STRAND I: Geometry and Trigonometry. UNIT 37 Further Transformations: Student Text Contents. Section Reflections. 37.
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1 MEP Jamaica: STRN I UNIT 7 Further Transformations: Student Tet ontents STRN I: Geometr and Trigonometr Unit 7 Further Transformations Student Tet ontents Section 7. Reflections 7. Rotations 7. Translations 7. ombined Transformations
2 MEP Jamaica: STRN I UNIT 7 Further Transformations: Student Tet 7 Further Transformations 7. Reflections Reflections are obtained when ou draw the image that would be obtained in a mirror. Ever point on a reflected image is alwas the same distance from the mirror line as the original. This is shown below. Note istances are alwas measured at right angles to the mirror line. Worked Eample raw the reflection of the shape in the mirror line shown.
3 7. MEP Jamaica: STRN I UNIT 7 Further Transformations: Student Tet Solution The lines added to the diagram show how to find the position of each point after it has been reflected. Remember that the image of each point is the same distance from the mirror line as the original. 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. 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.
4 7. MEP Jamaica: STRN I UNIT 7 Further Transformations: Student Tet The points can then be joined to give the reflected image. Eercises. op the diagrams below and draw the reflection of each object. (a) (d) Mirror Line (e) (f) Mirror Line
5 7. MEP Jamaica: STRN I UNIT 7 Further Transformations: Student Tet. op each diagram and draw the reflection of each shape in the mirror line shown. (a) (d) (e) (f). (a) raw a set of aes with and values from to. (d) Plot the points with coordinates (, ), (, ), (, ), (, ), (, ), (, ). 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 in the -ais. List the coordinates of the corners. (e) 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.
6 7. MEP Jamaica: STRN I UNIT 7 Further Transformations: Student Tet. op the diagram below. 8 8 raw in the mirror line for each reflection described. (a) (d). (a) 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. Repeat using the -ais. X (a) 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. 8. (a) Find the area of the shaded shape. op the diagram and draw the reflection of the shaded shape in the mirrorline.
7 MEP Jamaica: STRN I UNIT 7 Further Transformations: Student Tet 7. 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 9 anti-clockwise Original position entre of Rotation Rotation 8 Rotation 9 clockwise It is often helpful to use tracing paper to find the position of a shape after a rotation. Worked Eample Rotate the triangle shown in the diagram through 9 clockwise about the point with coordinates (, ). Solution entre of Rotation - - ' ' ' The diagram opposite shows how each verte can be rotated through 9 to give the position of the new triangle.
8 7. MEP Jamaica: STRN I UNIT 7 Further Transformations: Student Tet 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 verte of the shape was rotated. F E
9 7. MEP Jamaica: STRN I UNIT 7 Further Transformations: Student Tet Each rotation is now described. to : Rotation of 8 about the point (, ). to : Rotation of 8 about the point (, ). to : Rotation of 9 anti-clockwise about the point (, ). to E: Rotation of 8 about the point (, ). to F: Rotation of 9 anti-clockwise about the point (, ). Eercises. op the aes and triangle shown opposite. (a) Rotate through 9 clockwise around (, ) to obtain. Rotate through 9 anticlockwise around (, ) to obtain. Rotate through 8 around (, ) to obtain Repeat Question for the triangle with coordinates (, ), (, ) and (, ).. op the aes and triangle shown below Rotate the triangle through 8 using each of its vertices as the centre of rotation. 8
10 7. MEP Jamaica: STRN I UNIT 7 Further Transformations: Student Tet. op the aes and shape shown below (a) Rotate the original shape through 9 clockwise around the point (, ). Rotate the original shape through 8 around the point (, ). Rotate the original shape through 9 clockwise around the point (, ). (d) Rotate the original shape through 9 anti-clockwise around the point (, ).. Repeat Question for the triangle with vertices at (, ), (, ) and (, ).. The diagram shows the position of a shape labelled and other shapes which were obtained b rotating. 7 E (a) escribe how each shape can be obtained from b a rotation. Which shapes can be obtained b rotating the shape E? 9
11 7. MEP Jamaica: STRN I UNIT 7 Further Transformations: Student Tet 7. The shape has been rotated to give each of the other shapes shown. For each shape, find the centre of rotation E (a) escribe how each shape shown below can be obtained from b a rotation. E Which shapes cannot be obtained from b a rotation?
12 7. MEP Jamaica: STRN I UNIT 7 Further Transformations: Student Tet 9. On a set of aes with and values from to, draw the triangle with vertices at the points (, ), (, ) and (, ). (a) Rotate the triangle through 9 clockwise about the point (, ). Rotate the second triangle through 9 clockwise about the point (, ). escribe how to obtain the third triangle from the original triangle b a single rotation.. 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 what is called a vector 8 Further work on vectors is in Strand J, Unit 8. 8 Worked Eample escribe the translation which moves the shaded shape to each of the other shapes shown.
13 7. MEP Jamaica: STRN I UNIT 7 Further Transformations: Student Tet Solution 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 units to the right and units down. This is described b the vector. 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 units to the left and then units up. This is described b the vector. Worked Eample The shape shown in the diagram is to be translated using the vector. raw 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.
14 7. MEP Jamaica: STRN I UNIT 7 Further Transformations: Student Tet The points can then be joined to give the translated image. Eercises. 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
15 7. MEP Jamaica: STRN I UNIT 7 Further Transformations: Student Tet. escribe the translation which moves: (a) F E (d) (e) (f) E (g) F (h) F E. raw the shape shown and its image when translated using each of the following vectors. (a) (d). (a) raw the shape shown. Translate using the vector. Translate the image using the vector. (d) Which vector would be needed to translate the final image back to the position of the original?. (a) 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.
16 7. MEP Jamaica: STRN I UNIT 7 Further Transformations: Student Tet. The number can be formed b translation of the lines and. escribe the translations which need to be applied to and to form the number. 7. (a) raw 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. (d) (e) ompare the coordinates obtained in and. How do the change as a result of the translation? Repeat and (d) with a translation using the vector. 8. (a) raw a simple shape and translate it using the vector. Then translate the image using the vector. 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? (d) If a shape was translated using the vector and then the vector which single translation would be equivalent? 8, 9. The points,, and have coordinates (, 7), (, ), (, ) and (, 7). Find the vector which would be used to translate: (a) to to to (d) to. hallenge!. moving onl one coin in the pattern shown, make one row and one column, each containing coins.. Rearrange the 8 coins to form a square with coins on each side. rearranging coins, make a square with coins on each side.
17 MEP Jamaica: STRN I UNIT 7 Further Transformations: Student Tet 7. 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 raw 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. 7 8 Worked Eample entre of Rotation escribe two different was in which the shape marked can be moved to the position shown at. 8 8
18 7. MEP Jamaica: STRN I UNIT 7 Further Transformations: Student Tet Solution One wa, shown below, is to first n alternative approach is to rotate translate using the vector, shape through 8 around point (, ). and then reflect in the line =. This can then be reflected in the line = to obtain, as shown below. 8 = Mirror Line 8 = entre of Rotation Mirror Line 8 8 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. escribe full the transformations (a) J K. (X) 7
19 7. MEP Jamaica: STRN I UNIT 7 Further Transformations: Student Tet Solution (a) Rotate O b 9 clockwise, centre O. Reflect new shape in the -ais. Worked Eample On graph paper, taking cm to represent 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 (, ) '' (, ) (, ) '' (, 7) (, ) '' ( 7, ) raw the triangle '' '' '' (d) escribe the transformation Q in TWO different was. Solution (a),, as shown on the diagram below. (X) 7 '' '' ' '' ' ' (d) For eample, translate b the vector followed b Translate b the vector followed b.. 8
20 7. MEP Jamaica: STRN I UNIT 7 Further Transformations: Student Tet Eercises. (a) raw a set of aes with and values from to 9. Plot the points (, ), (7, ), (9, ) and (7, ). Join them to form a single shape. Reflect the shape in the line =. Translate the shape obtained in using the vector (d) Rotate the original shape through 8 about the point with coordinates (, ). (a) raw a set of aes with -values from to and -values from to. Join the points (, ) and (, ) to form a straight line. Rotate this line through 9 clockwise around the point (, ). 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 transformations to the solid line. Each transformation maps the solid line onto one of the dashed lines. escribe how this could be done using: (a) onl rotations onl reflections.. (a) raw a set of aes with values from to and values from to 8. Join the points with coordinates (, ), (, ) and (, ) to form a triangle. Enlarge this triangle with scale factor using the point (, ) as the centre of enlargement. (d) Rotate the new triangle through 8 about the point (9, ). (e) escribe full the transformations which map the final triangle back onto the original.. (a) raw a set of aes with and values from to 8. Plot and join the points (, ), (, ), (, 8) and (, 7). Rotate this shape through 9 clockwise around the point (, ). Then translate the new image using the vector. (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? 9
21 7. MEP Jamaica: STRN I UNIT 7 Further Transformations: Student Tet. (a) raw a set of aes with and values from to. Plot the points with coordinates (, ), (, ) 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? (d) Reflect the original triangle in the line = and then reflect this image in the line =. Which single transformation would map the final triangle back to its original position? 7. H G F E - - (a) The triangle can be mapped onto, and using single transformations. escribe full each transformation. The triangle can be mapped onto E, F, G and H using two transformations. escribe full each pair of transformations. 8. (a) (i) raw a simple shape and reflect it in an vertical line. (ii) Reflect the image in an horizontal line. (d) escribe 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. o ou obtain the same result in each case?
22 7. MEP Jamaica: STRN I UNIT 7 Further Transformations: Student 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) raw the reflection of the F in the -ais. Rotate the original F through 9 anticlockwise, with O as the centre of rotation. raw the image. Enlarge the original F with centre of enlargement O and scale factor.. (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. escribe the single transformation which would map triangle onto triangle.
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