STRAND J: TRANSFORMATIONS, VECTORS and MATRICES
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1 Mathematics SKE, Strand J UNIT J Further Transformations: Tet STRND J: TRNSFORMTIONS, VETORS and MTRIES J Further Transformations Tet ontents Section J.1 Translations * J. ombined Transformations
2 Mathematics SKE: STRND J UNIT J Further Transformations: Tet J Further Transformations J.1 Translations translation moves all the points of an object in the same direction and the same distance. The diagram shows a translation. 8 Here ever point has been moved 8 units to the right and units up. This translation is described b what is called a vector 8 Further work on vectors is in Strand J, Unit J. Worked Eample 1 Describe the translation which moves the shaded shape to each of the other shapes shown. D 1
3 J.1 Mathematics SKE: STRND J UNIT J Further Transformations: Tet Solution D 5 To move to, the shaded shape is moved units to the right (horizontall) and units up (verticall). This is described b the vector. To obtain, the shaded shape is moved 5 units to the right 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 D, the shaded shape is moved 5 units to the left and then units up. This is described b the vector Worked Eample 5. The shape shown in the diagram is to be translated using the vector. Draw the image obtained using this translation. Solution The vector describes a translation which moves an object units to the right and units down. This translation can be applied to each point of the original.
4 J.1 Mathematics SKE: STRND J UNIT J Further Transformations: Tet 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 D
5 J.1 Mathematics SKE: STRND J UNIT J Further Transformations: Tet. Describe the translation which moves: (a) F E D (e) D (f) E (g) D F (h) F D E. Draw the shape shown and its image when translated using each of the following vectors. (a) 1 5. (a) Draw the shape shown. Translate using the vector. Translate the image using the vector 1. Which vector would be needed to translate the final image back to the position of the original? 5. (a) Describe a translation which would move one onto another. Describe an other translations which would move a letter onto the same letter in a different position.
6 J.1 Mathematics SKE: STRND J UNIT J Further Transformations: Tet. The number 5 can be formed b translation of the lines and. Describe the translations which need to be applied to and to form the number (a) Draw a simple shape. Write down the coordinates of each corner of our shape. Translate the shape using the vector and write down the coordinates of the new shape. (e) ompare the coordinates obtained in and. How do the change as a result of the translation? Repeat and with a translation using the vector. 8. (a) Draw 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? 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? If a shape was translated using the vector and then the vector which single translation would be equivalent? 8, 9. The points,, and D have coordinates (, 7), (, ), (, ) and (, 7). Find the vector which would be used to translate: (a) to to D D to to D. hallenge! 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. 5
7 Mathematics SKE: STRND J UNIT J Further Transformations: Tet J. 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 Draw the image of the triangle shown if it is first reflected in the line = and then rotated clockwise about the point (, ). 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 (, ), as below Worked Eample entre of Rotation Describe two different was in which the shape marked can be moved to the position shown at
8 J. Mathematics SKE: STRND J UNIT J Further Transformations: Tet Solution One wa, shown below, is to first n alternative approach is to rotate translate using the vector, shape through 18 around point (, ). and then reflect in the line =. This can then be reflected in the line = to obtain, as shown below. 1 8 = Mirror Line 1 8 = entre of Rotation Mirror Line Worked Eample ' O ' ' In the diagram above, O = O', = ' ' and all angles are right angles. O can be mapped onto O' ' ' b a transformation, J, followed b another transformation, K. Describe full the transformations (a) J K. 7
9 J. Mathematics SKE: STRND J UNIT J Further Transformations: Tet Solution (a) Rotate O b 9 clockwise, centre O. Reflect new shape in the -ais. Worked Eample On graph paper, taking 1 cm to represent 1 unit on both the and aes, draw (a) the triangle formed b joining the points (, ), (, ) and (, ). the triangle ' ' ', the image of triangle, under a reflection in the -ais. transformation Q maps the image of triangle onto triangle '' '' '' such that (, ) '' (, ) (, ) '' ( 5, 7) (, ) '' ( 7, ) Draw the triangle '' '' '' Describe the transformation Q in TWO different was. Solution (a),, as shown on the diagram below. 7 '' '' 5 ' '' ' ' For eample, translate b the vector followed b Translate b the vector followed b 1.. 8
10 J. Mathematics SKE: STRND J UNIT J Further Transformations: Tet Eercises 1. (a) Draw a set of aes with and values from 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 =. Translate the shape obtained in using the vector Rotate the original shape through 18 about the point with coordinates (5, ). (a) Draw a set of aes with -values from to 1 and -values from to. Join the points (1, ) and (, 1) to form a straight line. Rotate this line through 9 clockwise around the point (, 1). Describe 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 transformations to the solid line. Each transformation maps the solid line onto one of the dashed lines. Describe how this could be done using: (a) onl rotations onl reflections.. (a) Draw a set of aes with values from to 1 and values from 1 to 8. Join the points with coordinates (, 1), (, 1) and (, ) to form a triangle. Enlarge this triangle with scale factor using the point (, 1) as the centre of enlargement. Rotate the new triangle through 18 about the point (9, ). (e) Describe full the transformations which map the final triangle back onto the original. 5. (a) Draw a set of aes with and values from to 8. Plot and join the points (1, 5), (, 5), (, 8) and (1, 7). Rotate this shape through 9 clockwise around the point (, 5). Then translate the new image using the vector. 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? 9
11 J. Mathematics SKE: STRND J UNIT J Further Transformations: Tet. (a) Draw a set of aes with and values from to 1. Plot the points with coordinates (, 1), (, 1) and (, ) and join them to form a triangle. 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? Reflect the original triangle in the line = and then reflect this image in the line = 11. Which single transformation would map the final triangle back to its original position? 7. H G F E - - D (a) The triangle can be mapped onto, and D using single transformations. Describe full each transformation. The triangle can be mapped onto E, F, G and H using two transformations. Describe full each pair of transformations. 8. (a) (i) Draw a simple shape and reflect it in an vertical line. (ii) Reflect the image in an horizontal line. Describe two other was in which the original image could have been mapped onto the final image. Repeat (a) and using an two lines which are perpendicular. Do ou obtain the same result in each case? 1
12 J. Mathematics SKE: STRND J UNIT J Further Transformations: Tet 9. op the diagram below and then show the answers to the questions on our cop of the diagram. (You are advised to use a pencil.) (a) Draw the reflection of the F in the -ais. Rotate the original F through 9 anticlockwise, with O as the centre of rotation. Draw the image. Enlarge the original F with centre of enlargement O and scale factor. 1. (a) 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. Triangle ma be mapped onto triangle b means of a single rotation. Find the coordinates of the centre of rotation Triangle is reflected in the line = to form triangle. Describe the single transformation which would map triangle onto triangle. 11
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