University:
High School
Course:
High School Mathematics
Academic year:

2024
Views:

316

Pages:

12
Author:

Camilla Dunn

u� Name 11111 Notetaking with Voca:;; Date - --- (continued) Core Concepts Vectors The diagram shows a vector. The initial point, or starting point, of the vector is P, and the terminal point, or ending point, is Q. The vector is named PQ, which is read as "vector PQ." The horizontal component of PQ is 5, and the vertical component is 3. The component form of a vector combines the horizontal and vertical components. So, the component form of PQ is (5, 3). Q /t3units L.._j up P 5 units right Notes: Translations A translation moves every point of a figure the same distance in the same direction. More specifically, a translation maps, or moves the points I' and Q of a plane figure along a vector (a, b) to the points P' and Q', so that one of the following statements is true. • PP' = QQ' and PP' II QQ', or X • PP' = QQ' and PP' and QQ' are collinear. Notes: Extra Practice In Exercises 1-3, name the vector and write its component form. 1. V- -r1 I I- N 1-- 1--1-- -- I- 2. 3. ,_ i-!-- �1--1----1--1- -- Name ------------------------- Date BII Notetaking with Vocabulary (continued) In Exercises 4-7, the vertices of b,.ABC are A(1, 2), 8(5, 1), C(5, 4). Translate !::1ABC using the given vector. Graph b,.ABC and its image. 4. 5. 6. (-4, o) �l\l'-A-(-2, -4) R-1d (o, -5) .,,��t ·s\� 7. (1, -3) 8. y '?)� _I - t -- I _i__J_ In Exercises 8 and 9, write a rule for the translation of quadrilateral PQRS to quadrilateral P'Q'R'S'. 2 P'_i __ s� I_-..__-+: . ! .!:..................... 5 -- f - - 7· I -8 . : -4 ] / ,� l x , s.'·- i-- --·R , 1 2 I In Exercises 10 and 11, use the translation. i--)(+I� y- d (_: J'{) ➔ (x, y) ➔ (x + 6, y - 3) 10. What is the image of J(4, 5)? j' ( to/2-) 11. What is the image of R'(0, -5)? 12. 'I (_ (I I-<() 12. In a video game, you move a spaceship 1 unit left and 4 units up. Then, you move the spaceship 2 units left. Rewrite the composition as a single transformation. ---- Name _____________ _ ________ __ Date ---- Notetaking with Vocabulary (continued) Core Concepts Coordinate Rules for Reflections • If (a, b) is reflected in the x-axis, then its image is the point ( a, -b). • If (a, b) is reflected in the y-axis, then its image is the point (-a, b). • If ( a, b) is reflected in the line y • If (a, b) is reflected in the line y = -x, then its image is the point (-b, -a). = x, then its image is the point (b, a). Notes: Postulate 4.2 Reflection Postulate A reflection is a rigid motion. Extra Practice In Exercises 1-4, graph !:,,ABC and its image after a reflection in the given line. 1. A(-1, 5), B(-4, 4), C(-3, l);y-axis 3. A(2, -1), B(-4, -2), C(-1, -3);y =1 2. A(O, 2), B(4, 5), C(5, 2); x-axis 4. A(-r, 3), B(-2, -2), C(O, -2); x = -3 Name---- ----------------------- Date ----- Notetaking with Vocabulary (continued) In Exercises 5 and 6, graph the poly on's image after a reflection in the given line. 5. y = 6. y = -x X D C y A In xercises 7 and 8, graph /J.JKL with vertices J(3, 1), K(4, 2), and L(1, 3) and its image after the glide reflection. 7. Translation: (x, y) ➔ (x - 6, y - l) 8. Translation: (x, y) ➔ (x, y - 4) Reflection: in the line x = I Reflection: in the line y = - x In Exercises 9-12, identify the line symmetry (if any) of the word. 9. MOON 10. wow 11. KID 12. DOCK 13. You are placing a power strip along wall w that connects to two computers. Where should you place the power strip to minimize the length of the connecting cables? Computer A Computer B w Name-- ------------------------ Date ----- Notetaking with Vocabulary (continued) Coordinate Rules for Rotations about the Origin When a point (a, b) is rotated counterclockwise about the origin, the following are true. • For a rotation of 90° , (a, b) ➔ (-b, a). • For a rotation of 180 ° , (a, b) ➔ (-a, -b). • For a rotation of 270 °, ( a, b) ➔ ( b, -a). y X Notes: (b, -a) Postulate 4.3 Rotation Postulate A rotation is a rigid motion. Extra Practice In Exercises 1-3, graph the image of the polygon after a rotation of the given number of degrees about the origin. 2. 90° 1. 180° 3. 270° •�Y A I r--- B 4 o~1--Jc -4 I 4\ 1V ' In Exercises 4-7, graph the image of 4. Reflection: x-axis Rotation: 180 ° about the origin I \ 'l} -4 I• ) I N • '� ,-�--4 l ,, 11- 4 1-:... ,:; 5. Rotation: 90 ° about the origin Translation: (x, y) ➔ (x + 2, y - 3) y I l4I \ \ �.Af':J M,_. 4 I .� I... I MN after the composition. "t y M \x -' J,� t)'I I� 4- ft- y-. ,, X Name ______ __________________ _ Date ---_ _ Notetaking with Vocabulary (continued) 6. Rotation: 270 ° about the origin 7. Rotation: 90 ° about the origin Translation: (x, y) ➔ (x - 5, y) Reflection: in the line y = x In Exercises 8 and 9, graph t.JKL with vertices J(2, 3), K(1, -1), and L(-1, 0) and its image after the composition. 8. Rotation: 180° about the origin 9. Translation: (x, y) ➔ (x - 4, y - 4) Rotation: 270 ° about the origin Reflection: x = 2 \\ y X In Exercises 10 and 11, determine whether the figure has rotational symmetry. If so, describe any rotations that map the figure onto itself. 11. 10. ) Name Date --- Notetaking with Vocabulary (continued) Extra Practice 1. Identify any c ongruent figures in the coordinate plane. Explam. • � fl IF I +.Y HK G L p �I/ A R D ' .... ,. 2 X C � - \ J 4 I/ p M 1......V ......"- I/ -2 -2 K ....... N 2 . Descn·be -..... i.... 0 .......� �V B 4 1 I .� u V T s 2 - -r-,... .I Vn ih 1 -eff .D s-ru v 1 e,tv,.J;). i'.> Ii n� Y=r<. o.. 1 e.f( ecfi bsr-'- 0 �c.D'. 0V' u f O f:: FG,lt ,·s /.t ; :--+i.. qo• rtrt,,Ai"& e. : .ti::r I'- L. v 0 n ;ts cktNiA- ofD A--BGD a congruem.:e transformation that maps b.PQR to b.STU. ,y u F6tt p81VV � n 6 l' /5 ,,.__ -fr-ans 1..Ji·l'A S M 1-10? -2 T 2 , L4 ' Q l , J "� p 2 I/ !--,.. ...,. X \ R ld· (/vl 5' tJyti'-h tl (.'ti0) f(-i..\1 0 5 ) l- -I , ) - 3 D escribe a cong,)f�nce transfo rmation that maps polygon ABCD to polygon EFGH. -4· , -. ' ,-- C 2 J'I\ ,_ IA� YI ----' . I 2 i..--rI/ F - I E ,....i.., H � :-,,iL B ·1 J) ......�.. G 4 , X � ie ff e c--fi c/Vl ·, V\- +1-<..e., I i vie. )H .f{i I l 1)\1(-d. b'.;'.J- ().__ �VI s k-·&' Vl,,i O jl V n 1'-!;, i 7:{j- t(pie,( S U f1 ,t, ck w11,. ( 5 I\{") )b) � \: 1i /) @l -\ / -s) (J I '() 7 (x < 'L )y _.---) Name-------------------------- Date _ __ _ _ Notetaking with Vocabulary (continued) In Exercises 4 and 5, determine whether the polygons with the given vertices are congruent. Use transformations to explain your reasoning. 4. A(2, 2), B(3, 1), C{l, 1) and 5. G(3, 3), K(2, -1), L(-3, -3), M(2, -2),N(2, -1) D(2, -2), E(3, -1), F{l, -1) 'i .es) y y -4 bD€F r e-rk eh" 6\,\ b 1+e<- -4 {, ' / � 2 -2 I 2 � 14 I In Exercises 6-9, k 4 . 2 j� CA-- or H(2, 1), I(6, 2), J(6, 3) and ' I/ ..,, -2 (.) ' 4 -4 X � 2 I 4 X 2 14 I I JI -2 - m, UV is reflected in line k, and U'V' is reflected in line m. G. A translation maps UV onto which segment? u"v ,, m 7. Which lines are perpendicular to UU"? r i "-e... � vt n.J u tt l,, qy'I 8. Why is V" the image of V? �xplain your reasoning. (; Ill.- ll-- {( I iVt� /}'111 9. If the distance between k and m is 5 inches, what is the length of VV"? lD in 10. What is the angle of rotation that maps A onto A"? A" Name-------------------------- Date ----- Notetaking with Vocabulary (continued) Coordinate Rule for Dilations If P(x, y) is the preimage of a point, then its image after a dilation centered at the origin (0, 0) with scale factor k is the point P'(kx, y ky). I I I eP'(kx, ky) iP(x, y) I X Notes: Extra Practice In Exercises 1-3, find the scale factor of the dilation. Then tell whether the dilation is a reduction or an enlargement. 3. P' '1 -- ---------------·-:' '' In Exercises 4 and 5, graph the polygon and its image after a dilation with scale factor k. 4. A(-3, 1), B(-4, -1), C(-2, -1); k = 2 - y 4 ,__----- - 0' -10 -a I " -- l I V -6 f [\- 14,) --- I,� - 2 2 -- 2 ( 1,..1 ,.. I I ,- i- 4 4 X ·-,- Name ------ ---------------- Date ---- Notetaking with Vocabulary (continued) Extra Practice In Exercises 1-3, graph the polygon with the given vertices and its image after the similarity transformation. lI 1. A(3, 6), B(2, 5), C(4, 3), D(5, 5) Translation: (x, y) ➔ (x - 5, y - 3) Dilation: (x, y) ➔ (3x, 3y) .I y 10 I/ ' r--....'"' ' / -......... / ," ' 8- \' � f -....... y 7 b /"'-.. ""I,I I4 �'7 I� �L rt}, -8 'l\. 2. R(12, 8), S(8, 0), T(O, 4) 1)1 I -'\V... -4, -2 -6 y � "\ (' ..... / • \� I I/ - � I V V --.....,..._ fi 4 r-.... ,1 ... 2 I'- I -2 V ' 2 ,·2-x \_,._,,, 4 6 - I_ _.,,.V I\ '-/ �- I I 8· ' --..... � I 10 6 �y 14 \ -1 rt- X I ....... 4 .,.\ l:J i l\ -: 'v, � 1\'� r-:::'LI I '1/" i\. V \ __.:..---- 4 -- --- - �--- / -. it \. 3. 2 l, �-'-8 Reflection: in the y-axis V _,,,v - 8 X Name________________________ Date ---- Notetaking with Vocabulary (continued) In Exercises 4-6, describe the similarity transformation that maps the preimage to the image. 4. - y N ....... �i.- M 2 ,_ _i.-.... M -2 -4 )(__� /2L{ o n ,1S 1)1 lA-1 \ u n;t ck-wn L' N' 2· -6 L _i- 6 4 X 2 I ·-- 5. 6 -� - -6 -4 I/ -4 A B I/ � -2 2 C' C / / 2 I A' B' 4 6 X I2 · 6. ia:1u& UVl �y -6 r/ S' -6 -4 R' I"\r-......I/ T' Q' Q I 2 � I/ I/ 1/R 2 4 6 ; X �-h W\ 1 � =- /3 Cfo O cc w