CHAPTER 5 Cartesian Coordinate System 136 E KL ² ² flXQLWV F JM ² ² XQLWV The area of triangle JKL ðflð XQLW O y H J I x –2 4 2 –4 –2 2 4 6 E F G 5HIHUULQJWRWKH&DUWHVLDQSODQHDVVKRZQ VWDWHWKHFRRUGLQDWHVRIWKHSRLQW (a) E E F F G (d) H (e) I (f) J 3ORW HDFK RI WKH IROORZLQJ SRLQWV LQ D &DUWHVLDQSODQHDQGKHQFHVWDWHLWVGLVWDQFHV from the yD[LVDQGxD[LVUHVSHFWLYHO\ (a) K ffi E L F M G N (e) P I Q O y x –2 4 2 –6 –4 –2 2 4 5HIHUULQJWRWKH&DUWHVLDQSODQHDVVKRZQ VWDWH WKH FRRUGLQDWHV RI DOO WKH YHUWLFHV RI WKHTXDGULODWHUDO 1DPH WKH VKDSH IRUPHG E\ WKH IROORZLQJ SRLQWV (a) A B ²² C ² DQGD E K L M ² DQGN ² F P ² Q ² DQGWKHRULJLQ )LQG WKH KRUL]RQWDO GLVWDQFH DQG YHUWLFDO GLVWDQFHEHWZHHQWKHIROORZLQJSRLQWV (a) A DQGB E C ²fl DQGD )LQGWKHGLVWDQFHIRUHDFKRIWKHIROORZLQJ SRLQWV D DQG E DQG F ²² DQG ²² G ²² DQG ² H DQG I fl DQG ffifl J ² DQG ² K ffi² DQG ²² )LQG WKH GLVWDQFH EHWZHHQ WKH IROORZLQJ SDLUVRISRLQWV*LYH\RXUDQVZHUFRUUHFWWR WZRGHFLPDOSODFHVLIQHFHVVDU\ D DQG E fl DQG ² F ² DQG ² G ² DQG ² H DQG I DQG ²²fl J ² DQG K ²ffi DQG fl² )LQG WKH PLGSRLQW RI WKH OLQH VHJPHQW FRQQHFWLQJHDFKSDLURIWKHIROORZLQJSRLQWV (a) A DQGB E C ² DQGD ffi F H ²²fl DQGK ² (d) P ² DQGQ ² *LYHQWKDWA DQGB ffi ,I SRLQWP GLYLGHVWKH OLQH VHJPHQW LQWKH UDWLR ffl ÀQGWKHFRRUGLQDWHVRISRLQWP Practice 5.2 Basic Intermediate Advanced ©Praxis Publishing_Focus On Maths
137 Cartesian Coordinate System CHAPTER 5 y x O R(–2, –1) P(4, 4) S Q 7KH GLDJUDP DERYH VKRZV D UHFWDQJOH PQRSRQD&DUWHVLDQSODQH)LQG (a) the length of QS E WKHDUHDRIUHFWDQJOHPQRS $ VWUDLJKW OLQH FRQQHFWV WKH SRLQW P k) DQGSRLQWQ(h²ffi *LYHQWKDWWKHPLGSRLQW of PQLV ² )LQGWKHYDOXHVRIh and k *LYHQ WKDWA ² ² DQGB ² ,IB LV WKHPLGSRLQWRIACÀQGWKHFRRUGLQDWHVRI SRLQWC $VWUDLJKWOLQHSDVVHVWKURXJKA ² DQG B ² 3RLQWKGLYLGHVWKHOLQHVHJPHQW AB ZLWK AB AK )LQG WKH FRRUGLQDWHV of K. 7KH GLDJUDP EHORZ VKRZV D VWUDLJKW OLQH CDE x y O –2 C(h, 1) D E(4, k) *LYHQWKDWCD DE)LQGWKH YDOXHVRI h and k *LYHQ WKDW P DQG Q ² ² 3RLQW Q GLYLGHV WKH OLQH VHJPHQW PR ZLWK PQfflQR ffl)LQGWKHFRRUGLQDWHVRIR 5HIHUULQJ WR WKH &DUWHVLDQ SODQH ÀQG WKH OHQJWK RI HDFK VLGH RI WKH TXDGULODWHUDO PQRSDQGKHQFHGHWHUPLQHWKHSHULPHWHU RIWKHTXDGULODWHUDO –1 –2 1 2 3 –3 1 2 3 4 5 –1–2 O y R Q P S x y x O U(–5, 2) T p S 7KH GLDJUDP DERYH VKRZV D ULJKWDQJOHG triangle STU RQ D &DUWHVLDQ SODQH *LYHQ that US XQLWVÀQG D WKHYDOXHRIp E WKHSHULPHWHURIWULDQJOHSTU y x O R(0, k) P(–8, 1) Q(8, 1) 7KH GLDJUDP DERYH VKRZV DQ LVRVFHOHV WULDQJOHSORWWHGRQD&DUWHVLDQSODQH*LYHQWKDW PR XQLWVÀQG D WKHYDOXHRIk E WKHSHULPHWHURIWULDQJOHPQR 7KHORFDWLRQVRIWKUHHWRZQVDUHSORWWHGRQ D&DUWHVLDQSODQHZKHUHXQLW UHSUHVHQWV NP 7RZQ A 7RZQ B and TRZQ C DUH ORFDWHG DW ² ² DQG ² UHVSHFWLYHO\ D &DOFXODWHWKHGLVWDQFHLQNPEHWZHHQ 7RZQBDQG7RZQC*LYH\RXUDQVZHU FRUUHFWWRWZRGHFLPDOSODFHV E :KLFK RI WKH WZR WRZQV 7RZQ B or 7RZQCLVIDUWKHUDZD\IURP7RZQA" ,WLVJLYHQWKDW ² DQG DUH WKUHHYHUWLFHVRIDUHFWDQJOH)LQG D WKHFRRUGLQDWHVRIWKHIRXUWKYHUWH[ E WKHOHQJWKRIGLDJRQDORIWKHUHFWDQJOH F WKHSHULPHWHURIWKHUHFWDQJOH 21 E ² LV WKH PLGSRLQW RI D VWUDLJKW OLQH MRLQLQJD ² DQGG F LVWKHPLGSRLQW of line EG)LQGWKHPLGSRLQWRIDF ©Praxis Publishing_Focus On Maths
138 CHAPTER 5 Cartesian Coordinate System The distance between two points • (0, 0) and (x, y) • (x1 , y1 ) and (x2 , y2 ) = x2 + y 2 = (x2 – x1 )2 + (y2 – y1 )2 Divisor of a line segment P = nx + mx m + n , ny + my m + n The midpoint between two points (x1 , y1 ) and (x2 , y2 ) = x1 + x2 2 , y1 + y2 2 Summary Cartesian Coordinate System Section A 1. A 1 2 3 4 5 B CD E 7KH GLDJUDP VKRZV WKH SRVLWLRQ RI WKH PDULQH OLIH LQVLGH DQ DTXDULXP DW DQ H[KLELWLRQ:KDWLVWKHORFDWLRQRIWKHWXUWOH" A % C ' B % D ' 2. 2 4 –2 –4 –2 O 2 4 x y A B C D 7KH GLDJUDP VKRZV IRXU SRLQWV RQ D &DUWHVLDQSODQH7KHyFRRUGLQDWHRIDSRLQW PLV:KLFKRIWKHSRLQWVODEHOOHGABC and DFRXOGEHSRLQWP" 3. x y O II I III IV 7KH GLDJUDP VKRZV WKH IRXU TXDGUDQWV RI D &DUWHVLDQ SODQH :KLFK RI WKH IROORZLQJ SRLQWVOLHLQTXDGUDQW,9" P ² Q ² R ²² S ² T ² A P and Q B R and S C P and S D Q and T ©Praxis Publishing_Focus On Maths
139 Cartesian Coordinate System CHAPTER 5 4. Point PLVXQLWVIURPWKHxD[LVDQGXQLWV from the yD[LV:KLFKRIWKHIROORZLQJFRXOG EHWKHFRRUGLQDWHVRIP" , ,, ² ,,, ² ,9 ²² A ,DQG,, C ,,,DQG,9 B ,,DQG,,, D ,DQG,9 5. :KLFK RI WKH SRLQWV P ² Q R ² DQG S ffi OLHV RQ WKH yD[LV" A P and Q C R and S B P and S D Q and R 6. 2 4 –2 –4 –2 O 2 4 6 x y R S 7KHGLDJUDPVKRZVD&DUWHVLDQSODQHZLWK WZR SRLQWVR and SRS and T are three YHUWLFHV RI DQ LVRVFHOHV WULDQJOH :KLFK RI WKHIROORZLQJFRXOGEHWKHFRRUGLQDWHVRIT" A C ² B ² D 7. 2 4 6 –2 –6 –4 –2 O 2 4 x y (–6, 0) (–5, 3) (2, 6) (4, 5) 7KH GLDJUDP VKRZV IRXU SRLQWV LQ WKH &DUWHVLDQ SODQH :KLFK RI WKH IROORZLQJ SRLQWVLVWKHQHDUHVWWRWKHRULJLQ" A ² C B ² D 8. O x y T(–2, 5) U(6, 5) W V(6, –2) 7KH GLDJUDP VKRZV D &DUWHVLDQ SODQH ZKHUHSRLQWVTUV and WDUHYHUWLFHVRI DUHFWDQJOH7KHDUHDRITUVWLQXQLW LV A C B D 9. T(10, 9) Q(6, 3) R(4, 9) S P ,Q WKH GLDJUDPPQ RS and T DUH ÀYH SRLQWVLQD&DUWHVLDQSODQH*LYHQWKDWRST LVDVWUDLJKWOLQHDQGSLVWKHPLGSRLQWRIWKH VWUDLJKWOLQHPQ7KHFRRUGLQDWHVRIP are A B C ffi D 10. $VWUDLJKWOLQHSDVVHVWKURXJKA ² DQG B ²ffi² 3RLQWPGLYLGHVWKHOLQHVHJPHQW AB ZLWK AB PB )LQG WKH FRRUGLQDWHV of P A ² B ² C ²² D ²² ©Praxis Publishing_Focus On Maths
140 CHAPTER 5 Cartesian Coordinate System Section B 1. 2QWKH&DUWHVLDQSODQHSURYLGHG D SORWWKHSRLQWVG DQGH ²² E +HQFH PDUN D SRLQW M DQG VWDWH LWV FRRUGLQDWHV VR WKDW GMH LV D ULJKW DQJOHGWULDQJOH 2 4 –2 –4 –4 –2 2 4 x y O 2. 2 4 –2 –4 –4 –2 O 2 4 6 x y C 7KHGLDJUDPVKRZVD&DUWHVLDQSODQH D 6WDWHWKHFRRUGLQDWHVRISRLQWC E 2QWKH&DUWHVLDQSODQHPDUNWKHSRLQW D ZKLFK LV XQLWV IURP C, ZLWK WKH FRQGLWLRQWKDWERWKFRRUGLQDWHVRID are LQWHJHUV 3. D 3ORWWKH SRLQWB RQWKH &DUWHVLDQ SODQHSURYLGHG E 6WDWHWKHGLVWDQFHEHWZHHQSRLQWA and SRLQWB 2 4 –2 –4 –4 –2 2 4 x y O A 4. *LYHQWKDWWKHSRLQWVM ² DQGN(pq ÀQGWKH YDOXHVRIp and q if MNLVSDUDOOHO to the xD[LVDQGKDVDOHQJWKRIffiXQLWV 5. 7KH FDOFXODWLRQ DERYH LV WKH VWHSV ZULWWHQ E\=XONLÁ\WRGHWHUPLQHWKHPLGSRLQWRIPQ 3DUWV RI WKH FDOFXODWLRQ KDYH EHHQ HUDVHG GXH WR HUURUV :KDW DUH VXSSRVHG WR EH ZULWWHQLQWKHSDUWVWKDWKDYHEHHQHUDVHG" 6. y O x P(–4, 6) S Q(–2, 6) R M(–4, 4) 7KHGLDJUDPDERYHVKRZVDUHFWDQJOHPQRS RQD&DUWHVLDQSODQHMLVWKHPLGSRLQWRI PS)LQGWKHFRRUGLQDWHVRISRLQWR 7. y x O E(2, 1) (m, 9) F(6, n) G 7KHGLDJUDPDERYHVKRZVWKHVWUDLJKWOLQH EFGRQD&DUWHVLDQSODQHZKHUHEF FG )LQGWKHYDOXHVRIm and n 8. (8, b) Cave Waterfall (a, 2) (5, 4) Camping site 7KHGLDJUDPVKRZVWKDWWKHFDPSLQJVLWHLV ORFDWHG KDOIZD\ EHWZHHQWKH FDYH DQGWKH ZDWHUIDOO)LQGWKHYDOXHVRIa and b ©Praxis Publishing_Focus On Maths
141 Cartesian Coordinate System CHAPTER 5 9. 7KH GLDJUDP EHORZ VKRZV D VWUDLJKW OLQH PQR y x O R(7, 6) P(–5, 8) Q(4, k) Point Q GLYLGHV WKH VWUDLJKW OLQH PR VXFK that PQfflPR mffln)LQGWKHYDOXHVRIm and n 10. y x O D(–4, 1) C(–4, q) E(8, 1) 7KH GLDJUDP DERYH VKRZV D ULJKWDQJOHG triangle CDE GUDZQ RQ D &DUWHVLDQ SODQH *LYHQ WKDW WKH DUHD RI WKH WULDQJOH LV XQLWV ÀQG D WKHYDOXHRIq E WKHOHQJWKRICE 11. y x O A C B(–4, k) 7KH GLDJUDP DERYH VKRZV D WUDSH]LXP OABC RQ D &DUWHVLDQ SODQH *LYHQ WKDW OC AB and BC XQLWV D )LQGWKHYDOXHRIk E &DOFXODWHWKHSHULPHWHURIWKHWUDSH]LXP 12. y x O M(4, –6) N(10, –6) P 7KHGLDJUDPDERYHVKRZVDWULDQJOHPMN Point PLVflXQLWVIURP xD[LV&DOFXODWHWKH area of triangle PMN 13. 7KH GLDJUDP EHORZ VKRZV D SHQWDJRQ ABCDE y x O 8 D E(–2, 1) A B(5, –3) C *LYHQ WKDW WKH VLGH AB LV SDUDOOHO WR WKH xD[LV DQG WKH DUHD RI WULDQJOH ADE LV XQLWV )LQG WKHFRRUGLQDWHVRID. 14. y x O Post office Madam Yeo’s house (3, 1) 5 School 7KH GLDJUDP DERYH VKRZV WKH ORFDWLRQ RI 0DGDP <HR·V KRXVH D SRVW RIÀFH DQG D VFKRROWKDWDUHGUDZQRQD&DUWHVLDQSODQH ,WLVJLYHQWKDWWKHDFWXDOGLVWDQFHEHWZHHQ 0DGDP <HR·V KRXVH DQG WKH VFKRRO LV NP D )LQGWKHDFWXDOGLVWDQFHLQNPEHWZHHQ 0DGDP<HR·VKRXVHDQGWKHSRVWRIÀFH E 0DGDP<HR·VZDQWVWRIHWFKKHUGDXJKWHU IURPVFKRRODIWHUSD\LQJELOOVDWWKHSRVW RIÀFHDQGJRHVKRPHXVLQJWKHVKRUWHVW GLVWDQFH)LQGWKHDFWXDOGLVWDQFHLQNP RIKHUZKROHMRXUQH\ 15. 7KH GLDJUDP EHORZ VKRZVWKH SRVLWLRQV RI &KDUOLHDQG0XVOLPDK y x O Charlie (–8, 2) Muslimah (7, 5) 7KH\PRYHWRZDUGVHDFKRWKHULQDVWUDLJKW OLQHVXFKWKDWWKHYHORFLW\RI&KDUOLHLVWZLFH WKHYHORFLW\RI0XVOLPDK)LQGWKHGLVWDQFH RI 0XVOLPDK IURP KHU LQLWLDO SRVLWLRQ ZKHQ VKHPHHWV&KDUOLH ©Praxis Publishing_Focus On Maths
6TRANSFORMATIONS Applications of this chapter Transformation has always been a key concept utilised by architects in their building designs. By combining various types of transformations, they can create distinct and unique shapes that offer dynamic and innovative spaces for users. The principle of transformation allows architects and designers to effectively leverage changes in the surrounding environment, providing users with more opportunities to utilise and interact with the space. &DQ \RX ÀQG VRPHEXLOGLQJV RQWKHLQWHUQHWWKDWDSSOLHGWKH FRQFHSW RI transformation? What is the transformation applied? 142 ©Praxis Publishing_Focus On Maths
Concept Map Learning Outcomes • Understand the concept of transformation. • Understand and use the concept of translation. • 8QGHUVWDQGDQGXVHWKHFRQFHSWRIUHÁHFWLRQ • Understand and use the concept of rotation. • Understand and use the concept of similarity. • Understand and use the concept of enlargement. • Understand and use the properties of quadrilaterals using concept of transformation. 143 Translation Enlargement Christian Felix Klein was a German mathematician and mathematics educator who made important contributions to the study of geometry by using transformations as a foundation. +HVWDUWHGE\H[SORULQJUHÁHFWLRQ • Clockwise • Anticlockwise • Direction • Orientation • Plane • Image • Object • One-to-one correspondence • Reflection • Symmetry • Rotational symmetry • Transformation • Translation • Rotation • Enlargement • Proportion • Scale factor • Similar Key Terms Maths History 5HÀHFWLRQ Transformations Point and Line Translation Scale Factor of Enlargement 5HÀHFWLRQRIX-axis, Y-axis and O 5HÀHFWLRQRIOLQHy = x and y = –x 5HÀHFWLRQRI/LQHx = a and y = b Plane Translation Centre of Enlargement V\PPHWU\DOVR NQRZQDV UHÁHFWLRQLQDOLQH RUDFURVV an axis. ©Praxis Publishing_Focus On Maths
CHAPTER 6 Transformations 144 Flashback 1. Draw the images of the letters in the mirror. A B C A piece of paper is folded into half and cut. (a) How does each paper look like when unfolded? Circle the correct answer. (b) What transformations are shown in the diagrams? 1 If you stand 3 m in front of a plane mirror, how far away would you see yourself in the mirror? Critical Thinking 2. Draw the direction of the key when it is turned (a) clockwise. (b) anticlockwise. ©Praxis Publishing_Focus On Maths
Transformations CHAPTER 6 145 6.1 Transformations A Explaining the idea of transformation The stated changes show a one-to-one correspondence between points in a plane where every point in the object undergoes a similar change. The change is known as a transformation. In a transformation, if A changes to B, then A is known as the object and B is known as the image of the transformation. Moves to Object Image B A A transformation is also known as a mapping. Thus, in a transformation with object A and image B, we can say that A is mapped onto B under the transformation. When an object XQGHUJRHVDWUDQVIRUPDWLRQWKHREMHFWFRXOGEHPRYHGÁLSSHGRUURWDWHG When an object undergoes a change, it could involve a change in position and orientation. For example, (a) A B (b) P A B Q (c) moving the pencil from A to B involves a change in position. ÁLSSLQJ WKH PDWFKVWLFN about line PQ from A to B involves changes in position and orientation. rotating someone on a Ferris wheel from A to B involves a change in position. EXAMPLE 1 Determine whether each of the following is a transformation. (a) Every point of a triangle is moved through the same distance in the same direction. (b) N M Movement of M to N (c) P Q Movement of P to Q ©Praxis Publishing_Focus On Maths
CHAPTER 6 Transformations 146 Solution: (a) A transformation. Every point in the triangle is moved in the same manner. (b) A transformation. Every point in M is moved in the same manner to reach N. (c) Not a transformation. The movement of points in the triangle does not correspond to a similar manner. )RUHDV\LGHQWL¿FDWLRQRIREMHFW and image, the apostrophe sign (' ) is usually used to denote the LPDJH RI DQ REMHFW SRLQW )RU instance, we use A' to denote the image of point A, B' to denote the image of point B and VRRQ Each of the following statements implies that M LV WKH REMHFW and N is the image under a WUDQVIRUPDWLRQ • M is mapped onto N under DWUDQVIRUPDWLRQ • A transformation maps M onto N • M undergoes a transformation to form N • N is the image of M under a WUDQVIRUPDWLRQ B Identifying the object and its image in a transformation When P undergoes a transformation to form Q, P is known as the object and Q is known as the image under the transformation. For example, A' A Triangle A undergoes a transformation to form A'. Therefore, A is the object and A' is the image. EXAMPLE 2 In each of the following transformations, determine the object and the image. (a) (b) Solution: (a) M is the object and N is the image. (b) Q is the object and P is the image. M N P Q ©Praxis Publishing_Focus On Maths
Transformations CHAPTER 6 147 &DUU\RXWWKHIROORZLQJDFWLYLW\ZLWK\RXUFODVVPDWHV /RRNIRU GHVLJQV DURXQG \RXWKDW FDQ EH GHVFULEHG XVLQJWUDQVIRUPDWLRQV&RS\ HDFK GHVLJQ'HVFULEH D SRVVLEOHVHWRIWUDQVIRUPDWLRQVIRUHDFKGHVLJQ team work Practice 6.1 Basic Intermediate Advanced Determine whether each of the following is a transformation. (a) A table is being moved from ground ÁRRUWRWKHWKLUGÁRRU (b) Movement of R to S P Q Movement of Q to P R S State the object and the image in each of the following transformations. (a) K L (b) M N (c) P Q (d) R S 6.2 Translation A Recognise a translation A translation is a transformation that takes place when all points on a plane are moved in the same direction through the same distance. For example, Every point on triangle A is moved in the direction equivalent to moving 4 units left followed by 3 units upwards to form triangle B. The movement of A to B is a translation. A B EXAMPLE 3 In each of the following transformations, determine whether it is a translation. (a) (b) Solution: (a) The transformation from P to Q is a translation. (b) The transformation from S to R is not a translation. P Q R S All the points on P are moved in the same direction through the same distance. The points on S are neither moved in the same direction nor through the same distance to form R. ©Praxis Publishing_Focus On Maths
CHAPTER 6 Transformations 148 B Describe a translation A translation is a transformation whereby only the position of the object changes. It is described by how far an object moves horizontally (left or right) and by far it moves vertically (up or down). Under a translation, the object and the image have the same shape, size and orientation. A translation can be described by stating the direction and distance of movement using graphical, verbal or symbolic method. (I) Using graphical method: by drawing an arrow EXAMPLE 4 In the diagram, R' is the image of R under a translation. Describe the translation using an arrow labelled with m to state the direction and distance of movement. Solution: R R m Translation of arrow m. R R By taking one object point as a key point, draw an arrow to the corresponding image. The arrow illustrates the direction and distance of movement of the translation. (II) Using verbal method: by stating horizontal movement followed by vertical movement EXAMPLE 5 In the diagram, U' is the image of U under a translation. Describe the translation using verbal method. Solution: U U 4 units 3 units Translation of moving 4 units to the right followed by 3 units downwards. U U By taking one object point as a key point, identify the distance towards left or right followed by the distance upwards or downwards to map it to the image. ©Praxis Publishing_Focus On Maths
Transformations CHAPTER 6 149 (III) Symbolic method: by stating the translation vector A translation can also be described using the symbol of a vector. Let arrow PQ represents the direction and distance of a translation, as shown in the diagram, then the translation can be represented as vector P ±AQ. A translation can also be represented using symbol of vector a b or using the symbolic representation (x, y) A (x + a, y + b), where a represents the horizontal movement (parallel to x-axis) and b represents the vertical movement (parallel to y-axis). The values of a and b can be positive or negative according to the direction of movement. x is negative x is positive y is positive y is negative EXAMPLE 6 In the diagram on the right, M' is the image of M under a translation. Describe the translation using symbolic representation. Solution: Translation R ±AS or translation 4 –5 or translation (x, y) A (x + 4, y – 5). EXAMPLE 7 Point M(2, –1) is mapped onto M' (–1, 3) under a translation. Describe the translation using (a) graphical representation, (b) verbal representation, (c) symbolic representation. M M The direction and distance of the translation can be represented by arrow RS, where each point is moved 4 units to the right followed by 5 units downwards. M M R S 5 units 4 units P Q In the translation of the form a b , if a = 0, then the REMHFWLV moved upwards or downwards RQO\,Ib = 0, WKHQWKHREMHFWLV PRYHG OHIW RU ULJKW RQO\ ©Praxis Publishing_Focus On Maths
CHAPTER 6 Transformations 150 How does the concept of translation (transformation) apply to everyday life situations or activities, and what DUHVRPHVSHFL¿FH[DPSOHVRILWVDSSOLFDWLRQ" C Determine the image and object under a translation (I) Determining the image The vertices of a diagram are usually used as key points to work out the image of a WUDQVIRUPDWLRQ EXAMPLE 8 Determine and draw the image of triangle R under the translation equivalent to moving 6 units to the left followed by 4 units upwards. Solution: R R' Using the vertices of the triangle as key points, move each point 6 units left followed by 4 units upwards. EXAMPLE 9 For each of the following, draw the image M' of object M under the stated translation. (a) q M R Translation of q. Solution: –2 –2 –4 2 2 4 4 y x p O M(–1, 3) M(2, –1) (a) Translation of arrow p. (b) Translation of moving 3 units to the left followed by 4 units upwards. (c) Translation –3 4 or translation (x, y) A (x – 3, y + 4). ©Praxis Publishing_Focus On Maths
Transformations CHAPTER 6 151 (b) M (c) M Solution: (a) q M M (b) M M (c) M M :KDWLVWKHGLIIHUHQFHEHWZHHQWUDQVODWLRQ –4 3 and 4 –3 ? Critical Thinking EXAMPLE 10 Find the image of point R(–2, –3) under translation 4 5 . Solution: y x –2 O –2 2 2 4 –6 –4 4 R(–2, –3) R 5 units 4 units R'(2, 2) Translation of moving 3 units to the left followed by 2 units upwards. Translation 4 –1 . The image of point (x, y) under translation a b can EH determined using the representation ( x, y) A(x + a, y + b For instance, in Example 11, R ' = (–2 + 4, –3 + 5) = (2, 2) ©Praxis Publishing_Focus On Maths
CHAPTER 6 Transformations 152 EXAMPLE 11 Determine the image P(5, –2) under the translation –4 3 . Solution: The image of P(5, –2) = [5 + (– 4), –2 + 3] = (1, 1) EXAMPLE 12 Copy and construct the image of shape M under translation P ±AQ. Solution: Construct a parallel line of the same length as PQ from every vertex of shape M. Join all the image points. M P Q M M P Q Plot the point P in a &DUWHVLDQ SODQH The image is (1, 1) as VKRZQ 2 2 –2 –2 4 y x P P O (II) Determining the object The concept of inverse is used to determine the object in a translation if the image is given. For instance, (a) if an object is mapped onto its image under translation P ±AQ , then the image will be mapped onto the object under translation Q ±AP . (b) if an object is mapped onto its image under translation a b, then the image will be mapped onto the object under translation –a –b . (c) if an object is mapped onto its image under the translation represented by moving a units to the right followed by b units upwards, then the image will be mapped onto the object under the translation represented as moving a units to the left followed by b units downwards. (d) if point (p, q) is mapped onto image (m, n) under translation (x, y) A (x + a, y + b), then (m, n) will be mapped onto (p, q) under translation (x, y) A(x – a, y – b). EXAMPLE 13 In the diagram, R is the image of object T under translation P ±AQ. Draw and label T. R Q P ©Praxis Publishing_Focus On Maths
Transformations CHAPTER 6 153 Solution: R T Q P ,IDQREMHFWLVPDSSHGRQWRLWVLPDJHXQGHUDWUDQVODWLRQ a b , what is the translation vector, in terms of a and b, ZKLFKPDSWKHLPDJHWRWKHREMHFW"'LVFXVVDQGH[SODLQ\RXUDQVZHUZLWKDGLDJUDP7KHQSUHVHQWWRWKHFODVV INTERACTIVE ZONE T is the image of R under translation Q ±A P. EXAMPLE 14 In the diagram, N is the image of object M under translation 2 –3 . Draw and label M. Solution: N M EXAMPLE 15 Given that (–2, 6) is the image of point M under translation 1 –5 ÀQGWKHFRRUGLQDWHVRISRLQWM. Solution: The coordinates of point M = (–2 + 1, 6 – 5) = (–1, 1) N M is the image of (–2, 6) under translation 1 –5 . M is the image of N under translation –2 3 . ©Praxis Publishing_Focus On Maths
CHAPTER 6 Transformations 154 EXAMPLE 16 Moving 4 units to the right followed by 2 units upwards maps (h, k) to (5, –3). Find the value of h and of k. Solution: (h, k) = (5 – 4, –3 – 2) = (1, –5) Thus, h = 1, k = –5. Moving 4 units to the ULJKWIROORZHGE\XQLWV upwards is equivalent to translation 4 2 Thus, (h, k) is the image of (5, –3) under translation –4 –2 Objective: To explore and understand the concept of translation. Instruction: Carry out this activity in pairs. 1. Open the Translation in Action activity ÀOH XVLQJ GeoGebra. 2. Click on the points A, B or C individually and drag it to change the shape of the object ABC if necessary. 3. Drag the points D and E to manipulate the length and direction of the arrow. 4. &OLFN RQ WKH ҊAnimation’ button and observe how the image of object ABC is constructed. 5. 5HSHDW 6WHSV WR D IHZ WLPHV &OLFN RQ WKH ҊShow traces of movement ’ if necessary. (a) What is the relationship between the direction and the length of the arrow DE with the transformation shown? (b) Does every point A, B, and C move in the same way to produce its image? (c) Do the object ABC and its image have the same shape, size and orientation? 6. Discuss the questions with your partner. What conclusion can you make? 1 From Activity 1, we found that object ABC undergoes a translation and produces image . ©Praxis Publishing_Focus On Maths
Transformations CHAPTER 6 155 D Solve problems involving translations EXAMPLE 17 The diagram shows a helicopter moving from position I to position II. Describe the translation representing the movement of the helicopter. Solution: II I Translation –4 5 . II 5 u nits 5 units 4 units 4 I EXAMPLE 18 In the Cartesian plane shown, N is the image of M under a transformation. Describe the transformation. Solution: EXAMPLE 19 The diagram as shown is drawn on a grid. Given that triangle R is the image of triangle P under a translation, which of the triangles labelled A, B, C and D is the image of triangle M under the same translation? x y 4 2 –4 –2 2 4 O –2 –4 M N x y 4 2 –4 –2 2 4 O –2 –4 M N It is a translation –5 5 . The transformation LVDWUDQVODWLRQ A B C M D R P ©Praxis Publishing_Focus On Maths
CHAPTER 6 Transformations 156 Determine whether each of the following transformations is a translation. (a) (b) (c) (d) F F Copy each of the following and draw the image of the shape M under the stated translation. (a) M (b) M (c) M (d) 3 units to the left followed by 2 units upwards. 6 units to the right followed by 1 units upwards. M 5 units to the right followed by 1 units downwards. Practice 6.2 Basic Intermediate Advanced 2 units to the left followed by 4 units downwards. Solution: A B C M D R P EXAMPLE 20 On the Cartesian plane, point B is the image of point A under a translation. Draw the image of quadrilateral P under the same translation. Solution: 2 –2 –4 4 –6 –4 2 4 6 x y O A B P –2 P Answer: C 2 –2 –4 4 –6 –4 2 4 6 x y O A B –2 P ©Praxis Publishing_Focus On Maths
Transformations CHAPTER 6 157 In each of the following, P' is the image of P under a translation. Describe the translation (i) using left or right movement followed by upward or downward movement, (ii) in the form a b . (a) P P (b) P P (c) P P (d) P P Determine whether each of the following statements is true or false. (a) If P is the image of Q under a translation, then P and Q have the same size. (b) Under a translation, the object has a larger area than the image. (c) A translation could map an equilateral triangle onto an isosceles triangle. Find the image of point P(5,–8), under each of the following translations. (a) 3 –2 (b) –5 –4 (c) 8 15 (d) –12 4 For each of the following, find the coordinates of the object that is mapped onto the given coordinates of the image under the translation stated. (a) (–5, 1); 2 –3 (b) (8, –9); 11 –3 (c) (–10, –7); –6 7 (d) (4, 2); –7 –9 P A B C D Copy the diagram and draw the image of the shape P under the translation equivalent to the translation of (a) A to B (b) B to C (c) C to D If (5, –6) is mapped onto (–4, 9) under a WUDQVODWLRQÀQGWKHFRRUGLQDWHVRIWKHLPDJH of (–7, 11) under the same translation. If (–7, 10) is the image of (–5, –2) under a WUDQVODWLRQÀQGWKHFRRUGLQDWHVRIWKHREMHFW that is mapped onto (8, 6) under the same translation. K In the international chess game, one move of the knight is equivalent to moving two units horizontally followed by one unit vertically or one unit horizontally followed E\WZRXQLWVYHUWLFDOO\0DUNZLWK¶X·DOOWKH SRVVLEOH PRYHV RI WKH NQLJKW ¶K· RQ WKH chessboard shown. Hence, represent each move by a translation in the form a b . ©Praxis Publishing_Focus On Maths
158 CHAPTER 6 Transformations ffl9LÅLJ[PVU A 9LJVNUPZLHYLÅLJ[PVU A UHÁHFWLRQ LVDWUDQVIRUPDWLRQWKDWWDNHVSODFHZKHQDOOSRLQWVLQDSODQHDUHÁLSSHGRYHU in the same plane in a line known as the D[LVRIUHÁHFWLRQ. For example, The triangle PLVÁLSSHGRYHUDORQJWKHOLQHAB to form triangle P'. Triangle P is said to have undergone a UHÁHFWLRQLQWKHOLQHAB to form the image P'. Notice that if triangle P, triangle P' and the line AB are traced onto a piece of tracing paper, and the tracing paper is folded along the line AB, triangle P and triangle P' overlap perfectly. EXAMPLE 21 Determine whether each of the following transformations is DUHÁHFWLRQ (a) (b) Solution: D $UHÁHFWLRQ E 1RWDUHÁHFWLRQ 'LVFXVVZLWK\RXUIULHQGVDQGVWDWHVRPHH[DPSOHVRIUHÀHFWLRQLQ\RXUGDLO\OLIH INTERACTIVE ZONE B +LZJYPILYLÅLJ[PVU $ UHÁHFWLRQ LV GHVFULEHG E\ VWDWLQJ WKH axis of UHÁHFWLRQ. )RULQVWDQFHҊUHÁHFWLRQRQOLQHMN ҋ if line MN is the D[LVRIUHÁHFWLRQ P Axis of reflection P R Q Q R P P A B EXAMPLE 22 On the Cartesian plane as shown, H is the image of G under a reflection. Describe WKHUHÁHFWLRQ x y –2 2 O 2 4 H G –4 4 ©Praxis Publishing_Focus On Maths
Transformations CHAPTER 6 159 Solution: x y –2 2 O 2 4 3 units 3 units H G –4 4 5HÁHFWLRQRQWKHOLQHSDVVLQJWKURXJK ² EXAMPLE 23 The diagram is drawn on a square grid. Triangle N is the image of triangle M XQGHU D UHÁHFWLRQ 'HVFULEH WKH UHÁHFWLRQ Solution: 5HÁHFWLRQRQOLQe PQ. C +L[LYTPUL[OLPTHNLHUK[OLVIQLJ[\UKLYHYLÅLJ[PVU (I) Determining the image EXAMPLE 24 In the diagram, draw the image of shape MXQGHUWKHUHÁHFWLRQRQOLQHPQ. Solution: ,QDUHÀHFWLRQLISRLQWAOLHVRQWKHD[LVRIUHÀHFWLRQKRZZRXOG\RXGHWHUPLQHWKHLPDJH"'LVFXVVZLWK\RXU FODVVPDWHV Critical Thinking Find the perpendicular ELVHFWRU RIWKH OLQH MRLQLQJ DQ REMHFW SRLQW DQG LWV LPDJH M N 'UDZ WKH D[LV RI UHÀHFWLRQ DQG ODEHO ZLWK VXLWDEOH OHWWHUV Q P M N P Q M (DFK REMHFW SRLQW and its image have the same perpendicular distance from the D[LV RI UHÀHFWLRQ P Q M M There are many ways to VWDWH WKH D[LV RI UHÀHFWLRQ For instance in Example 23, WKH UHÀHFWLRQ FDQ DOVR EH GHVFULEHG DV ҊUHÀHFWLRQ RQ WKH line passing through (–1, 2) and parallel to the y-axis’ and HWF ©Praxis Publishing_Focus On Maths
CHAPTER 6 Transformations 160 EXAMPLE 25 Find the coordinates of image of point (–2, 3) under a UHÁHFWLRQRQWKHy-axis. Solution: y O x 2 2 –2 –4 4 –4 –2 4 (–2, 3) (2, 3) 2 units 2 units Image is (2, 3). EXAMPLE 26 Copy and construct the image of shape A XQGHU D UHÁHFWLRQ RQ OLQH PQ. Solution: Select a suitable vertex of shape A and construct the perpendicular line from the vertex to PQ. Mark the image of the vertex so that the object and image have the same distance from PQ. Repeat steps and for all other vertices. Join all the image points to complete the image. A Q P A Q P A (II) Determining the object Concept of inverseLVXVHGWRGHWHUPLQHWKHREMHFWRIDUHÁHFWLRQLIWKHLPDJHLVJLYHQ,IA is mapped onto BXQGHUDUHÁHFWLRQWKHQB is mapped onto AXQGHUWKHVDPHUHÁHFWLRQ EXAMPLE 27 On the Cartesian plane, find the coordinates of object M which is mapped onto point N unGHUWKHUHÁHFWLRQRQWKH y-axis. O –2 –2 2 2 N –4 4 x y ©Praxis Publishing_Focus On Maths
Transformations CHAPTER 6 161 Objective:7RH[SORUHWKHFRQFHSWRIUHÁHFWLRQ Instruction: Carry out this activity in pairs. 1. Trace the following diagram onto a piece of tracing paper and fold along the line MN. R D Q E P B C A F G H M N 2. Determine the image of P, Q and RUHVSHFWLYHO\XQGHUDUHÁHFWLRQLQWKHOLQHMN. 3. Study the diagram below. 2 Solution: O –2 –2 2 2 N M(3, 1) –4 4 x y Thus, the coordinates of object M = (3, 1). EXAMPLE 28 Find the coordinates of the object that is mapped onto ² XQGHU WKH UHÁHFWLRQ RQ WKH OLQH MRLQLQJ SRLQW and point (–3, –3). Solution: y O x 2 2 –2 –4 4 –4 –2 4 (–4, 2) (2, –4) The coordinates of the object is (2, –4). M is the image of N under WKH VDPH UHÀHFWLRQ ©Praxis Publishing_Focus On Maths
CHAPTER 6 Transformations 162 4. Which of the triangles labelled A, B, C and D cannot be the image of triangle M under a UHÁHFWLRQ" 5. Discuss your answers with your partner. M D C B A Objective:7RH[SORUHDQGXQGHUVWDQGWKHFRQFHSWRIUHÁHFWLRQ Instruction: Carry out this activity in pairs. Material: Mirror 1. Open the 5HÁHFWLRQLQ$FWLRQDFWLYLW\ÀOHXVLQJGeoGebra. 2. Click on the points A, B or C individually and drag it to change the shape of the object ABC if necessary. 3. Drag the points D and E to manipulate the orientation of the line DE if necessary. 4. &OLFN RQWKH ҊAnimation’ button and observe how object ABC is mapped onto its image. 5. 5HSHDW 6WHSV WR D IHZ WLPHV &OLFN RQ WKH ҊShow traces of movement’ if necessary. Discuss the following questions with your partner. (a) What is the relationship between object ABC and its image with line DE? (b) Does every point A, B, and C move in the same way to produce its image? (c) Do the object ABC and its image have the same shape, size and orientation? 6. Discuss with your partner. What conclusion can you make? 3 From Activity 3, we found that object ABC undergoes a on the line and produces image . ©Praxis Publishing_Focus On Maths
Transformations CHAPTER 6 163 Determine whether each of the following WUDQVIRUPDWLRQVLVDUHÁHFWLRQ (a) (b) (c) (d) Determine whether each of the following statements is true or false. D 8QGHU D UHÁHFWLRQ WKH REMHFW DQG WKH image have the same orientation. (b) If the line MN is mapped onto MN' XQGHU D UHÁHFWLRQ WKHQ SRLQW M lies RQ WKH D[LV RI UHÁHFWLRQ (c) If ABCD is mapped onto A'B'C'D' under D UHÁHFWLRQ LQ WKH OLQH MN, then AB is perpendicular to MN. (d) If triangle A'B'C' is the image of triangle ABC XQGHU D UHÁHFWLRQ WKHQ ABC = A'B'C'. Copy each of the following and construct the image of shape PXQGHUWKHUHÁHFWLRQLQWKH line MN. (a) P M N (b) (c) (d) M N P M N P P M N Practice 6.3 Basic Intermediate Advanced D :VS]LWYVISLTZPU]VS]PUNYLÅLJ[PVU EXAMPLE 29 In the diagram, trapezium P'Q'R'S' is the image of trapezium PQRS under a UHÁHFWLRQRQWKHx-axis. Find the coordinates of image M under WKHVDPHUHÁHFWLRQ. Solution: 2 –2 –4 4 –4 2 4 6 8 x y O M P Q R M –2 S P Q R S M' (–4, 3) 2 –2 –4 4 –4 2 4 6 8 x y O M P Q R –2 S P Q R S ©Praxis Publishing_Focus On Maths
164 CHAPTER 6 Transformations Each pair of the following diagrams UHSUHVHQWV D UHÁHFWLRQ &RQVWUXFW WKH D[LV RIUHÁHFWLRQIRUHDFKUHÁHFWLRQ (a) (b) (c) (d) Find the image of M(9, –2) under the UHÁHFWLRQ (a) in the x-axis, (b) in the y-axis, (c) in the line passing through (3, –5) and parallel to the y-axis, (d) in the line passing through (–1, 4) and parallel to the x-axis, (e) in the line passing through (3, 3) and the origin. Find the point that is mapped onto N(–2, – 6) XQGHUWKHUHÁHFWLRQ (a) in the x-axis, (b) in the y-axis, (c) in the line passing through (– 4, –1) and parallel to the x-axis, (d) in the line passing through (3, 2) and parallel to the y-axis, (e) in the line passing through (2, –1) and (–2, 3). –2 2 –2 2 4 6 –4 4 6 8 O y P Q R T S x 5HIHUULQJWRWKH&DUWHVLDQSODQHÀQG (a) the coordinates of the image of P under D UHÁHFWLRQ LQ WKH y-axis, (b) the coordinates of the object of Q under D UHÁHFWLRQ LQWKH OLQH SDVVLQJWKURXJK (5, 3) and parallel to the x-axis, (c) the coordinates of U if it is mapped onto R XQGHU D UHÁHFWLRQ LQ WKH OLQH passing through (1, 1) and parallel to the y-axis, (d) the coordinates of the image of S under D UHÁHFWLRQ LQWKH OLQH SDVVLQJWKURXJK (1, 0) and (4, 3), (e) the coordinates of the object of T under D UHÁHFWLRQ LQWKH OLQH SDVVLQJWKURXJK (2, –1) and R. –2 2 –2 2 4 6 8 –4 O 468 y x P Q R M S Referring to the Cartesian plane, describe WKHUHÁHFWLRQWKDWPDSVTXDGULODWHUDOM onto quadrilateral (a) P (b) Q (c) R (d) S If A(–3, 6) is mapped onto A' (5, 6) under a UHÁHFWLRQÀQG (a) the coordinates of the image of B ² ² XQGHU WKH VDPH UHÁHFWLRQ (b) the coordinates of the object that is mapped onto C(6, 2) under the same UHÁHFWLRQ 10 P T V W R S U Q M The diagram as shown is drawn on a grid. Copy the diagram and draw the image of triangle MXQGHUWKHUHÁHFWLRQLQWKHOLQH (a) PQ (b) RS (c) TU (d) V W ©Praxis Publishing_Focus On Maths
Transformations CHAPTER 6 165 6.4 Rotation A Recognise a rotation A rotation is a transformation that takes place when all points in a plane are rotated about DÀ[HGSRLQWLQWKHVDPHGLUHFWLRQWKURXJKWKHVDPHDQJOH7KHÀ[HGSRLQWLVNQRZQDVWKH centre of rotation. The angle through which the object is rotated is known as the angle of rotation. The direction in which the object is rotated is stated as clockwise or anticlockwise. For example, Point P moves in a circular motion to P'. The movement of P to P' is a rotation. In the rotation, C is the centre of rotation. The angle of rotation is 100° rotating in the clockwise direction. 100 P P C EXAMPLE 30 Determine whether each of the following transformations is a rotation. (a) C P P (b) C R R Solution: (a) A rotation. (b) Not a rotation. B Describe a rotation A rotation can be described by stating the centre of rotation, the direction of rotation and the angle of rotation. ,I WKH REMHFW DQG WKH LPDJH RI D URWDWLRQ DUH JLYHQ WKH URWDWLRQ FDQ EH GHVFULEHG E\ ÀUVW determining the centre of rotation followed by the angle of rotation and the direction of rotation. ©Praxis Publishing_Focus On Maths
CHAPTER 6 Transformations 166 Object Image Notice that in a rotation, an object point is moved along an arc of a circle to map onto its image. This means, the line joining an object point and its image is a chord of the circle. Thus, the centre of rotation is the intersection point of the perpendicular bisectors of these chords. EXAMPLE 31 In the diagram, E'F'G' is the image of EFG under a rotation. Determine the centre of rotation and hence describe the rotation. Solution: y x 2 2 4 –2 –2 –4 O 4 E F E F G G EXAMPLE 32 y x O 2 2 4 –2 –2 –4 4 C AB C D A D B In the diagram, A'B'C'D' is the image of ABCD under a rotation. Determine the centre of rotation and hence describe the rotation. y x 2 2 4 –2 –2 –4 O 4 E F E F G G Rotation of 90° clockwise about the centre (–1, –2). 7KHUHDUHIHZZD\VWRGHVFULEH a rotation, as long as the angle, direction and centre are stated FOHDUO\ For instance, D D URWDWLRQ RI ffi DERXW (3, 4) clockwise, E D URWDWLRQ RI ffi FORFNZLVH DERXW SRLQWP, (c) a rotation anticlockwise DERXW SRLQW WKURXJK ffi (i) Rotation of 90° clockwise is the same as rotation of DQWLFORFNZLVH (ii) Rotation of 90° anticlockwise is the same as rotation of FORFNZLVH Thus, the rotation in Example FDQ DOVR EH GHVFULEHG DV “Rotation of 270° anticlockwise DERXW WKH FHQWUH ± ± ´ ©Praxis Publishing_Focus On Maths
Transformations CHAPTER 6 167 Objective: : To explore and understand the concept of rotation. Instruction: : Carry out this activity in pairs. 1. Open the Rotation in Action DFWLYLW\ ÀOH XVLQJ GeoGebra. 2. Click on the points A, B or C individually and drag it to change the shape of the object ABC if necessary. 3. Click on the point P and drag it to change the centre of rotation if necessary. 4. Select the rotation direction, angle view and traces of movement on the drop down list. 5. &OLFN RQ WKH ¶Animation· EXWWRQ DQG REVHUYH KRZ object ABC is mapped onto its image. 6. Repeat Steps 2 to 5 several times. Click on the ¶Show traces of movement· LI QHFHVVDU\ 'LVFXVV the following questions with your partner. (a) How is the image of the object ABC related to point P? (b) Does every point A , B , and C move in the same way to produce its image? (c) Do the object ABC and its image have the same shape, size and orientation? 7. Based on the questions in 6(a), (b), (c), what conclusion can you make? 4 If Av and Bv are the images of A and B respectively under a rotation and intersection point of line AAv and line BBv is the midpoint of AAv and also the midpoint of BBv, then the intersection point is the centre of rotation and the angle of URWDWLRQ LV fl Solution: Rotation of 180° about the centre (0, 1). y x O 2 2 4 –2 –2 –4 4 C AB C D A D B Rotation of 180°, whether clockwise or anticlockwise, SURGXFHV WKH VDPH LPDJH Thus, it does not need to state the direction of rotation ZKLOHGHVFULELQJWKH URWDWLRQ RI fl ©Praxis Publishing_Focus On Maths
CHAPTER 6 Transformations 168 C Determine the image and the object under a rotation (I) Determining the image EXAMPLE 33 Copy and construct the image of triangle ABC under the rotation of 180° about point P. Solution: Join point A to the centre of rotation P. Draw an arc at centre P with radius AP. Measure and mark image A' on the arc so that the angle of rotation APA' = 180°. Repeat steps to for point B and point C respectively. Join the points A'ʍB' and C' to complete the image. B C P A A C B EXAMPLE 34 Copy and draw the image of triangle R under rotation of 90° anticlockwise about centre P. Solution: Join a vertex of the triangle to the centre of rotation P. Draw the image of the vertex for the angle of rotation 90°. Repeat steps and for two other vertices. Join all the image points to complete the image. R R P B C P A R P A square grid FDQ EH XVHG as a guide to draw angle of 90° with sides of an equal OHQJWK ©Praxis Publishing_Focus On Maths
Transformations CHAPTER 6 169 EXAMPLE 35 Find the coordinates of image of M(–1, 2) under a rotation (a) 90° clockwise about point C(1, –1), (b) 270° anticlockwise about point D(0, –2). Solution: (a) –4 –2 2 –2 2 4 O 4 y M M x C Coordinates of the image is (4, 1). (b) –4 –2 –2 –4 2 O 4 y M x 2 D M –6 Coordinates of the image is (4, –1). Objective: To explore the concept of rotation. Instruction: Carry out this activity. 1. Which of the triangles A, B and C is the image of triangle P under a 90° clockwise rotation about the point O? Step Choose a suitable vertex of the triangle P and draw a line joining O to the vertex. Step Draw another line passing through O so that the angle at O is 90°. Step Place a piece of tracing paper on the diagram and trace the triangle P and the line joining O to the vertex of P onto the tracing paper. A P O B C A P O B C A P O B C 5 ©Praxis Publishing_Focus On Maths
CHAPTER 6 Transformations 170 Step Hold the tracing paper in place and place the point of a sharp pencil at O. Then, rotate the tracing paper clockwise as required until the line on the tracing paper coincides with the other line on the diagram. 2. What do you notice about triangle P? 3. What conclusion can you make? A P O C P (II) Determining the object If point Q is mapped onto point R under a rotation, then R is mapped onto Q under the same rotation in an opposite direction. 30° R Q P Rotation of 30° anticlockwise about P 30° R Q P Rotation of 30° clockwise about P EXAMPLE 36 Find the coordinates of object Q which is mapped onto Q' (3, 2) under a rotation of 90° clockwise about C(1, –2). Solution: Thus, the coordinates of object Q is (–3, 0). y x 0 2 2 4 –2 –2 –4 4 Q C Q D Solve problems involving rotations EXAMPLE 37 In the diagram, triangle B is the image of triangle A under a rotation through 90°. 2 4 6 8 –4 2 4 6 8 x y O B –2 A P ©Praxis Publishing_Focus On Maths
Transformations CHAPTER 6 171 State (a) the direction of the rotation, (b) the coordinates of the center of the rotation, (c) the coordinates of the image of point P under the same rotation. Solution: 2 4 6 8 –4 2 4 6 8 x y O B P –2 A P Centre of the rotation (a) Anticlockwise (b) (2, 7) (c) (5, 8) 12 1 2 O P3 4 5 6 7 8 9 10 11 The diagram shows a clock face with the minute hand OP at a FHUWDLQ LQVWDQFH 'HVFULEH WKH rotation OP has undergone after PLQXWHV Identify whether each of the following changes is a rotation. (a) 12 1 2 3 4 5 6 7 8 9 10 11 Movement of the minute hand. (b) kg 1 2 3 4 5 0 Movement of the reading needle when a load is added. (c) Running one lap on the running track. Determine whether each of the following statements is TRUE or FALSE. (a) If line PQ' is the image of PQ under a rotation, then P is the centre of rotation. (b) Under a rotation, if point P is mapped onto P', then the perpendicular bisector of line PP' passes through the centre of rotation. (c) Quadrilateral ABCD is mapped onto A'B'C'D' under a rotation. If vertices A, B, C and D are arranged anticlockwise, then vertices A', B', C' and D' are arranged clockwise. (d) Point P and Q are mapped onto P' and Q' respectively under a rotation. If line PP' and line QQ' intersect at the centre of rotation, then the angle of rotation is 180°. Practice 6.4 Basic Intermediate Advanced ©Praxis Publishing_Focus On Maths
172 CHAPTER 6 Transformations For each of the following, M' is the image of M under a rotation. Copy and construct the centre of rotation and label with P. Hence, describe the rotation in full. (a) M M (b) (c) M M (d) M M In each of the following diagrams, A'B'C'D' is the image of ABCD under a rotation. Determine the centre of rotation and hence describe the rotation. (a) y x O 2 2 4 –2 –2 –4 4 A D C B A B C D (b) y x 2 2 4 –2 O –2 4 A A D C B B C D y x O 2 2 –2 –4 –6 4 6 –4–6–8 –2 4 6 8 P R Q S W V U T The diagram above shows the Cartesian plane. Describe in full the rotation that maps (a) P onto Q (b) R onto S (c) T onto U (d) V onto W M M P I H J Copy and construct the image of triangle HIJ under a rotation of 90° clockwise about point P. Copy each of the following diagrams on a square grid and draw the image of shape T under the rotation as stated with O as the centre of rotation. (a) T O E 90° anticlockwise 180° (c) O T (d) 90° clockwise 270° clockwise Copy and construct the image of triangle ABC under rotation of 90° anticlockwise about point (1, –2). Find the coordinates of the image of each of the following points under the rotation as described. (a) P(3, 4); 90° clockwise about (0, –2). (b) Q(–3, 5); 180° about (–1, 1). (c) R(5, –1); 90° anticlockwise about (1, 2). (d) S(–4, –2); 270° anticlockwise about (0, 3). T O O T x y –2 O –2 2 4 2 4 A B C –4 –4 ©Praxis Publishing_Focus On Maths
173 Transformations CHAPTER 6 M P Copy and construct the object which is mapped onto shape M under rotation of 90° anticlockwise about point P(0, 3). Point P is mapped onto (3, 1) under rotation anticlockwise about (0, 2). Find the coordinates of point P. 6.5 Enlargement A Explaining the meaning of similarity of geometric objects Two geometrical shapes are similar if (a) the corresponding angles are equal and (b) the corresponding sides are proportional. For example, (i) A = P, B = Q, C = R, D = S BC 2 CD 2.4 2 (ii) = , = = , QR 3 RS 3.6 3 DA 1.4 2 AB 1.6 2 = = , = = SP 2.1 3 PQ 2.4 3 Therefore, quadrilaterals ABCD and PQRS are similar. If the corresponding angles of two triangles are equal, then the corresponding sides are proportional. C G A BE F For 6ABC and 6EFG, A = E, B = F and C = G. AB BC AC Therefore, = = . EF FG EG Copy each of the following diagrams on a square grid and draw the object which is mapped onto shape W under the rotation as stated with O as the centre of rotation. (a) W O (b) O W 90° anticlockwise 180° 2.4 cm 2 cm 3.6 cm 3 cm 2.4 cm 2.1 cm 1.6 cm 1.4 cm C S P Q R D A B ©Praxis Publishing_Focus On Maths
CHAPTER 6 Transformations 174 B Identifying an enlargement Enlargement is a type of transformation where the image size is changed. All the sides of the triangle X'Y'Z' are twice as long as the sides of the original triangle XYZ. The triangle XYZ has been enlarged by a scale factor of 2. To enlarge a shape, a centre of enlargement is required. When a shape is enlarged from a centre of enlargement, the distances from the centre to each point are multiplied by the scale factor. In an enlargement, (a) the corresponding angles between the object and image are equal. (b) the ratios of the corresponding sides are constant. Both object and image in an enlargement are similar. The diagram shows an enlargement. X X Z Z Y Y Object Image Centre of enlargement n m A A C B O B C The ratio of the distance of centre of enlargement from a point on the image to the distance of centre of enlargement from the corresponding point on the REMHFWLVNQRZQDVVFDOHIDFWRU. EXAMPLE 38 A D B C 1 cm 1.6 cm H G E F 3.2 cm 2 cm Determine if rectangles ABCD and EFGH as shown are similar. Solution: A = E, B = F, C = G, D = H AB 1.6 1 BC 1 CD 1.6 1 AD 1 = = , = , = = , = EF 3.2 2 FG 2 GH 3.2 2 EH 2 Therefore, ABCD and EFGH are similar. ©Praxis Publishing_Focus On Maths
Transformations CHAPTER 6 175 C Scale factor of enlargement Scale factor, k, of an enlargement can be determined as follows: k = GLVWDQFHRISRLQWRILPDJHIURP2 GLVWDQFHRISRLQWRIREMHFWIURP2 = n m = OB' OB or k = length of side of image OHQJWKRIVLGHRIREMHFW = A'B' AB For scale factor, k, within the range 0k < 1, the size of the image formed is smaller than the object as shown in the diagram below. Scale factor, k = 10 cm 20 cm = 1 2 When the scale factor is negative value, the image formed under enlargement is located on the opposite side of the object as shown in the diagram below. Scale factor, k = – 20 cm 10 cm = –2 Image 20 cm 10 cm Object Centre of enlargement O Image 20 cm 10 cm Object Centre of enlargement O EXAMPLE 39 Each diagram below shows the mapping of an object onto its image by a transformation. Identify which transformation is an enlargement. Solution: (a) An enlargement. The image is larger than the REMHFW DQG WKH\ DUH VLPLODU (b) Not an enlargement. The image and the REMHFW DUH QRW VLPLODU (c) Not an enlargement. The corresponding sides RIWKH REMHFW DQG LWV LPDJH DUH QRW SDUDOOHO (a) (b) (c) A A B S R V W T U P Q C B S W V U T R P Q C ©Praxis Publishing_Focus On Maths
CHAPTER 6 Transformations 176 The following table shows the size of image and the position of image for the scale factor in different range. Scale factor, k Size of image Position of image to the centre of enlargement O k > 1 larger than the size of object Be on the same side as the object 0 < k < 1 smaller than the size of object Be on the same side as the object ïfik < 0 smaller than the size of object Be on the opposite side as the object kfiï larger than the size of object Be on the opposite side as the object k = 1 or k ï equal in the size to the object When k = 1, the image is on the same side as the object. When k ïWKHLPDJHLVRQWKH opposite side as the object. EXAMPLE 40 In each of the following diagrams, triangle A'B'C' is the image of the object ABC under an enlargement. Describe the enlargement in each diagram. (a) A B C 2 2 4 6 8 10 –4–6 –2 O 4 y x A B C (b) A B C 2 6 –8 O y x –6 A B C 2 4 –2 –2 –4 –4 ©Praxis Publishing_Focus On Maths
Transformations CHAPTER 6 177 Solution: (a) Scale factor = PA' PA The ratio of the distance of point on image from P to the distance of point on REMHFWIURPP = 6 units 2 units = 3 A'B'C' is the image of ABC under an enlargement at centre P(–4, 1) with scale factor 3. (b) Scale factor = B'C' BC The ratio of the length of the side of image to the OHQJWKRIVLGHRIREMHFW = 3 units 6 units = 1 2 A'B'C' is the image of ABC under an enlargement at centre P(0, –3) with scale factor 1 2 . EXAMPLE 41 In the diagram, PQRS is the image EFGH under an enlargement. Describe the enlargement. Solution: Scale factor = – distance of S from centre distance of H from centre = – 3 units 2 units = – 3 2 PQRS is the image of EFGH under an enlargement at centre (2, 3) with scale factor – 3 2 . A B C P(0, –3) O y x A B C 2 6 –8 –6 2 4 –2 –2 –4 –4 Q R S P G F H E 2 2 4 6 –2 O –2 –4 –4 4 y x 6 8 Q R S P G F H E 2 2 4 6 –2 O –2 –4 –4 4 y x 6 8 A B C 10 –4 –2 O P(–4, 1) 42 y x –6 A B C 2 4 6 8 10 Negative sign shows that the image is on the opposite VLGHDVWKHREMHFWV ©Praxis Publishing_Focus On Maths
CHAPTER 6 Transformations 178 D Determining the centre of enlargement C A B C O A B The centre of enlargement can be determined by drawing lines that join up points in the object to the corresponding points in the image. The point of intersection of these lines is the centre of enlargement, O. EXAMPLE 42 The diagram shows an enlargement of quadrilateral PQRS to quadrilateral P'Q'R'S'. Mark and label the centre of enlargement as O. Solution: R Q P S P Q R S R Q P S P Q R S O E Determining the image and object of an enlargement 7KHÁRZPDSEHORZVKRZVWKHVWHSWRGHWHUPLQHWKHLPDJHRUREMHFWRIDQHQODUJHPHQW Identify the centre of enlargement. Identify the scale factor. Draw the projection line from the centre of enlargement. Draw the image or object which has the similar shape according to the scale factor. EXAMPLE 43 Construct the image of quadrilateral ABCD under an enlargement at centre O with scale factor 2. A B O C D ©Praxis Publishing_Focus On Maths
Transformations CHAPTER 6 179 F Relationship between the areas of the image and its object Under an enlargement at centre O with scale factor k, the area of the image is k 2 times that of the object, that is, Area of the image = k 2 $UHD RI WKH REMHFW EXAMPLE 44 6OAB is the imageRI¨OCD under an enlargement with scale factor k. Given that the areas of 6OABDQG¨OCD are 16 cm2 and 4 cm2 UHVSHFWLYHO\ÀQGWKH value of k. Solution: Area of 6OAB k 2 = Area of 6OCD 16 = 4 = 4 k = ± 4 = –2 A B O C D k EHFDXVH WKH LPDJH DQG WKH REMHFW DUH RQ RSSRVLWH sides of the centre of enlargement, O Solution: Step Draw a line joining the point A and the centre of enlargement, O. Extend the line OA to A' so that OA' = 2OA. Step Repeat Step to obtain the images of points B, C and D. Step Join up the points A', B', C' and D' to obtain the quadrilateral A'B'C'D'. A O B C D A B C D ©Praxis Publishing_Focus On Maths
CHAPTER 6 Transformations 180 EXAMPLE 45 Quadrilateral PQRS is mapped onto quadrilateral ABCD under an enlargement. Given that PQ = 6 cm, AP = PQ = QB and the area of quadrilateral ABCD is 45 cm2 ÀQG (a) the scale factor of enlargement, (b) the area of quadrilateral PQRS. Solution: AB (a) Scale factor= PQ 6 + 6 + 6 = 6 = 3 Area of ABCD (b) = 32 Area of PQRS 45 = 9 Area of PQRS 45 Area of PQRS = 9 = 5 cm2 EXAMPLE 46 The editor of a school bulletin has a picture measuring 12 cm by 8.4 cm. He wants to insert the picture in his report but has to reduce it by a scale factor of 1 2 so that he can place it in a space measuring 6.6 cm by 4.8 cm. Find (a) the area of the picture in the report, E WKH DUHD RIWKH VSDFH QRW ÀOOHG XS E\WKH SLFWXUH Solution: (a) Area of the original picture = 12 × 8.4 = 100.8 cm2 Area of the picture in the report Area of the original picture = 1 2 2 Area of the picture in the report 100.8 = 1 4 Area of the picture in the report = 1 4 =100.8 = 25.2 cm2 A P Q B S R D C ©Praxis Publishing_Focus On Maths
Transformations CHAPTER 6 181 (b) 6.6 cm 4.8 cm Area of the space = 6.6 × 4.8 = 31.68 cm2 $UHD RIWKH VSDFH QRW ÀOOHG XS = 31.68 – 25.2 = 6.48 cm2 Based on each pair of object and its image given below, determine whether the transformation involved is an enlargement (a) (c) (b) (d) Find the scale factor of each of the following enlargements. (a) (b) (c) (d) A B D C C B D A In each diagram below, mark and label the centre of enlargement as O. (a) A B B′ A′ (b) V W T S U U′ V′ W′ S′ T′ C A B A B C A D C B B C D A B C E D D E C B A (c) (d) Copy the following diagrams, and with C as the centre of enlargement, construct the image of each diagram with the scale factor given. (a) Scale factor = 2 (b) Scale factor = 3 (c) Scale factor = –1 (d) Scale factor = – 2 (e) Scale factor = 1 2 (f) Scale factor = – 1 2 S Q P R S′ P′ Q′ R′ Z Y X Z′ X′ Y′ C C C C C C Practice 6.5 Basic Intermediate Advanced ©Praxis Publishing_Focus On Maths
182 CHAPTER 6 Transformations C B A P R Q In the diagram, 6PQR is mapped onto 6ABC under an enlargement. State (a) three pairs of parallel lines, (b) three pairs of equal lines, (c) the ratio of BC QR , AB PQ , AC PR , (d) two similar triangles. P A BA B C C An enlargement at centre P with scale factor k maps 6ABC onto 6A'B'C' as shown. Given that PA = 10 cm, AB = 6 cm, BA' = 4 cm and A'C' FP ÀQG (a) the value of k, (b) the length of A'B', (c) the length of AC. E A B C D 10 m 4 m 18 m 6 m In the diagram, 6ABE is mapped onto 6DBC under an enlargement at centre B. Find (a) the scale factor of enlargement, (b) the length of BD, (c) the length of BE. In the diagram, quadrilateral P'Q'R'S' is the image of quadrilateral PQRS under an enlargement at centre O. Given that OQ = 12 cm, OR' = 9 cm and OR = 18 cm, ÀQG (a) the scale factor of enlargement, (b) the length of S'R', (c) the length of QQ'. P P Q R S O Q R S 14 cm Complete the table below. (a) (b) (c) (d) (e) (f) Scale factor of enlargement 2 3 1 –— 2 —3 2 Area of the image (cm2 ) 108 16 3 50 Area of the object (cm2 ) 10 24 27 8 An enlargement with a scale factor of 3 maps rectangle ABCD onto rectangle AB'C'D'. Given that the area of ABCD is 4 cm2 ÀQG (a) the area of AB'C'D', (b) the area of the shaded region. IQ WKH GLDJUDP ¨ABC is mapped RQWR ¨A'B'C' under an enlargement at centre P. Given that AB = 12 cm, A'B' FP DQGWKH DUHD RI ¨A'B'C' is 14 cm2 ÀQG (a) the scale factor of enlargement, E WKH DUHD RI ¨$%& (c) the area of the shaded region. Rashid wants to enlarge a portrait of our ÀUVWSULPHPLQLVWHUE\DVFDOHIDFWRURI for his history project. If the area of the portrait in his project is 54 cm2 , what is the area of the original portrait? The map of Asean countries is enlarged by a scale factor of 4. If the area of the original map is 42 cm2 ÀQGWKHDUHDRIWKHHQODUJHG map. D B C A D B C C A B C A P B ©Praxis Publishing_Focus On Maths
183 Transformations CHAPTER 6 Translation • A transformation which takes place when all points in a plane are moved in a same direction through a same GLVWDQFH P P 6 units 2 units $ WUDQVODWLRQ LV GHVFULEHG E\ VWDWLQJ WKH KRUL]RQWDO and vertical movement (or using arrows to indicate the movement) or in the form a b 7KH REMHFW DQG WKH LPDJH KDYH WKH VDPH VKDSH VL]H DQG RULHQWDWLRQ Rotation • A transformation which takes place when all points in a plane are URWDWHG DERXW D SRLQW LQ WKH VDPH GLUHFWLRQ WKURXJK D VDPH DQJOH R R Centre of rotation 90° $ URWDWLRQ LV GHVFULEHG E\ VWDWLQJ the centre, angle and direction of URWDWLRQ 7KHREMHFWDQGWKHLPDJHKDYHWKH VDPH VKDSH VL]H DQG RULHQWDWLRQ 7KH SHUSHQGLFXODU ELVHFWRU RI WKH OLQH MRLQLQJ DQ REMHFW SRLQW WR LWV image passes through the centre RI URWDWLRQ Summary 5HÀHFWLRQ • A transformation which takes place when all points in a plane DUH ÀLSSHG RYHU LQ WKH VDPH plane in a line known as axis RI UHÀHFWLRQ Q Q Axis of reflection $ UHÀHFWLRQ LV GHVFULEHG E\ VWDWLQJ WKH D[LV RI UHÀHFWLRQ 7KH REMHFW DQGWKH LPDJH KDYH WKH VDPH VKDSH DQG VL]H EXW WKH\ DUH ODWHUDOO\ LQYHUWHG 7KH D[LV RI UHÀHFWLRQ LV WKH SHUSHQGLFXODU ELVHFWRU RI WKH OLQH MRLQLQJ DQ REMHFW SRLQW WR LWV LPDJH 6LPLODUVKDSHV • Two shapes are similar if D WKHLU FRUUHVSRQGLQJ DQJOHV DUH HTXDO E WKHLU FRUUHVSRQGLQJ VLGHV DUH SURSRUWLRQDO For example, T U S V C B A D ABCD and STUV are similar if A = S , B = T, C = U, D = V and ST TU UV SV —– = —– = —– = —– AB BC CD AD • If the corresponding angles of two triangles are equal, then the corresponding sides are SURSRUWLRQDO Hence, two triangles are similar if the corresponding angles are equal or the corresponding sides are SURSRUWLRQDO Enlargement O R Q P R Q P • An enlargement at centre O with scale factor k is a WUDQVIRUPDWLRQ ZKLFK HQODUJHV DQ REMHFW k times its RULJLQDO VL]H IURP FHQWUHO • Under an enlargement, the centre of enlargement is DQ LQYDULDQW SRLQW 'LVWDQFH RI LPDJHIURPWKH centre of enlargement • Scale factor, k = —————————————— 'LVWDQFH RI REMHFW IURPWKH centre of enlargement For example, OP' OQ' OR' k = —— = —— = —— OP OQ OR or Length of side of the image Scale factor, k = —————————————— Length of corresponding side RIWKH REMHFW For example, P'Q' Q'R' P'R' k = —— = —— = —— PQ QR PR • Area of image = k2 î$UHD RI REMHFW Transformations $WUDQVIRUPDWLRQ LV D RQHWRRQH FRUUHVSRQGHQFH EHWZHHQ SRLQWV LQ D SODQH ©Praxis Publishing_Focus On Maths
184 CHAPTER 6 Transformations Section A 1. Which of the following movement of points is not a transformation? A B C D 2. The image of the point (3, 4) under the translation 5 – 2 is A (8, 2) C (1, –5) B (2, –6) D (9, –1 3. P K Q A B D C 7KH GLDJUDP LV GUDZQ RQ D VTXDUH JULG Triangle Q is the image of triangle P under DWUDQVODWLRQ:KLFKRIWKHWULDQJOHVODEHOOHG A, B, C and D is the image of triangle K under the same translation? 4. 2 –2 4 6 2 4 6 8 x y –4 –2 O P Q 7KH GLDJUDP VKRZV D &DUWHVLDQ SODQH Shape Q is the image of shape P under a WUDQVODWLRQ7KHWUDQVODWLRQ LV A 8 4 C 8 – 4 B 4 8 D –4 –8 5. M A B C D 7KH GLDJUDP LV GUDZQ RQ D VTXDUH JULG :KLFKRIWKHTXDGULODWHUDOVODEHOOHGA, B, C and D is the image of quadrilateral M under the translation –2 –3 ? 6. M A B C D P Q 7KH GLDJUDP LV GUDZQ RQ D VTXDUH JULG :KLFK RIWKH SRLQWV ODEHOOHGA, B, C and D is the image of point M XQGHU D UHÀHFWLRQ in the line PQ? 7. P A B C D 7KH GLDJUDP LV GUDZQ RQ D VTXDUH JULG :KLFK RIWKHWULDQJOHV ODEHOOHGA, B, C and D is not the image of triangle P under a UHÀHFWLRQ" ©Praxis Publishing_Focus On Maths
185 Transformations CHAPTER 6 8. On the Cartesian plane, point P is the image RI D SRLQW XQGHU D UHÀHFWLRQ RQ WKH OLQH x 0 2 –2 –4 4 –2 4 2 6 y x B D C A P Which of the point A, B, C and D, is the REMHFW RIWKH UHÀHFWLRQ" 9. C4 B4 D4 C3 D3 D2 D1 C1 C2 E4 E3 E2E1 E A B3 B2 C B1 D B 7KHGLDJUDPVKRZV¿YHSRO\JRQVGUDZQRQ DJULG7KHSRVVLEOHLPDJHRIABCDE under an enlargement is A AB1 C1 D1 E1 B AB2 C2 D2 E2 C AB3 C3 D3 E3 D AB4 C4 D4 E4 10. 2 4 6 8 O 2 4 6 8 x y P Q Q P R In the diagram, triangle P'Q'R' is the image of triangle PQRXQGHUDQHQODUJHPHQW7KH coordinates of R' are A (8, 5) B (5, 7) C (7, 5) D (6, 5) Section B 1. 2 4 –2 –4 –4 –2 4 x y O M N 2 In the diagram, N is the image of M under a transformation. Describe the transformation. 2. –2 2 –2 2 4 6 –4 4 6 8 y O x S R In the diagram, triangle S is the image of triangle R under a translation. Describe the translation. ©Praxis Publishing_Focus On Maths