems by understanding and applying circle theorems - Prove and use the alternative segment theory T and T are are tangents to a circle. O is the centre of the circle. ngle T = 40º Diagram not accuartely drawn. Work out the size of angle T. Give a reason for your answer. Work out the size of angle O. Give a reason for your answer. (c) Work out the size of angle. Give a reason for your answer. Grade (c) O 40º T - Perpendicular line from the centre of a chord bisects the chord 2. P and Q are points on the circumference of a circle. O is the centre of the circle. M is the point where the perpendicular line from O meets the chord PQ Prove that M is the midpoint of the chord PQ (3 marks) 2. P M Q O 162 GSE Revision 2006/7 - Mathematics Lnet

60 33. ircle Theorems - nswers Grade 116º ngle O - at centre of circle - is twice as big as the angle at the circumference ( = 58º) 90º ngle in a semi-circle is a right angle ( is a diameter) (c) 58º ngles in the same segment are equal and angle = 58º 2. (i) a = 55º 180º - 125º = 55º (opposite angles in a cyclic quadrilateral add up to 180º) (ii) b = 70º 180º - 110º = 70º (opposite angles in a cyclic quadrilateral add up to 180º) Grade Triangle T = isosceles (T = T) ngle T = (180º - 40º) 2 = 70º ngle OT = 90º (angle between tangent and radius is equal to 90º) ngle O = 90º - 70º = 20º (c) ngle = ngle T lternate segment theory = 70º 2. OP = OQ (both are radii) OM = OM (OM is common) ngle OMP = ngle OMQ = 90º Triangle OMP = Triangle OMQ PM = QM M is the midpoint of PQ (iii) c = 55º 180º - 125º = 55º (angles on a straight line add up to 180º) (i) p = 27º 180º - 153º = 27º (angles in a triangle add up to 180º) (ii) q = 33º (angles in the same segment are equal) (iii) r = 27º (angles in the same segment are equal) º - (angle at the centre is twice the angle at the circumference) 50º 2 tangents drawn to a circle from an outside point are equal in length and have formed 2 congruent right-angled triangles. OXT and OYT are right angles 360º - 90º - 90º - 130º 360º - 310º = 50º Lnet GSE Revision 2006/7 - Mathematics 163