14 Loci and Transformations

Transcription

1 1 Loci and Transformations 1.1 rawing and Smmetr 1. raw accuratel rectangles with the following sizes: cm b 5 cm 9 cm b.5 cm. Make accurate drawings of each of the shapes below and answer the question below each shape. 3 cm 8 cm cm cm cm cm cm What is the length of the sloping side? What is the length of one of the sloping sides? (c) cm 3 cm 8 cm 5 cm 10 cm What is the length of the longest straight line which can be drawn inside the shape? 3. Each shape below includes a semi-circle. Make an accurate drawing of each shape and state the radius of the semi-circle. 3 cm 3 cm 8 cm 5 cm cm 55

2 1.1. For each shape below: (i) state the order of rotational smmetr, (ii) cop the shape and draw an line of smmetr. (c) (d) (e) 5. Make copies of this shape. Shade part of the shape to produce shapes with: eactl lines of smmetr eactl 1 line of smmetr (c) rotational smmetr of order 1 (d) rotational smmetr of order.. Which of the shapes below have: rotational smmetr of order 1 no lines of smmetr (c) more than lines of smmetr (d) rotational smmetr of order (e) rotational smmetr of order greater than? E 5

3 7. The diagram shows part of a design. The dotted lines are lines of smmetr of the whole design. op and complete the design. Write down the order of the rotational smmetr of the completed design. (OR) 8. The properties of quadrilaterals are shown inthe table. Shape Lines of smmetr Order of rotational smmetr iagonals of equal length Yes Yes No 0 No op and complete these statements b filling in the letter in each case. square is described b shape..... parallelogram is described b shape..... (c) rectangle is described b shape..... (Q) 9. Write down the name of this quadrilateral. Three of these statements are true for a kite. On a cop of the diagram, draw arrows from the statements that are true to the picture of the kite. One of them has been done for ou. Two pairs of sides are equal It has lines of smmetr It has rotational smmetr of order The diagonals cross at right angles Opposite angles are equal Kite One pair of opposite angles are equal (Q) 57

4 1. Scale rawings 1. The scale drawing of a shop, shown below, has been drawn on a scale of 1 : 00. hanging Rooms ounter Stock Room Find : the actual sizes of the stock room and each changing room. the area of the counter (c) the width of the shop entrance.. rough sketch is shown below of the ground floor of a house. 3.5 m m.5 m m 3.5 m Use the information given to produce a scale drawing with a scale of 1 : The drawing opposite represents a scale drawing of a garden. It is drawn with a scale of 1 : 10. Find: the dimensions of the shed Pond Grass Grass Flower ed the area of the vegetable garden (c) (d) (e) the dimensions of the flower bed the radius of the pond. the area of the land grassed. Vegetable Garden Shed 58

5 . room is rectangular, with width 5 m and length m. What would be the size of the rectangle on a scale drawing with a scale of: 1 : 50 1 : 100 (c) 1 : 00? 5. The sketch drawing shows the plan of a field. 75 m Not to scale 0 m 15 0 m 98 Using a scale of 1 centimetre to 10 metres, make an accurate scale drawing. Write down the length of the side to the nearest metre. (MEG). The scale drawing shows the position of an airport tower, T, and a radio mast, M. 1 cm on the diagram represents 0 km. North T M (i) Measure, in centimetres, the distance TM. (ii) Work out the distance in km of the airport tower from the radio mast. (i) Measure and write down the bearing of the airport tower from the radio mast. (ii) Write down the bearing of the radio mast from the airport tower. plane is 80 km from the radio mast on a bearing of 0. (c) On a cop of the digram, plot the position of the plane, using a scale of 1 cm to 0 km. (LON) 59

6 1. 7. Not to scale tall fence is supported b a post at an angle as shown. The foot of the post is 1.1 m from the fence. The post makes an angle of 3 with the ground. 1.1 m 3 omplete a scale drawing to show the angled post. The fence and the ground have been drawn for ou to cop. Use a scale of cm to 1 metre. How far up the fence does the post reach? (OR) 0

7 8. The diagram shows a rough sketch of a triangular field. 5 m 55 m 75 m Using ruler and compasses onl, make an accurate scale drawing of the field. Use a scale of 1 cm to represent 10 m. You must show clearl all our construction arcs. The length of one side of the field is 75 metres. This length is measured to the nearest metre. What is the smallest possible length of this side? (Q) 1.3 onstructing Triangles and Other Shapes 1. raw triangles with sides of the following lengths: 1 cm, 8 cm, 7 cm 8 cm, 5 cm, cm.. raw accuratel the triangles shown in the rough sketches below and then answer the question about each sketch. cm 35 7 cm What is the length of? (c) 5 (d) What is the size of the angle? 7 cm 8 cm 95 cm 5 How long are the sides and? 7 cm 5 cm What is the length of? cm 1

8 raw accuratel an equilateral triangle with sides of length cm.. n isosceles triangle has a base of length 8 cm and base angle of 37. Make an accurate drawing of the triangle and use it to estimate the lengths of the other sides of the triangle. 5. n isosceles triangle has sides of length cm and one side of length 10 cm. Find the sizes of all the angles in the triangle.. For each rough sketch below, draw accuratel two possible triangles. cm 9 cm 30 8 cm 5 10 cm 7. raw a rhombus,, with = cm and ˆ = 0. What are the lengths of the diagonals? cm 0 cm small window is made of a semicircle of radius 30 cm and a straight section of height 50 cm. onstruct an accurate drawing of this window. 1. Enlargements 1. 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) (d) entre of Enlargement entre of Enlargement

9 . In each diagram below, the smaller shape has been enlarged to obtain the larger shape. For each eample, cop the diagram on to squared paper, state the scale factor and find the centre of enlargement. (c) (d) 3. entre of Enlargement op the diagram opposite and enlarge it with a scale factor of 3.. On a cop of the grid below, enlarge the shaded shape b a scale factor of 3. Start our enlargement at point. (LON) 3

10 1. 5. Enlarge the shape with scale factor and centre (0, 3) (SEG). Enlarge the shaded figure, using scale factor 3 and centre of enlargement E. E (MEG) 7. In the diagram, triangle T is an enlargement of triangle S from a centre. T S On a cop of the diagram, mark and label the centre of enlargement. Write down the scale factor of the enlargement. (MEG)

11 8. On a cop of the grid, draw an enlargement of this shape. Use a scale factor of shape has been drawn on a grid of one centimetre squares. (OR) Work out the area of the shape. 1.5 Reflections On a cop of the grid, enlarge the shape with a scale factor of. (OR) 1. op each diagram below and draw the reflection of each object. Mirror Line Mirror Line 5

12 1.5 (c) (d) Mirror Line Mirror Line (e) Mirror Line (f) Mirror Line. op the diagram below raw the mirror line for each of the following reflections. (c) (d) 3. op the diagram opposite. raw the mirror line for each of the following reflections: (c) (d) 0 8

13 . Mirror Line op the ais and the shape shown opposite (i) Reflect the shape in the mirror line shown. (ii) Reflect the new shape in the -ais. (iii) Reflect the new shape in the -ais. 8 (iv) Reflect the new shape in the -ais. (c) escribe two reflections needed to take the shape defined in (iv) above back to the original shape. 5. Reflect each of the shapes below in the mirror line shown. Mirror Line Mirror Line. The diagram shows a triangle drawn on a centimetre square grid O Write down the coordinates of. On a cop of the diagram, draw the reflection of the triangle in the -ais. (Q) 7

14 1. onstruction of Loci 1. The line is 5 cm long. raw the locus of a point which is alwas cm from the line.. op the diagram opposite. onstruct the locus of a point which is equidistant from both lines Two oil rigs, and, are in the sea near the port of. N N Sea Land third oil rig is the same distance from and from. op the diagram and construct the locus of possible positions of the third oil rig on the diagram. The third oil rig,, is also the same distance from as it is from and. Mark with a cross the position of. (SEG). The diagram shows a field,, drawn to a scale of 1 cm to 10 m. Treasure is hidden in the field. 8

15 The treasure is at equal distances from the sides, and. op the diagram and construct the locus of points for which this is true. The treasure is also 0 m from the corner,. onstruct the locus of points for which this is true. (c) Mark with an X the position of the treasure. (SEG) 5. The diagram below is a scale drawing of the floor of a room. In the diagram, cm represents 1 m. X marks the position of an electric socket. X vacuum cleaner is attached b a cable to the socket and can clean the floor up to 3 metres from the socket. op the diagram and shade the part of the floor which can be cleaned b the vacuum cleaner. (MEG). Jason has to sail his ship between two rocks so that his ship is alwas the same distance from Point on the first rock and Point on the second rock. The diagram on the following page shows the rocks. On a cop of the diagram, construct accuratel the path along which Jason must sail his ship. 9

16 1. Rock The Smphleglades Rocks 7. Laton Moorb Newdon The map above, drawn to a scale of cm to represent 1 km, shows the positions of three villages, Laton, Moorb and Newdon. Simon's house is the same distance from Moorb as it is from Laton. The house is also less than 3 km from Newdon. raw a cop of the map and mark on our drawing the possible positions of Simon's house. Show our construction lines clearl. (MEG) 8. Signals from a radio mast, M, can be received up to a distance of 100 km. Use a scale drawing of 1 cm to represent 0 km to answer the following questions. Shade the region in which signals from the radio mast can be received. The distance of a helicopter from the radio mast is 70 km, correct to the nearest kilometre. Write down (i) (ii) the maimum distance the helicopter could be from the radio mast the minimum distance the helicopter could be from the radio mast. (LON) 70

17 9. X Y The diagram represents a bo which is to be moved across a floor XY. = 30 cm and = 0 cm. First the bo is rotated about the point so that becomes vertical. Then the bo is rotated about the new position of the point so that becomes vertical. op the diagram and draw the locus of the point. alculate the maimum height of above the floor. Give our answer correct to one decimal plece. ( measurement from the scale drawing is unacceptable.) (LON) 10. (i) raw a straight line,, 8 cm long. (ii) raw the locus of points, P, which lie above the line such that the area of triangle P is 1 cm. On the same diagram, construct the locus of points, Q, which lie above the line such that angle Q is 90. (c) Hence draw all triangles which have above, an area of 1 cm and an angle of 90. (MEG) 11. The diagram shows the wall of a house drawn to a scale of cm to 1 m. dog is fastened b a lead 3 m long to a point X on a wall. On a cop of the diagram shade the area that the dog can reach. Scale: cm to 1 m House X (OR) 71

18 1. 1. The diagram shows an L shape. On a cop of the diagram, draw the locus of all points cm from the L shape. (Q) 13. The diagram shows a quadrilateral PQRS. P Q S R On a cop of the diagram, draw the locus of points that are the same distance from P as from Q. 7

19 Shade the region inside the quadrilateral which is less than 7 cm from S and nearer to Q than to P. (Q) 1.7 Enlargements which Reduce 1. For each pair of objects, state the scale factor of the enlargement which produces the smaller image from the larger one. (c) (d). For each pair of objects below, the smaller shape has been obtained from the larger shape b an enlargement. For each eample, state the scale factor and give the coordinates of the centre of enlargement (d) 8 (c)

20 op each shape and enlarge using the given centre of enlargement and the specified scale factor. entre of Enlargement Scale factor 1 entre of Enlargement Scale factor 1 3 (c) (d) entre of Enlargement entre of Enlargement Scale factor 1 Scale factor 3. The larger rectangle is reduced in size to the smaller rectangle. 9 cm ' ' cm cm ' ' What is the scale factor of the enlargement? What is the length of? 7

21 5. Each diagram below shows a shape and its image after enlargement. In each case, state the scale factor and find the unknown lengths in the image. 15 cm 7.5 cm 1 cm 0 cm 5 cm 1 cm 1 cm z 15 cm 5 cm (c) (d) 1 cm 1 cm 7 cm 1 cm 1 cm 1 cm. Shape is shown in the diagram. Shape is enlarged to obtain the shape. One side of shape has been drawn. Write down the scale factor of the enlargement. op the drawing and complete the shape on our diagram. 1.8 Further Reflections (LON) 1. op the diagrams below and draw the reflection of each shape in the mirror line. Mirror Line Mirror Line 75

22 1.8 (c) Mirror Line (d) (e) Mirror Line (f) Mirror Line. The diagram shows the positions of the shapes,,, and E. Mirror Line 8 E What is the equation of the line of reflection required to transform from: to to (c) to (d) to (e) to E? 3. op the diagrams and reflect each of the shapes in the mirror lines given. Mirror Line Mirror Line (LON) 7

23 . op the set of aes opposite and the shape labelled. (c) (d) Reflect in the line = 5 to obtain. Reflect in the line = to obtain. Reflect in the line = 3 to obtain. Reflect in the line = to obtain E (e) Name two reflections of E which would bring the shape back to. 5. The diagram shows a number of shapes, some of which have been reflected in various lines. 8 F E G 8 State whether each mapping is a reflection and if so, give the equation of the mirror line. to to (c) to (d) to E (e) E to F (f) to G (g) to G (h) F to (i) to E. 77

24 1.8. O On a cop of the diagram: reflect flag in the line =. Label the image. enlarge flag with scale factor 1 and centre ( 3, ). Label the image. (OR) 7. Triangle and triangle have been drawn on the grid. On a cop of the diagram, reflect triangle in the line =. Label this image. 5 3 escribe full the single transformation which will map triangle onto triangle. 1 O (Edecel) 1.9 Rotations 1. op the aes and shape shown. 8 Rotate the shape through 90 anticlockwise around (0, 0) to obtain. Rotate the shape through 180 clockwise around (0, 0) to obtain (c) What rotation is needed to obtain the shape from? 78

25 . op the aes and shape shown below Rotate the original shape through 90 clockwise about the point (1, 0). Rotate the original shape through 180 about the point (5, ). (c) escribe the rotation that takes the shape in to the shape in. (d) Rotate the original shape through 90 anti-clockwise about the point (, 1). 3. The diagram shows the position of a shape labelled and other shapes which were obtained b rotating. 8 E escribe how each shape can be obtained from b a rotation. Which shapes can be obtained b rotating the shape? 79

26 escribe full the single transformation which will transform the shape labelled to the shaded shape. On a cop of the grid, draw the shaded shape after it has been reflected in the line =. (NE) = = 0 The square is reflected in the line = 1. What are the new coordinates of? The square is rotated through 180 about. What are the new coordinates of? (SEG). op the diagram below. Mirror Line Reflect the shape in the mirror line. Label the reflection. 80

27 escribe full the transformation which maps the triangle onto the triangle. (LON) 7. The sketch shows the position of a rectangle. (, 11) 1 10 (, 8) 8 (, 3) (0, 0) The rectangle is reflected in the line = to give rectangle What are the coordinates of 1? The rectangle is rotated about anticlockwise through 90 to give. What are the coordinates of? (c) The rectangle is enlarged b scale factor, centre. What are the coordinates of the new position of? (SEG) 81

28 O On a cop of the diagram, reflect the shaded triangle in the line =. Label this new triangle with the letter. Rotate the original shaded triangle b a quarter-turn anticlockwise about (0, ). Label this new triangle with the letter. (OR) O escribe the transformation that maps triangle to triangle. Triangle is rotated 90 anti-clockwise about (0, 1). On a cop of the diagram, draw the image of after this transformation. (c) Triangle is an enlargement of triangle. (i) Write down the scale factor of the enlargement. (ii) Write down the coordinates of the centre of the enlargement. (Q) 8

29 10. O (i) escribe full the single transformation that maps flag onto flag. (ii) escribe full the single transformation that maps flag onto flag. alculate the area of this flag. 15. cm 1.10 Translations 13. cm (OR) 1. The shaded shape has been moved to each of the other positions b a translation. Give the vector used for each translation.. For the shapes shown, describe the translation which maps: (c) (d) E (e) E (f). E 83

30 op this diagram. raw the image of this shape when translated using: (c) 7.. op the shape opposite. Translate the shape using the vector 7 0. Translate the new shape using the vector 0. 3 (c) What translation is needed to translate the original shape to the final shape? 5. The points, and have coordinates (3, ), (, 5) and ( 3, 1) respectivel. Find the vector needed to translate: to to (c) to.. Find the centre of the rotation which maps triangle on to triangle. O escribe the single transformation which maps triangle on to triangle. (OR) 7. The diagram shows a shaded flag. On a cop of the diagram: rotate the shaded flag 90 anti-clockwise about the origin. Label this new flag with the letter. (c) translate the original shaded flag units to the right and 3 units down. Label this new flag with the letter. reflect the original shaded flag in the line = 1. Label this new flag with the letter O (Q)

31 1.11 ombined Translations 1. op the set of aes and shape shown below. 0 8 Reflect this shape in the line = 5. Rotate the original shape 90 clockwise about (5, 0). (c) Reflect the shape in about the line =. (d) escribe a single transformation of the shape in part to the shape formed in part (c).. F E 0 8 escribe a single transfomation from (i) to (ii) F to (iii) to (iv) E to. escribe two combined transformations for (i) to (ii) to E (iii) to F. 85

32 The point is reflected in the -ais. The image is the point. Write down the coordinates of.. The point is rotated through 90 anticlockwise about O. The image is the point. Write down the coordinates of. (c) The point can be mapped onto point b a translation. Write down the column vector of this translation (MEG) 5 = op the diagram and reflect the triangle in the -ais. Label the reflection. Reflect the triangle in the line =. Label the reflection. (c) escribe full the single transformation which maps triangle onto triangle. (d) Write down the equation of the line which is parallel to = and which passes through the point (0, 8). This line crosses the -ais at the point P. (e) alculate the coordinates of P. (LON) 8

33 5. The parallelogram has vertices at (, 3), (9, 3), (1, 9) and (9, 9) respectivel n enlargement, scale factor 1 3 and centre (0, 0), transforms parallelogram onto op the diagram and draw the parallelogram The parallelogram is translated b the vector. What are the coordinates of? 0 onto (c) The parallelogram can be transformed onto b an enlargement. Give the scale factor and centre of this enlargement. (SEG). The diagram shows triangles T, S and U. S is the image of T under a reflection in the line = 3. U is the image of S under reflection in the line =. reflection in the line = 3 n, where n is an integer, is denoted b R n, T S U So S is the image of T under R 1 and U is the image of T under R 1 followed b R. V is the image of T under the successive transformations R 1 followed b R followed b R 3. raw V on a cop of the diagram above. escribe full the single transformation that will map T to V. W is the image of T under the successive transformation R 1, followed b R, followed b R 3 and so on to R n. (c) escribe full the single transformation that will map T to W: (i) when n is even (ii) when n is odd. (LON) 87

34 88 E F G H K L M N P J M L N F E m n l Z Y X J K W I H G 3 5 Q 5 R P 5 S T U m n l 1.1 ongruence 1. From the shapes in the diagram, write down the letters of three pairs of congruent shapes. (LON). State which pairs of triangles are congruent.

35 3. For each question below, determine whether the triangles are congruent. If the triangles are congruent, justif our answer. (c) (d) (e) (f). Pthagoras wrote about a triangle where = 1unit, = 1 unit, and ˆ = Pthagoras claimed that the length of is a rational number. Was he correct? Eplain our answer. 1 The triangle PQR has PQ = 1 unit, QR = 3 units and QPR ˆ = 90, as shown. P 1 Q R Is the triangle PQR congruent to triangle? Eplain our answer. (SEG) 89

36 Here are si L-shapes on a unit square grid. E F (i) Which two L-shapes are congruent? (i) Which L-shape is an enlargement of? Here is a different L-shape. Show on a cop of the grid opposite how four of these L-shapes can be fitted together to make a square. (c) On a cop of the grid below, draw the L-shape after a rotation of 180. (Q) 90

37 1.13 Similarit 1. The diagram shows two similar triangles. 5. cm 0 E 70 cm 9 cm What is: F the size of angle FE the length of E (c) the ratio : F? 3. clinder has a height of 10 cm and has volume 300 cm. Find the volumes of similar clinders of heights: 5 cm 0 cm. 3. Two cubes have volumes 79 cm 3 and 1331 cm 3. What is the ratio of: the side length of the cubes the surface areas of the cubes?. In the diagram, = metres, E = 3 metres and = 5 metres. is parallel to E. E and are straight lines. 5 m Eplain wh triangle is similar to triangle E. m 3 m alculate the length of. E (LON) 5. Show clearl that these two triangles are similar. alculate the value of cm 3 cm cm 0 70 cm (NE) 91

38 1.13. ll these triangles are similar. alculate the length. alculate the length. F H G 3 9 z 3 E 9 I (SEG) 7. F E 3.5 m 5. m 15 m In the diagram, FG = 5. metres, EH = 35. metres and H = 15 metres. EH is parallel to FG. H G FE 50 and cm HG are straight lines. 8. alculate the length of G. O 85 cm h cm (LON) Floor 75 cm The diagram shows a simplified ironing board. The feet, and, are 75 cm apart. The supports at and are 50 cm apart. The legs, and, are equal in length and are pivoted at O. is parallel to. The height of the ironing board above the floor is 85 cm. Use similar triangles to calculate the height, h cm, of O above the floor. 9

39 9. The diagram shows a smmetrical framework for a bridge. = 100 m = = 70 m E F (i) alculate the angle. (ii) alculate the length E. similar framework is made with the length corresponding to (i) alculate the length corresponding to. (ii) What is the size of the angle corresponding to angle? 10. The model of the cross-section of a roof is illustrated below. = 180 m. (SEG) = cm = 9 cm E ˆ = cm 9 cm (i) (ii) alculate the length of E. Triangles E and E are similar triangles with angle E equal to angle E. alculate the length of. E 19.5 When the roof is constructed, the actual length of is.5 m. alculate the area of the cross-section of the actual roof space. (SEG) 11. Paul's ladder is m long. (i) Paul leans his ladder against a vertical wall, with the end, on horizontal ground. The angle between the ladder and the ground is 70. alculate the distance of from the wall. m 70 (ii) Pamela moves the ladder and uses it to reach a windowsill which is 3.8 m above the gournd. For safet, the angle between the ladder and the ground should be within of 70. m 3.8 m Is the ladder safel placed? (You must show some calculation to eplain our answer.) 93

40 1.13 Ranjit has a stepladder. S P is the midpoint of RS. Q is the midpoint of TS. The length of PQ is 0 cm. (i) (ii) Eplain wh triangles SPQ and SRT are similar. alculate the length of RT. R P 0 cm Q T (SEG) 1. child builds a tower from three similar clindricular blocks. The smallest block,, has radius.5 cm and height cm. Find the volume of the smallest block. lock is an enlargement of and block is an enlargement of, each with a scale factor of 1 3. Find the total height of the tower. (MEG) 13. Two similar solid shapes are made. The height of the smaller shape is 7 cm. The width of the smaller shape is cm. The width of the larger shape is 9. cm. Not to scale alculate the height of the larger shape. The volume of the larger shape is 95 cm 3. Find the volume of the smaller shape. (SEG) 1. This is a scale model of the proposed Millennium Tower. θ The model is a cone of height m and base radius 0.1 m. The angle between the slanting side and the vertical is θ. 0.1 m m (c) What is the value of θ, to the nearest degree? The proposed Millennium Tower is 1000 m high. What is its base radius? alculate, in cubic metres, the volume of the model. (d) How man times larger than the scale model is the volume of the proposed Millennium Tower? (SEG) 9

41 15. and PQR are similar triangles. Q. cm 1.9 cm Not to scale.5 cm P 7.5 cm R Find the length marked (i) (ii) (rea of triangle PQR) = n (rea of triangle ) Find the value of n. (OR) 1.1 Enlargements with Negative Scale Factors 1. The diagram below shows the original shape (shaded) and the images obtained b enlargements with different scale factors. State the scale factor for each enlargement. 95

42 1.1. The shaded shape has been enlarged to give the other images. For each image, find the scale factor and the coordinates of the centre of enlargement. 3. op each diagram below. Enlarge each shape using the scale factor and centre of enlargement given. Scale factor (c) Scale factor 1 (d) Scale factor 1 Scale factor 3 9

43 . op the aes and shape shown below raw the image obtained when the original shape is enlarged with: scale factor, centre of enlargement (0, 0) scale factor 1, centre of enlargement (0, ) (c) scale factor, centre of enlargement (, 0). 5. Triangle P is mapped to triangle Q b an enlargement of scale factor 0.. P E Q F If is of length 5 cm, what is the length of F? 97