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

2020
Views:

21

Pages:

21
Author:

Marianna B

�:x Name Date ----- - Notetaking with Vo::;; {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 Q /hunits up � P 5 units right 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). 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 P 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. - N - --·-- - J ,_ 2. I I I I.. II' A - -- L....- 3. -· 1,, -- -- -,_�,- - - -,- - ___ Date ---- Name _______________________ - Notetaking with Vocabulary (continued) In Exercises 4-7, the vertices of l::,.ABC are A(1, 2), 8(5, 1), C(5, 4 ). Translate l::,.ABC using the given vector. Graph l::,.ABC and its image. ,.. , . . •, . 4. 5. 6. 7. (-4, o) (-2, -4) (o, -5) - -I YJ - --4 ,- (1, -3) 4 4 1- ·- 4 I rI - X ,_ - In Exercises 8 and 9 , write a rule for the translation of quadrilateral PQRS to quadrilateral P'Q'R'S'. - y 8. - 4 =j ' •. ,· ,_ 2 ,- - --2· P' -1- - p - - ·- r 2 - .. -· .. -S' • -:---i---- s ,_,._ Q' , �L_k Ll '-°. 6 R' X 9. y Q 4 . s ·- Q' X 2 -- 1 In Exercises 10 and 11, use the translation. (x, y) ➔ (x + 6, y - 3) 10. What is the image of J( 4, 5)? 11. What is the image of R'(o, -5)? 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) In Exercises 4-7, the vertices of 6ABC are A 1, 2 , 8 5, 1 , C 5, 4 . ( ) ( ) ) ( Translate 6ABC using the given vector. Graph 6ABC and its image. 4. 5. 6. 7. (-4, o) �,\,\-A.- (-2, -4) (o, -5) (1, -3) tt-'d ., \�1v--\- 'Bl� -, -. -_:-1 - <2>� In Exercises 8 and 9, write a rule for the translation of quadrilateral PQRS to quadrilateral P'Q'R'S'. 8. ,--· ,_ s ] _ P' y -4 2 -� p -- =f Q' -� 6 - 9. y -� R'x X -· In Exercises 10 and 11, use the translation. (x, y) ➔ (x + 6, y - 3) 10. What is the image of J( 4, 5)? jf ( lo/L) 11. What is the image of R'(O, -5)? g_''(y,-<() 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 = x, then its image is the point (b, a). • If (a, b) is reflected in the line y = - 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 6ABC and its image after a reflection in the given line. 1. A(-1, 5), B(-4, 4), C(-3, l);y-axis 2. A( 0, 2), B(4, 5), C(5, 2); x-axis � -II ..-111'' 3. A(2, -1), B(-4, -2), C(-1, -3);y •'I .. l-+-t---t--•"- c. Draw 6.A'B'C'. Are its side lengths the same as those of 6.ABC? Justify your answer. - -3 -1 -,__l3 15 I y 8 ,,,., ,- I/ - - 1 ' ,-� ' \ -,� - ' Sx - - (,_ 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. y (a, b) ➔ (-b, a). • For a rotation of 90°, • For a rotation of180 ° , (a, b) ➔ (-a, -b). • For a rotation of 270° , (a, b) ➔ (b, -a). Notes: 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 ·4 - X I- -4 A I_,....8 D I-Jc . h')�t j,\x . '-- I --4\ Iv ,. ,:; In Exercises 4-7, graph the image of MN after the composition. 5. Rotation: 90 ° about the origin 4. Reflection: x-axis Rotation: 180 ° about the origin Translation: (x, y) ➔ (x + 2, y - 3) ,y -f1-4 t \ \ t!..-N X ,..... \" r4 ' 4 . ,, 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 �- y 4 . . •:. .. - M - - 1N- y 4 1 _,_ I-- 4 I-- -4 -- -4 -- - X N --, -4 4 4 r'9M In Exercises 8 and 9, graph 6JKL 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) Reflection: x = 2 �.• • • . ; . . ,.-( I Rotation: 270 ° about the origin y y •, -,- ' •• • 1 - � - -1-f-f-f-f-----l X ,_ ----+---+--+--+----+---+·-- . ·, ,. ··, - • . :• • In Exercises 10 and 11, determine whether the figure has rotational symmetry. If so, describe any rotations that map the figure onto itself. 10. 11. Name ---------------------------- Date ---- Notetaking with Vocabulary (continued) I 6. Rotation: 270 ° about the origin 7. Rotation: 90° about the origin Translation: ( x, y) ➔ (x - 5, y) Reflection: in the line y = x y In Exercises 8 and 9, graph D..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 X In Exercises 10 and 11 , determine whether the figure has rotational symmetry. If so, describe any rotations that map the figure onto itself. 10. 11. , 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 270 °, (a, b) ➔ (b, -a). • y For a rotation of180 °,(a,b) ➔ (-a,-b). Notes: 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. 1. 180° y ✓I �4 S· J V� �K< )M -�LIV -- 3. 270° 2. 90° .. 4 X ,-r <. 4 1 1 4 y ,__ ,� 4 -4 __ , X ;_ -4 •�Y r�-...tc A D.., 4 4 -T if X ,- In Exercises 4-7, graph the image of MN after the composition. 4. Reflection: x-axis Rotation: 180° about the origin ,Y M ' \ -4 4 1- ... ·N -4 I 5. Rotation: 90 ° about the origin Translation: (x, y) ➔ (x + 2, y - 3) 4 ,_ ,- -- -,___ 4 X ,__ M ,.,. I · y i,,..N 4 4 1--I·- • -,� X Name �------Notetaking Date ---- with Vocabulary (continued) Extra Practice 1. Identify any congruent fiigures in the coordinate plane. Explain. E F H G K L \17 J . A .J' N Q ' ...... R ..... .J B 4 i I 1 :: D M 1,10'? 1titt ps:rvV f Cl 6 X - u V T s > -' c,.._ -/ra,l,s tcd,· ,.,._ i5 f o l'I l'Y1 0 'f- L 1 /:::, 'j> lH. :'. fJ D ch;tNIA- of- D �c u'ts <.n .7 VI-J et \In ih 1-d'-f 1:oR..lf ;:_-+t._. d uf ,rv, cf,· e ..({ ,e .D s-ruv ,� ii. 0 ro-tiuficJ1Ao+ 'Cfo a is /::;s'Fli"-L t=F1 ......2.........._ C / ...... ➔Q_ / .,, .J' I"D -- p ....... M --........... .,, p L.. 2 -2 4 D Jl-&:-D �y • /jp�R- 2. Describe a congruence transformation that maps !}.PQR to !}.STU. 1(Nl.9 Id' ,y 2- 4 - \--, 2 p ,- 2 /IT2• - I \ , 17 --,4 s I -1/ Q .... ..... tJ y1. ih c.) ? ,hi II OvJr d bj I gn ° fl!W· vtA 4 u (/VI � X R 3. Describe a cong ruence transformation that maps polygon ABC'D t0 polygon EFGH. I �y C 4 - 2. 4 - -2 F - -- E ' /,,,,- H ' ie ff ed _ ..... DKi--.._ G - 4 . A� (/VI I V\. 4't<.e,, ¾tlo�d bit<-- �vi I I vie. x� ( k_;&·� ;l V YI 1-fs ( i� 'i{_f- (,{t,,,.J S- I)f1 f'/3 ck WII_ Name------ - ------------------- Date ----- Notetaking with Vocabulary For use after Lesson 4.4 In your own words, write the meaning of each vocabulary term. congruent figures congruence transformation Theorems Theorem 4.2 Reflections in Parallel Lines Theorem If lines k and m are parallel, then a reflection in line k followed by a reflection in line m is the same as a translation. If A" is the image of A, then 1. AA" is perpendicular to k and m, and 2. AA" = 2d, where dis the distance between k and m. Proof Ex. 31. p. 206 Notes: Theorem 4.3 Reflections in Intersecting Lines Theorem If lines k and m intersect at point P, then a reflection in line k followed by a reflection in line m is the same as a rotation about point P. The angle of rotation is 2x0, where x0 is the measure of the acute or right angle formed by lines k and m. Proof Ex. 31. p. 206 Notes: f----- d -----j Name ------------------ - - - - - - - - - Date ----- Dilations For use with Exploration 4.5 .. ,. Essential Question What does it mean to dilate a figure? .. . .. . ... . • • •. I EXPLORATION: Dilating a Triangle in a Coordinate Plane Go to BigldeasMath.com for an interactive tool to investigate this exploration. Work with a partner. Use dynamic geometry software to draw any triangle and label it t::J.ABC. a. Dilate t::J.ABC using a scale factor of 2 and a center of dilation at the origin to form l:!..A'B'C'. Compare the coordinates, side lengths, and angle measures of t::J.ABC and l:!..A'B'C'. Sample :-�:;c. C' • � •• ,. �- j ,· ,.1 • .•,- B ... , o . .. .. . .. D 0 A 1 b. Repeat part (a) using a scale factor of½· c. What do the results of parts (a) and (b) suggest about the coordinates, side lengths, and angle measures of the image of t::J.ABC after a dilation with a scale factor of k? 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 y at the origin (0, 0) with scale factor k is the point P'(kx, ky). I I •P'(kx, ky) /'P{x, y) 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', -- .----------------,' ' ' : ' C 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 - -- � ,_ ',· y -4 - � -r /, I \ 1, .:,o -s I -6 1\� 1-o - -,7 -2.__- --. �,_ =2 � ..) -2 l4 I 2 4 X -- - ,- ' : Name --------------------------- Date ---,I Notetaking with Vocabulary (continued) 5. P(-10, 0), Q(-5, 0), R(O, 5), S(-5, 5); k = ½ . .. ,. ,y �4 2-,- .. ... �10 -8 -6 -4 -2 r -1� 2· 2 4 ·x 4 l. " • •· • '•' I .:•·..-.. --1 . : . . .. ,. In Exercises 6 and 7, find the coordinates of the image of the polygon after a dilation with scale factor k. 6. A(-3, 1), B(-4, -1), C(-2, -1); k = -6 7. P(-8, 4), Q( 20, -8), R(16, 4), S(0, 12); k = -0.25 8. You design a poster on an 8. 5-inch by 11-inch paper for a contest at your school. The poster of the winner will be printed on a 34-inch by 44-inch canvas to be displayed. What is the scale factor of this dilation? ·. ,. . .' 9. A biology book shows the image of an insect that is 10 times its actual size. The image of the insect is 8 centimeters long. What is the actual length of the insect? Name ------------------ ------- Date ---- Notetaking with Vocabulary (continued) 5. P(-10, 0), Q(-5, 0), R(O, 5), S(-5, 5); k = / - 1) -8 ;,;;- D/ -6 y ) I/ / t V / -2· s // ,- -1.,-2 ,. ' ... l I 2 4 X - 2 L4 I ' In Exercises 6 and 7, find the coordinates of the image of the polygon after a dilation with scale factor k. 6. A(-3, 1), B(-4, -1), C(-2, -1); k = -6 7. P(-8, 4), Q(20, -8), R(l6, 4), S(O, 12); k = -0.25 Pl11-rJ G'(-\ 1i-J '2- c {-� 1 -1) s'(0)-3) 8. You design a poster on an 8.5-inch by 11-inch paper for a contest at your school. The poster of the winner will be printed on a 34-inch by 44-inch canvas to be displayed. What is the scale factor of this dilation? 9. A biology book shows the image of an insect that is 10 times its actual size. The image of the insect is 8 centimeters long. What is the actual length of the insect? Date ----- Name___________________________ 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 ' .. . y at the origin ( 0, 0) with scale factor k is the point F'(k:x, ky). I I I eP'(kx, ky) lP(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'; \------------; '' '' C 1. 27 .- .. p 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 - -.:-,u - ~4 - ,_ - -8 -6 -4 -2 -2 2 -2 14 - 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. Translation: (x, y) ➔ (x - 5, y - 3) Dilation: (x, y) ➔ (3x, 3y) l I/r---.. ,{ 1. A(3, 6), B(2, 5), C(4, 3), D(5, 5) 'I t,., V I/ .._ "' � I"-. ... \J I'-.. � r:: I I-4 I 'b / ,'"-. _ -4 '- i"'--K; 2 4 I K: I, I'-,,._ '-,: 7 ,v,, [}.. -2 I.I X " y - 8 - �6- I- () �-\ ' ' " '\ / -, - �I.---- _, � /I t/ r-,.__ (\' ' � 7/ ,__ I (2 ...___,__ 4 ::, 6 10 -�'-- I I I\� I- 8�'-- I,, _._, 1, ...,___ l'\ ' r-;:: '--4 '.n I/ j I ' - 1-: I -2 I / '--. y \' I I - F K. ,,\ r (.I\ l" ........ _I.---- �........... �-'""" I- Reflection: in the y-axis -8 \' �· I I� I -6 -8 2. R(12, 8), S(8, 0), T(O, 4) (¼x, ¼Y) I'::::t-..._ A" � Dilation: (x,y) ➔ 10 r---........ 12 X - I_ �17 ),/ '\_...V' 2 '\ V )C 2 4 6 8 ·x