14 Loci and Transformations

Transcription

1 MEP Pupil Tet 1 1 Loci and Transformations 1.1 rawing and Smmetr This section revises the ideas of smmetr first introduced in Unit and gives ou practice in drawing simple shapes. Worked Eample 1 escribe the smmetries of each shape below. Solution This shape has 6 lines of smmetr, This shape has one line of as shown in the diagram. smmetr as shown below. It has rotational smmetr of order 6 as it can be rotated about its centre to 6 different positions. It has rotational smmetr of order 1, since it can rotate a full 60 back to its original position. Worked Eample raw accuratel a rectangle with sides of length 8 cm and 5 cm. Solution First draw a line 8 cm long. 79

2 1.1 MEP Pupil Tet 1 Then draw lines 5 cm long at each end, making sure the are at right angles to the base line. Finall, join these two lines to complete the rectangle. Measure the diagonals and check that the are both the same length. Eercises 1. raw accuratel rectangles with the following sizes. cm b 8 cm 10 cm b cm (c) 6 cm b 7 cm (d) 6 cm b cm For each rectangle check that both diagonals are the same length.. Make accurate drawings of each of the shapes shown below and answer the question below each shape. 5 cm cm cm 5 cm 8 cm What is the length of the sloping side? 6 cm What is the length of the longest diagonal? 80

3 MEP Pupil Tet 1 (c) cm cm 5 cm cm 7 cm What is the length of the sloping side? (d) 6 cm cm 5 cm cm What is the length of the longest straight line which can be drawn inside the shape?. Each shape below includes a semi-circle. Make an accurate drawing of each shape and state the radius of the semi-circle. 5 cm 5 cm cm cm cm (c) (d) cm cm cm 10 cm cm 8 cm 8 cm Investigation re human beings smmetrical? o ou know of an animal that is not smmetrical? 81

4 1.1 MEP Pupil Tet 1. For each shape below: (i) (ii) state the order of rotational smmetr, cop the shape and draw an line of smmetr. (c) (d) (e) (f) 5. op and shade part of the shape below so that it has lines of smmetr. What is the order of rotational smmetr of the shape? 6. State the number of lines of smmetr and the order of rotational smmetr for each shape below. (c) (d) 8

5 MEP Pupil Tet 1 7. Which of the shapes below have: rotational smmetr of order 1 no lines of smmetr (c) more than two lines of smmetr (d) rotational smmetr of order (e) rotational smmetr of an order greater than? E F 8. Make copies of the shape opposite. Shade triangles in the shape to produce shapes with: lines of smmetr, one line of smmetr, (c) rotational smmetr of order, (d) rotational smmetr of order. 1. Scale rawings Scale drawings are often used to produce plans for houses or new kitchens. For eample, a scale of 1: 0 means that 1 cm on the plan is equivalent to 0 cm in realit. Worked Eample 1 The diagram shows a scale drawing of the end wall of a house. The scale used is 1:100. Find: the height of the top of the chimne, the height of the door, (c) the width of the house. 8

6 1. MEP Pupil Tet 1 Solution s the scale is 1:100, ever 1 cm on the drawing represents 100 cm or 1 m in realit. (c) The height of the top of the chimne is 7.5 cm on the drawing. This corresponds to = cm or 7.5 m in realit. The height of the door is cm on the drawing and so is m in realit. The width of the house is 6 cm on the drawing and so is 6 m in realit. Worked Eample The diagram shows a rough sketch of the laout of a garden. 1 m Flower ed m m 6 m Patio Lawn m Shed 10 m Produce a scale drawing with a scale of 1: 00. Solution scale of 1: 00 means that 1 cm on the plan represents 00 cm or m in realit. The table below lists the sizes of each part of the garden and the size on the scale drawing. rea Real Size Size on rawing Garden 10 m 6 m 5 cm cm Patio 5 m m 5. cm 15. cm Lawn 7 m 5 m 5. cm 5. cm Flower ed 10 m 1 m 5 cm 0. 5 cm Shed m m 15. cm 1cm scale drawing can now be produced and is shown opposite drawn on squared paper. Flower ed Patio Lawn Shed 8

7 MEP Pupil Tet 1 Eercises 1. The scale drawing below of a flat has been drawn on a scale of 1: 00. Lounge athroom Kitchen edroom Hallwa (c) Find the actual sizes of the lounge and bathroom. Find the area of the bedroom floor. Find the length of the hallwa.. Work Top Work Top The scale drawing shown a plan of a kitchen on a scale of 1: 60. (c) (d) What are the length and width of the kitchen? What is the size of the cooker? What is the size of the sink? Find the area of the worktops in the kitchen.. rough sketch is made of a set of offices. It is shown below..5 m 5 m.5 m.5 m m m m Use the information given to produce a scale drawing with a scale of 1:

8 1. MEP Pupil Tet 1. Hannah produces a scale drawing of her ideal bedroom on a scale of 1: 50. The plan is shown below. Wardrobe ed ressing Table Shelf rm hair esk What is the size of the room? How long is her bed? (c) (d) What is the area of the top of her desk? What is the floor area of the room? 5. The diagram shows a rough sketch of the end wall of a house. 5 m m m Produce a scale drawing using a scale of 1:100. Use the drawing to find the sloping lengths of both sides of the roof. 6 m 8 m 6. The diagram shows a scale drawing of a garden. It is drawn with a scale of 1: 80. Flower ed (c) How wide is the garden? How long is the path? shed with a base of size m b 1.5 m is to be added to the plan. Find the size of the rectangle that should be drawn on the plan. Path Lawn (d) The garden also contains a pond of radius 0.6 m. What would be the radius of the circle which should be added to the plan? 86

9 MEP Pupil Tet 1 7. classroom is rectangular with width m and length 5 m. What would be the size of the rectangle used to represent the classroom on plans with a scale of: 1: 50 1: 5 (c) 1: 100? 8. The diagram shows a scale plan of a small industrial estate drawn on a scale of 1: 500. UNIT UNIT Road UNIT UNIT 5 UNIT 1 Find the floor area of each unit. Re-draw the plan with a scale of 1: The plan of a house has been drawn using a scale of 1: 0. (i) On the plan, the length of the lounge is 5 cm. What is the actual length of the lounge in metres? (ii) The actual lounge is. m wide. How wide is the lounge on the plan? The actual kitchen is.6 m wide. Estimate, in feet, the width of the actual kitchen. (SEG) 10. classroom is drawn on a plan using a scale of 1: 50. On the plan, how man centimetres represent one metre? The width of the classroom is 6.7 m. How man centimetres represent this width on the plan? (SEG) 87

10 1. MEP Pupil Tet ron and Shelle are two dogs. ron's lead is 1 m long. One end of the lead can slide along a railing, which is fied to the wall of the house. Shelle's lead is 1.5 m long. One end of this lead is attached to a post,, at the corner of his kennel. The scale diagram below represents the fenced garden, PQRS, where the dogs live. P House Q Scale cm represents 1 m Kennel S R op the diagram and show on our drawing all the possible positions of each dog if the leads remain tight. Not to scale 60 cm 80 cm 100 cm The diagram represents the cross-section of Shelle's kennel. alculate the area of this cross-section, giving our answer in cm. (MEG) Investigation Look at an atlas and find out the scales used in maps of the UK, Europe and the world. re the same scales used for all the different maps? If not, wh not? 88

11 MEP Pupil Tet 1 1. The diagram is an isometric drawing of a kitchen with cupboards at floor level. The kitchen is a cuboid 00 cm wide, 50 cm long and 50 cm high. The work top above the cupboards is 100 cm above the floor and 50 cm wide. Work Top upboards upboards 50 z oor O nother cupboard is to be fied above the work top on the shorter wall. It is a cuboid 00 cm long, 50 cm high and 5 cm from back to front. Its top is to be 50 cm below the ceiling. raw this cupboard on a cop of the diagram. es, O, O, Oz are taken as shown in the diagram. Write down the coordinates of (i) the point (ii) the point. (c) This is a scale drawing of the floor of the kitchen upboards The part of the floor within 50 cm of the cupboards must be kept clear upboards 00 On a cop of the diagram, show clearl and accuratel this part of the floor O (MEG) 89

12 MEP Pupil Tet 1 1. onstructing Triangles and Other Shapes protractor and a compass can be used to produce accurate drawings of triangles and other shapes. Worked Eample 1 raw a triangle with sides of length 8 cm, 6 cm and 6 cm. Solution First draw a line of length 8 cm. Then set the distance between the point and pencil of our compass to 6 cm and draw an arc with centre as shown below. The arc is a distance of 6 cm from. 8 cm With our compass set so that the distance between the point and the pencil is still 6 cm, draw an arc centred at, as shown below. 8 cm The point,, where the two arcs intersect is the third corner of the triangle. 90

13 MEP Pupil Tet 1 The triangle can now be completed. 6 cm 6 cm 5 cm 8 cm Worked Eample The diagram shows a rough sketch of a triangle. 7 cm 8 6 cm Make an accurate drawing of the triangle and find the length of the third side. Solution First draw a line of length 6 cm and measure an angle of cm Then measure 7 cm along the line and the triangle can be completed. 91

14 1. MEP Pupil Tet 1 7 cm 8 6 cm The third side of the triangle can then be measured as. cm. Worked Eample The diagram shows a rough sketch of a triangle. 5 cm Make an accurate drawing of the triangle. Solution 0 8 cm First draw the side of length 8 cm and measure the angle of 0, as shown below. 0 8 cm Set the distance between the point and pencil of our compass to 5 cm. Then draw an arc centred at, which crosses the line at 0 to. 0 8 cm 9

15 MEP Pupil Tet 1 s the arc crosses the line in two places, there are two possible triangles that can be constructed as shown below. 5 cm 0 8 cm oth triangles have the lengths and angle specified in the rough sketch. 5 cm Note n arc must be taken when constructing triangles to ensure that all possibilities are considered. Eercises 0 8 cm 1. raw triangles with sides of the following lengths. 10 cm, 6 cm, 7 cm 5 cm, cm, 6 cm (c) cm, 7 cm, 6 cm. raw accuratel the triangles shown in the rough sketches below and answer the question given below each sketch. 5 cm 0 8 cm 5 6 cm 0 How long is the side? How long are the sides and? 9

16 1. MEP Pupil Tet 1 (c) (d) 10 5 cm 7 cm 6 cm 7 cm 8 cm How long is the side? What is the size of the angle? (e) (f) 80 8 cm 70 7 cm 6 cm 0 How long is the side? How long is the side?. n isosceles triangle has a base of length 6 cm and base angles of 50. Find the lengths of the other sides of the triangle.. n isosceles triangle has sides of length 8 cm and one side of length cm. Find the sizes of all the angles in the triangle. 5. raw an equilateral triangle with sides of length 5 cm. 6. For each rough sketch shown below, draw two possible triangles. 5 cm.5 cm 5 6 cm 60 5 cm (c) (d) 5 cm.5 cm 0 7 cm 50 cm 9

17 MEP Pupil Tet 1 7. raw accuratel the parallelogram shown below. 5 cm 60 cm Measure the two diagonals of the parallelogram. 8. raw the kite shown in the rough sketch opposite. cm cm heck that the diagonals of the kite are at right angles. 5 cm 5 cm 0 9. pile of sand has the shape shown below. Using an accurate diagram, find its height. m m 6 m 10. raw accuratel the shape shown opposite. Find the size of the angle marked θ. 7 cm θ 7 cm cm cm 11. The sketch shows the design for a church window. and E are perpendicular to. is part of a circle, centre E. E is part of a circle, centre. Not to scale E 8 cm 8 cm Using a ruler and compasses draw the design accuratel. 6 cm (SEG) 95

18 1. MEP Pupil Tet 1 1. John is required to construct a pramid with a square base, as shown opposite. 7 cm Each sloping face is a triangle with base angles of cm 55 onstruct one of these triangles accuratel and to full size. onstruct the square base of the pramid accuratel and to full size. (MEG) 1. rectangle has sides of 8 cm and 5 cm. alculate the perimeter of the rectangle. onstruct the rectangle accuratel. (MEG) 1. onstruct a rhombus with the line as base and with  = 50. (MEG) 1. Enlargements n enlargement is a transformation which enlarges (or reduces) the size of an image. Each enlargement is described in terms of a centre of enlargement and a scale factor. '' ' O entre of Enlargement Scale Factor Scale Factor The eample shows how the original,, was enlarged with scale factors and. line from the centre of enlargement passes through the corresponding corners of each image. 96

19 MEP Pupil Tet 1 Note The distances, O and O, are related to O: O O = = The same is true of all the other distances between O and corresponding points on the images. Worked Eample 1 Enlarge the triangle shown using the centre of enlargement marked and scale factor. Solution The first step is to draw lines from the centre of enlargement through each corner of the triangle as shown below. O O entre of Enlargement O ' ' entre of O Enlargement ' s the scale factor is, then O = O O = O O = O The points, and have also been marked on the diagram. Once these points have been found the can be used to draw the enlarged triangle. 97

20 1. MEP Pupil Tet 1 ' ' entre of O Enlargement ' Worked Eample Enlarge the pentagon with scale factor using the centre of enlargement marked on the diagram. Solution E entre of Enlargement O The first step is to draw lines from the centre of enlargement which pass through the five corners of the pentagon. ' s the scale factor is the distances from the centre of enlargement to the corners of the image will be O = O O = O O = O O = O E' E entre of Enlargement ' O ' ' OE = OE. These points can then be marked and joined to give the enlargement. ' E' entre of Enlargement E ' O ' ' 98

21 MEP Pupil Tet 1 Worked Eample The diagram shows the square which has been enlarged to give the squares and. '' ' ' ' Find the scale factor for each enlargement. Find the centre of enlargement. ' ' Solution '' The sides of the square are each 1.5 cm. The sides of the square are cm. s these are twice as long as the original, the scale factor for this enlargement is. The sides of the square are 6 cm, which is times longer than the original square. So the scale factor for this enlargement is. To find the centre of enlargement draw lines through, and, then repeat for, and,, and and, and. ' '' '' ' ' entre of Enlargement ' ' '' '' These lines cross at the centre of enlargement as shown in the diagram. Investigation Points X and Y lie on a straight line. Given that X : X = 1 : and Y : YX = :, write down the ratio Y : X. 99

22 1. MEP Pupil Tet 1 Eercises 1. Enlarge the triangle shown with scale factor and the centre of enlargement shown. entre of Enlargement. op the diagrams below on to squared paper. Enlarge each shape with scale factor using the point marked as the centre of enlargement. entre of Enlargement entre of Enlargement (c) entre of Enlargement (d) entre of Enlargement (e) (f) entre of Enlargement entre of Enlargement. bo writes his initials as shown below. Use the marked centre of enlargement to enlarge his initials with scale factor. entre of Enlargement Repeat part using our own initials. 100

23 MEP Pupil Tet 1. In each diagram below, the smaller shape has been enlarged to obtain the larger shape. For each eample state the scale factor. (c) (d) (e) (f) 5. op each diagram on to squared paper. Then find the centre of enlargement and the scale factor when the smaller shape is enlarged to give the bigger shape. 101

24 1. MEP Pupil Tet 1 (c) (d) (e) (f) 6. op each diagram below and enlarge it with scale factor. entre of Enlargement entre of Enlargement 7. raw a set of aes with and values from 0 to 15. Plot the points (, ), (6, ), (6, 6) and (, 6), then join them to form a square. (c) Enlarge the square with scale factor, using the point with coordinates (, ) as the centre of enlargement. 8. raw a circle with centre at (, ) and radius. Enlarge the circle with scale factor and centre of enlargement (, ). (c) Enlarge the circle with scale factor and centre (5, 5). 10

25 MEP Pupil Tet 1 9. triangle with vertices at the points with coordinates (, 1) (7, 1) and (7, 6) is enlarged to give the triangle with coordinates at the points (6, ), (1, ) and (1, 18). (c) raw both triangles. What is the scale factor of the enlargement? What are the coordinates of the centre of enlargement? 10. On a set of aes draw a triangle. (c) (d) (e) Enlarge the triangle with scale factor using the point (0, 0) as the centre of enlargement. Write down the coordinates of both triangles. How do the compare? What would ou epect to be the coordinates of our triangle if it were to be enlarged with scale factor using (0, 0) as the centre of enlargement? heck our answer b drawing the triangle. Enlarge our original triangle with a different centre. Is there a simple relationship between the coordinates of the original and the enlargement, when the centre of enlargement is used? 11. Using the point O as the centre of enlargement, enlarge the rectangle with scale factor. O (MEG) 1. The shaded square has sides of length 1 cm. It is enlarged a number of times as shown.. op and complete the table below Length of Side of Square 1 cm cm cm cm Perimeter of Square cm 8 cm 1 cm rea of Square 1 cm cm 16 cm The shaded square continues to be enlarged. op and complete the following table. Length of Side of Square Perimeter of Square rea of Square 6 cm (SEG) 10

26 MEP Pupil Tet Reflections Reflections are obtained when ou draw the image that would be obtained in a mirror. Mirror Line Ever point on a reflected image is alwas the same distance from the mirror line as the original. This is shown below. Mirror Line Note istances are alwas measured at right angles to the mirror line. Worked Eample 1 raw the reflection of the shape in the mirror line shown. Mirror Line 10

27 MEP Pupil Tet 1 Solution The lines added to the diagram show how to find the position of each point after it has been reflected. Mirror Line Remember that the image of each point is the same distance from the mirror line as the original. Mirror Line The points can then be joined to give the reflected image. If the construction lines have been drawn in pencil the can be rubbed out. Worked Eample Reflect this shape in the mirror line shown in the diagram. Mirror Line Solution The lines are drawn at right angles to the mirror line. The points which form the image must be the same distance from the mirror lines as the original points. The points which were on the mirror line remain there. Mirror Line 105

28 1.5 MEP Pupil Tet 1 The points can then be joined to give the reflected image. Mirror Line Eercises 1. op the diagram below and draw the reflection of each object. Mirror Line Mirror Line (c) (d) Mirror Line Mirror Line (e) (f) Mirror Line Mirror Line 106

29 MEP Pupil Tet 1. op each diagram and draw the reflection of each shape in the mirror line shown. (c) (d) (e) (f). raw a set of aes with and values from 5 to 5. (c) (d) (e) Plot the points with coordinates (1, 1), (1, 5), (, 5), (, ), (, ), (, 1). Join the points in that order to form a shape. Reflect the image in the -ais. Write down the coordinates of the corners of this shape. Reflect the image obtained in (c) in the -ais. List the coordinates of the corners. Reflect the image obtained in (d) in the -ais. escribe how this shape could have been obtained directl from the original shape.. student reflected his two initials, the first in the -ais and the second in the -ais, to obtain the image opposite. op the diagram and show the original position of the initials. 107

30 1.5 MEP Pupil Tet 1 5. op the diagram below raw in the mirror line for each reflection described. (c) (d) 6. op the aes and shape shown. (i) raw the reflection of the shape in the -ais. (ii) ompare the coordinates of each shape. (iii) escribe what happens to the coordinates of a point when it is reflected in the -ais. (c) Repeat using the -ais X 7. op the diagram and draw the reflection of in the mirror line XY. Y has rotational smmetr. Mark with a cross its centre of rotation. (SEG) 8. Find the area of the shaded shape. op the diagram and draw the reflection of the shaded shape in the mirrorline. Mirror Line (NE) 108

31 1.6 onstruction of Loci MEP Pupil Tet 1 When a person moves so that the alwas satisf a certain condition, their possible path is called a locus. For eample, consider the path of a person who walks around a building, alwas keeping the same distance awa from the building. uilding Path of the person The dotted line in the diagram shows the path taken i.e. the locus. Worked Eample 1 raw the locus of a point which is alwas a constant distance from another point. Solution The fied point is marked. The locus is a circle around the fied point. Worked Eample raw the locus of a point which is alwas the same distance from as it is from. Solution The locus of the line will be the same distance from both points. mid-point However, an point on a line perpendicular to and passing through the mid-point of will also be the same distance from and. Locus The diagram shows how to construct this line. This line is called the perpendicular bisector of. 109

32 1.6 MEP Pupil Tet 1 Worked Eample raw the locus of a point that is the same distance from the lines and shown in the diagram below. Solution The locus will be a line which divides the ˆ into two equal angles. The diagram below shows how to construct this locus. This line is called the bisector of ˆ. Locus Eercises 1. raw the locus of a point which is alwas a distance of cm from a fied point.. The line is cm long. raw the locus of a point which is alwas cm from the line.. op the diagram and draw in the locus of a point which is the same distance from the line as it is from.. raw an equilateral triangle with sides of length 5 cm. raw the locus of a point that is alwas 1 cm from the sides of the triangle. 5. op the triangle opposite. raw the locus of a point which is the same distance from as it is from. 5 cm cm cm

33 MEP Pupil Tet 1 6. raw the locus of a point that is the same distance from both lines shown in the diagram below. 7. raw parallel lines. raw the locus of a point which is the same distance from both lines. 8. The diagram below shows the boundar fence of a high securit arm base. (c) Make a cop of this diagram. securit patrol walks round the outside of the base, keeping a constant distance from the fence. raw the locus of the patrol. second patrol walks inside the fence, keeping a constant distance from the fence. raw the locus of this patrol. 9. The points and are cm apart. raw the locus of a point that is twice as far from as it is from. 10. ladder leans against a wall, so that it is almost vertical. It slides until it is flat on the ground. raw the locus of the mid-point of the ladder. 11. The points and are cm apart. raw the possible positions of the point, P, if (i) P = cm and P = 1 cm (ii) P = cm and P = cm (iii) P = cm and P = cm (iv) P = 1 cm and P = cm raw the locus of the point P, if P + P = 5 cm. (c) raw the locus of the point P, if P + P = 6 cm. 111

34 MEP Pupil Tet Enlargements which Reduce When the scale factor of an enlargement is a fraction, the size of the enlargement is reduced. The image of the original is then between the centre of enlargement and the original. Worked Eample 1 The diagram shows three triangles. was enlarged with different scale factors to give and. '' ' Find the centre of enlargement. '' ' Find the scale factor for each enlargement. '' ' Solution To find the centre of enlargement, lines should be drawn through the corresponding points on each figure. ' '' entre of Enlargement '' '' ' ' To find the scale factors, compare the lengths of sides in the different triangles. First consider triangles and : = 6 cm and = 1 so =, cm, which means that the scale factor is 1. 11

35 MEP Pupil Tet 1 For triangles and, = 6 cm and = 1 so = 15. cm, which means that the scale factor is 1. Worked Eample Enlarge the triangle below with scale factor 1 and centre of enlargement as shown. entre of Enlargement Solution The first stage is to draw lines from each corner of the triangle through the centre of enlargement. ' O ' ' Then the corners of the image should be fied so that O = 1 O = 1 O = 1 O O O. 11

36 1.7 MEP Pupil Tet 1 These points can then be joined to give the image. ' O ' ' Eercises 1. For each pair of objects, state the scale factor of an enlargement which produces the smaller image from the larger one. (c) (d) (e) (f) 11

37 MEP Pupil Tet 1. In each eample below, the smaller shape has been obtained from the larger shape b an enlargement. For each eample, state the scale factor and the coordinates of the centre of enlargement (c) op each shape and enlarge using the centre of enlargement shown and the scale factor specified. entre of Enlargement entre of Enlargement Scale factor 1 Scale factor (c) (d) entre of Enlargement entre of Enlargement Scale factor 1 and Scale factor and

38 1.7 MEP Pupil Tet 1. For each of to (d) below: (i) draw the triangle which has corners at the points given; (ii) enlarge each triangle using the scale factor and centre of enlargement given; (iii) write down the coordinates of the corners of the image. (1, ), (, 6), (7, ) Scale factor 1 entre of enlargement (, 0) (, ), (6, 7), (1, ) Scale factor 1 entre of enlargement (0. 1) (c) (1, 1), (7, 1), (7, 7) Scale factor 5 6 entre of enlargement (1, 1) (d) (, 0), (, ), (6, ) Scale factor 1 entre of enlargement (8, 6) 5. Plot the points: (, ), (, 6), (5, 9), (8, 9), (11, 6), (11, ), (8, 0) and (5, 0). Join the points to form an octagon. (c) (d) When the octagon is enlarged, the corners move to the points with coordinates: (, ), (, ), (5, 5), (6, 5), (7, ), (7, ), (6, ) and (5, ). raw this octagon. Find the centre of enlargement and the scale factor. Use the same centre of enlargement to enlarge the original octagon with scale factor. 6. The larger triangle shown below is reduced in size b using a photocopier to give the smaller triangle. 0 cm 50 cm Not to scale ' 10 cm 5 cm ' ' What is the scale factor of the enlargement which took place? What are the lengths of and? 116

39 MEP Pupil Tet 1 7. Each diagram below shows a shape and its image after an enlargement. In each case, state the scale factor and the lengths of each side of the image. 15 cm 10 cm 18 cm 50 cm 9 cm z 1 cm 0 cm 0 cm 0 cm (c) (d) 6 cm 81 cm 6 cm 0 cm 6 cm 5 cm 9 cm z 8 cm cm 8. The parallelogram has vertices (6, ), (9, ), (1, 9) and (9, 9) respectivel. (cm) (cm) n enlargement scale factor 1 and centre (0, 0) transforms parallelogram onto parallelogram. (i) raw the parallelogram. (ii) alculate the area of parallelogram. The side has length cm. The original shape is now enlarged with a scale factor of 5 to give. alculate the length of the side. (SEG) 117

40 1.7 MEP Pupil Tet 1 9. raw a rectangle and write down the coordinates of each corner. Enlarge the rectangle with scale factor 1 using (0, 6) as the centre of enlargement. Write down the coordinates of each corner of the rectangle ou obtain. (c) (d) (e) How are the coordinates of the image related to the coordinates of the original? What would ou epect the coordinates of the smaller rectangle to be if the 1 original was enlarged with scale factor? heck our answer b drawing this enlargement. Find the area of each rectangle. How does the scale factor of an enlargement affect the area of a shape? 1.8 Further Reflections This section considers reflections in lines which are not simpl horizontal or vertical. It also makes more use of equations to describe mirror lines. Mirror Line Worked Eample 1 raw the reflection of the shape in the mirror line shown. Solution The image of each point will be the same distance from the mirror line as the original points. Mirror Line These distances should be measured so that the are perpendicular to the mirror line. 118

41 MEP Pupil Tet 1 These points can then be joined to give the reflected image. Mirror Line Worked Eample The diagram shows a number of triangles which can be obtained from each other b reflections in suitable lines E Find the equation of the mirror line which reflects: to to (c) to (d) to E. 119

42 1.8 MEP Pupil Tet 1 Solution The diagram shows the mirror line. This vertical line passes through 5. on the -ais and has equation = 5.. Mirror Line Mirror Line The diagram shows the mirror line. 9 This line passes through points with coordinates (1, 1), (, ), (, ), etc. and has equation = (c) The diagram shows the sloping mirror line which is used to reflect on to. It passes through the points with coordinates ( 1, 1), (, ), (, ), etc. and has equation =. Mirror Line

43 MEP Pupil Tet 1 9 (d) The diagram shows the horizontal mirror line which is needed for this reflection. It passes through 1 on the -ais and so has equation = Mirror Line E Eercises 1. op the diagrams below and draw the reflection of each shape in the mirror line shown. Mirror Line Mirror Line (c) Mirror Line (d) Mirror Line 11

44 1.8 MEP Pupil Tet 1 (e) (f) Mirror Line Mirror Line. The diagram shows the positions of the shapes,,, and E. 5 E The table below gives the equations of lines of reflection needed to obtain one shape from another. The smbol means that it is impossible to obtain one shape from the other b a reflection. E X =15 op and complete the table.. raw a set of aes with -values from 0 to 15 and -values from 0 to 5. (c) (d) Join in order the points with the following coordinates: (1, 1), (1, 5), (, 5), (, ) and (1, ). raw the line = 5 and reflect the original shape in this line. raw the line = 10 and reflect the image obtained in (c) in this line. 1

45 MEP Pupil Tet 1. op the set of aes below and the shape labelled Reflect in the line = to obtain. Reflect in the line = to obtain. (c) Reflect in the line = to obtain. (d) Reflect in the line = to obtain F. 5. The diagram shows a number of shapes which have been reflected in various lines G 5 F H 5 E

46 1.8 MEP Pupil Tet 1 Find the equation for the reflection which maps to to (c) to (d) to F (e) to G (f) to H (g) to H. 6. raw the triangle which has corners at (1, 1), (, 1) and (1, ) on a set of aes with and values from to 8. Reflect the triangle in the lines: (i) = + (ii) = 7. raw a set of aes with and values from 0 to 6. Join the points (, 1), (6, 1), (6, ), (5, ), (5, ), (, ), (, ), (, ) and (, 1). (c) raw the line = and the reflection of this shape in this line. (d) omplete a cop of the table below. oordinates of the oordinates of the original points reflected points (, 1) (?,?) (6, 1) (?,?) (6, ) (?,?) (5, ) (?,?) (5, ) (?,?) (, ) (?,?) (, ) (?,?) (, ) (?,?) (e) (f) (g) escribe what happens to the coordinates of a point when it is reflected in the line =. The coordinates of the corners of a rectangle are (, ), (5, ), (, ) and (5, ). If the rectangle was reflected in the line =, what would be the coordinates of the corners of the image? raw both rectangles and check our answers. Information The ablonians (around 000 ) were the first to divide a circle into 60 parts called 'degrees'. This was the consequence of the ablonians' estimation of 60 das in a ear. The ablonians used a seagesimal sstem (base 60). Thus angles at a point equal 60, one degree equals 60 minutes and one minute equals 60 seconds. 1

47 MEP Pupil Tet Rotations Rotations are obtained when a shape is rotated about a fied point, called the centre of rotation, through a specified angle. The diagram shows a number of rotations. Rotation 90 anti-clockwise Original position entre of Rotation Rotation 180 Rotation 90 clockwise It is often helpful to use tracing paper to find the position of a shape after a rotation. Worked Eample 1 Rotate the triangle shown in the diagram through 90 clockwise about the point with coordinates (0, 0). Solution entre of Rotation The diagram opposite shows how each corner can be rotated through 90 to give the position of the new triangle. 15

48 1.9 MEP Pupil Tet 1 Worked Eample The diagram shows the position of a shape and the shapes,,,, E and F which are obtained from b rotation. F E 7 8 escribe the rotation which moves onto each other shape. Solution The diagram shows the centres of rotation and how one corner of the shape was rotated. F E

49 MEP Pupil Tet 1 Each rotation is now described. to : Rotation of 180 about the point (5, 6). to : Rotation of 180 about the point (, ). to : Rotation of 90 anti-clockwise about the point (0, 0). to E: Rotation of 180 about the point (0, 0). to F: Rotation of 90 anti-clockwise about the point (0, ). Eercises 1. op the aes and triangle shown opposite. 6 5 Rotate through 90 clockwise around (0, 0) to obtain. Rotate through 90 anticlockwise around (0, 0) to obtain (c) Rotate through 180 around (0, 0) to obtain Repeat Question 1 for the triangle with coordinates (, 1), (6, ) and (0, ).. op the aes and triangle shown below Rotate the triangle through 180 using each of its corners as the centre of rotation. 17

50 1.9 MEP Pupil Tet 1. op the aes and shape shown below Rotate the original shape through 90 clockwise around the point (1, ). Rotate the original shape through 180 around the point (, ). (c) Rotate the original shape through 90 clockwise around the point (1, ). (d) Rotate the original shape through 90 anti-clockwise around the point (0, 1). 5. Repeat Question for the triangle with corners at (, ), (1, ) and (, 5). 6. The diagram shows the position of a shape labelled and other shapes which were obtained b rotating E escribe how each shape can be obtained from b a rotation. Which shapes can be obtained b rotating the shape E? 18

51 MEP Pupil Tet 1 7. The shape has been rotated to give each of the other shapes shown. For each shape, find the centre of rotation F E escribe how each shape shown below can be obtained from b a rotation. 10 F E 5 Which shapes cannot be obtained from b a rotation? 19

52 1.9 MEP Pupil Tet 1 9. On a set of aes with and values from to 1, draw the triangle with corners at the points (0, 0), (, 5) and (1, ). Rotate the triangle through 90 clockwise about the point (5, 6). Rotate the second triangle through 90 clockwise about the point (, ). (c) escribe how to obtain the third triangle from the original triangle b a single rotation. 10. The shape can be obtained from b two rotations. escribe these rotations Translations translation moves all the points of an object in the same direction and the same distance. The diagram shows a translation. Here ever point has been moved 8 units to the right and units up. This translation is described b the vector 8. 8 Worked Eample 1 escribe the translation which moves the shaded shape to each of the other shapes shown. 10

53 MEP Pupil Tet 1 Solution To move to, the shaded shape is moved 6 units to the right and units up. 6 This is described b the vector. To obtain, the shaded shape is moved 5 units across and 5 units down. 5 This is described b the vector. 5 To obtain, the shaded shape is moved units to the left and units down. This is described b the vector. To obtain, the shaded shape is moved 5 units to the left and then units up. 5 This is described b the vector. Worked Eample The shape shown in the diagram is to be 6 translated using the vector. raw the image obtained using this translation. Solution 6 The vector describes a translation which moves an object 6 units to the right and units down. This translation can be applied to each point of the original. 11

54 1.10 MEP Pupil Tet 1 The points can then be joined to give the translated image. Eercises 1. The shaded shape has been moved to each of the other positions shown b a translation. Give the vector used for each translation. G F E 1

55 MEP Pupil Tet 1. escribe the translation which moves: (c) F E (d) (e) (f) E F (g) (h) F E. raw the shape shown and its image when translated using each of the following vectors. 1 (c) 5 (d). raw the shape shown. Translate using the vector. (c) Translate the image using the vector 1. (d) Which vector would be needed to translate the final image back to the position of the original? 5. escribe a translation which would move one onto another. escribe an other translations which would move a letter onto the same letter in a different position. 1

56 1.10 MEP Pupil Tet 1 6. The number 5 can be formed b translation of the lines and. escribe the translations which need to be applied to and to form the number raw a simple shape. Write down the coordinates of each corner of our shape. (c) Translate the shape using the vector and write down the coordinates of the new shape. (d) (e) ompare the coordinates obtained in and (c). How do the change as a result of the translation? Repeat (c) and (d) with a translation using the vector. 8. raw a simple shape and translate it using the vector 5. Then translate the image using the vector 1. Which single translation would map the original shape to its final position? (c) Translate our shape using the vector 7. Then translate the image using the vector. Which single translation would move the original shape to its final position? (d) If a shape was translated using the vector and then the vector which single translation would be equivalent? 6 8, 9. The points,, and have coordinates (, 7), (, 6), (, 6) and (0, 7). Find the vector which would be used to translate: to to (c) to (d) to. Just for Fun 1. moving onl one coin in the pattern shown, make one row and one column, each containing 5 coins.. Rearrange the 8 coins to form a square with coins on each side. rearranging coins, make a square with coins on each side. 1

57 MEP Pupil Tet ombined Transformations n object can be subjected to more than one transformation, so when describing how a shape is moved from one position to another it ma be necessar to use two different transformations. Worked Eample 1 raw the image of the triangle shown if it is first reflected in the line = and then rotated clockwise about the point (, 0). Solution The diagram below shows the line = and the image of the triangle when it has been reflected in this line The new image can then be rotated about the point (, 0), as below Worked Eample escribe two different was in which the shape marked can be moved to the position shown at entre of Rotation

58 1.11 MEP Pupil Tet 1 Solution One wa, shown below, is to first n alternative approach is to rotate 6 translate using the vector 0, shape through 180 around point (, 6). and then reflect in the line = 6. This can then be reflected in the line = 6 to obtain, as shown below = 6 Mirror Line = 6 entre of Rotation Mirror Line Eercises 1. raw a set of aes with and values from 0 to 9. Plot the points (5, 1), (7, ), (9, ) and (7, 1). Join them to form a single shape. Reflect the shape in the line =. (c) Translate the shape obtained in using the vector (d) Rotate the original shape through 180 about the point with coordinates (5, ). raw a set of aes with -values from 0 to 1 and -values from 0 to. Join the points (1, ) and (, 1) to form a straight line. Rotate this line through 90 clockwise around the point (, 1). escribe two was in which the shape ou have obtained could be transformed into a 'W' shape.. The letter P is to be formed b appling a number of transfomations to the solid line. Each transformation maps the solid line onto one of the dashed lines. escribe how this could be done using: onl rotations onl reflections

59 MEP Pupil Tet 1. raw a set of aes with values from 0 to 16 and values from 1 to 8. Join the points with coordinates (, 1), (6, 1) and (6, ) to form a triangle. (c) Enlarge this triangle with scale factor using the point (0, 1) as the centre of enlargement. (d) Rotate the new triangle through 180 about the point (9, ). (e) escribe full the transformations which map the final triangle back onto the original. 5. raw a set of aes with and values from 0 to 8. Plot and join the points (1, 5), (, 5), (, 8) and (1, 7). Rotate this shape through 90 clockwise around the point (, 5). (c) Then translate the new image using the vector 0. (d) Reflect this image in the line =. (e) How could the final image be mapped back to the position of the original with a single transformation? 6. raw a set of aes with and values from 0 to 10. Plot the points with coordinates (, 1), (, 1) and (, ) and join them to form a triangle. (c) Reflect this triangle in the line = and then reflect the image in the line = 7. Which single transformation would map the original triangle onto its final position? (d) Reflect the original triangle in the line = 6 and then reflect this image in the line = 11. Which single transformation would map the final triangle back to its original position? 7. The triangle can be mapped onto, and using single transformations. H 6 escribe full each transformation. G The triangle can be mapped onto E, F, G and H using two transformations. escribe full each pair of transformations. 8 6 F E

60 1.11 MEP Pupil Tet 1 8. (i) raw a simple shape and reflect it in an vertical line. (ii) Reflect the image in an horizontal line. (c) (d) escribe two other was in which the original image could have been mapped onto the final image. Repeat and using an two lines which are perpendicular. o ou obtain the same result in each case? 9. op the diagram below and show the answers to the questions on this diagram. (You are advised to use a pencil.) 0 raw the reflection of the F in the -ais. Rotate the original F through 90 anticlockwise, with O as the centre of rotation. raw the image. (c) Enlarge the original F with centre of enlargement O and scale factor. (MEG) 10. Triangle is mapped onto triangle b means of an anticlockwise rotation, centre the origin, followed b another translation. (i) Write down the angle of rotation. (ii) Find the column vector of the translation. 0 Triangle ma be mapped onto triangle b means of a single rotation. Find the coordinates of the centre of rotation. (c) Triangle is reflected in the line = to form triangle. escribe the single transformation which would map triangle onto triangle. (MEG) 18

61 MEP Pupil Tet ongruence Two shapes are said to be congruent if the are identical in ever wa. Shapes which are different sizes but which have the same shape are said to be similar. These triangles are congruent. 5 cm 7 cm 7 cm cm 5 cm cm The triangle below is similar to the triangles above but because it is a different size it is not congruent to the triangles above..5 cm 1.5 cm.5 cm There are four tests for congruence which are outlined below. TEST 1 (Side, Side, Side) If all three sides of one triangle are the same as the lengths of the sides of the second triangle, then the two triangles are congruent. This test is referred to as SSS. TEST (Side, ngle, Side) If two sides of one triangle are the same length as two sides of the other triangle and the angle between these two sides is the same in both triangles, then the triangles are congruent. This test is referred to as SS. 19

62 1.1 MEP Pupil Tet 1 TEST (ngle, ngle, Side) If two angles and the length of one side are the same in both triangles then the are congruent. This test is referred to as S. TEST (Right angle, Hpotenuse, Side) If both triangles contain a right angle, have hpotenuses of the same length and one other side of the same length, then the are congruent. This test is referred to as RHS. Worked Eample 1 Which of the triangles below are congruent to the the triangle? F cm cm.6 cm 5 cm E.6 cm 5. I L cm 5. 5 cm cm G 5.5 H J.6 cm 5. K Solution onsider first the triangle EF: = E = EF = F s the sides are the same in both triangles the triangles are congruent. (SSS) 10

63 MEP Pupil Tet 1 onsider the triangle GHI: = HI ˆ = GHI ˆ ˆ = G ˆIH s the triangles have one side and two angles the same, the are congruent. (S) onsider the triangle JKL: Two sides are known but not the angle between, so there is insufficient information to show that the triangles are congruent. Worked Eample F is a square and = EF. Find the pairs of congruent triangles in the diagram. G Solution F E onsider the triangles and FE: = F (F is a square.) ˆ = FE (This is given in the question.) = FE ˆ = 90 (The are corners of a square.) The triangles and FE have two sides of the same length and also have the same angle between them, so these triangles are congruent. (SS) onsider the triangles G and EG: = E ( and FE are congruent.) G = G (The are the same line.) EG ˆ = G ˆ = 90 (This is given in the question.) oth triangles contain right angles, have the same length hpotenuse and one other side of the same length. So the triangles are congruent. (RHS) Investigation 1. How man straight lines can ou draw to divide a square into two congruent parts?. How man lines can ou draw to divide a rectangle into two congruent parts?. an ou draw two straight lines through a square to divide it into four congruent quadrilaterals which are not parallelograms? 11

64 1.1 MEP Pupil Tet 1 Eercises 1. Identif the triangles below which are congruent. F 5 cm 5 cm 8 cm cm 8 cm cm E G 69 K 66 6 cm.61 cm.61 cm 6.08 cm I cm H J L O 10 cm R M 85 0 N cm T 0 Q P.61 cm S 6 cm 5 U 1

65 MEP Pupil Tet 1. If O is the centre of the circle, prove that the triangles O and O are congruent. O. O If O is the centre of both circles, prove that the triangles O and O are congruent.. If O is the centre of the circle, prove that the triangles O and O are congruent. O 5. F E When = EF, this rhombus contains two congruent triangles. Identif the triangles and prove that the are congruent. 1

66 1.1 MEP Pupil Tet 1 6. If O is the centre of the circle and =, show that and are congruent triangles. O 7. Two triangles have sides of lengths 8 cm and 6 cm, and contain an angle of 0. Show that it is possible to draw different triangles, none of which are congruent, using this information. 8. The diagram shows the parallelogram with diagonals which cross at X. Prove that the parallelogram contains pairs of congruent triangles. X 9. The diagram shows a cuboid. The mid-point of the side is X. Show that the triangles HG and F are congruent. H X G Show that the triangles X and G are congruent. E F Investigation 1. In how man was can ou cut a square piece of cake into two congruent parts?. How can ou use eight straight lines of equal length to make a square and four congruent equilateral triangles? Just for Fun Stud the diagram opposite and then find the ratio of the area of the large triangle to that of the small triangle. 1

67 MEP Pupil Tet Similarit Similar shapes have the same shape but ma be different sizes. The two rectangles shown below are similar the have the same shape but one is much smaller than the other. cm 6 cm cm The are similar because the are both rectangles and the sides of the larger rectangle are three times longer than the sides of the smaller rectangle. It is interesting to compare the area of the two rectangles. The area of the smaller rectangle is 6 cm and the area of the larger rectangle is 5 cm, which is nine times greater. Note In general, if the length of the sides of a shape are increased b a factor k, then the area is increased b a factor k. 9 cm ( ) These two triangles are not similar The lengths of the sides of the triangles are not in the same ratio and so the triangles are not similar. For two triangles to be similar, the must have the same internal angles, as shown below

68 1.1 MEP Pupil Tet 1 The diagrams below show similar cubes. 1 cm cm cm 1 cm 1 cm cm cm cm cm Length rea of of side one face Surface rea Volume 1 cm 1 cm 6 cm 1 cm cm cm cm 8 cm cm 9 cm 5 cm 7 cm The table gives the lengths of sides, area of one face, total surface area and volume. omparing the larger cube with the 1 cm cube we can note that: For the cm cube For the cm cube The lengths are times greater. The areas are = times greater. The volume is 8 = times greater. The lengths are times greater. The areas are 9 = times greater. The volume is 7 = times greater. Note If the lengths of a solid are increased b a factor, k, its surface area will increase b a factor k and its volume will increase b a factor k. Worked Eample 1 Which of the triangles shown below are similar? 6.1 cm cm 1.6 cm cm cm 16 cm 16

69 MEP Pupil Tet cm.5 cm 6 cm 70 How do the areas of the triangles which are similar compare? cm Solution First compare triangles and. Here all the lengths of the sides are twice the length of the sides of triangle, so the two triangles are similar. Then compare triangles and. Here all the angles are the same in both triangles, so the triangles must be similar. Finall, compare triangles and Note that = 8 and 5. = 90., but So these triangles are not similar. The lengths of triangle are times greater than the lengths of triangle, so the area will be = times greater. The lengths of triangle are of the lengths of triangle. So the area will be 9 = of the area of triangle. 16 The ratio of the areas of triangles : : can be written as: 9 16 : 1: or 9: 16: 6. Worked Eample Eplain wh the triangles E and are similar. 6 Find the lengths of and. (c) Find the ratio of the area of E to E. Solution s the lines E and are parallel, E 6 9 E ˆ = ˆ and E ˆ = ˆ. 17

70 1.1 MEP Pupil Tet 1 s the corner,, is common to both triangles, ˆ = E ˆ. So the three angles are the same in both triangles and therefore the are similar. omparing the sides E and, the lengths in the larger triangle are 1.5 times the lengths in the smaller triangle. lternativel, it can be stated that the ratio of the lengths is :. So will be 1.5 times bigger than. = = 9. So =. In the same wa, = 15. E, so + = 15. = 05. = 8. (c) s the lengths are increased b a factor of 1.5 or for the larger triangle, the areas will be increased b a factor of 1. 5 or. We can sa that the ratio of the areas of the triangles is 1 :.5 or 1 : 9 or : 9. If the area of the triangle E is k, then the area of the triangle is 9k and hence the area of the quadrilateral E is 5k. So the ratio of the area of E to E is : 5. Worked Eample The diagrams show two similar triangles. cm 5 cm cm F E If the area of the triangle EF is 6.6 cm, find the lengths of its sides. 18