CHAPTER 7. Think & Discuss (p. 393) m Z m Z m Z 90 QR 2 RP 2 PQ 2 QR QR QR AB QR 7.

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1 HPTER 7 Think & Discuss (p. 393). The image in bo is flipped to get the image in bo. The image in bo is turned to get the image in bo D.. Sample answer: If ou look at the picture as a whole, the right half is the image of the left half flipped over the center vertical red line. Skill Review (p.394) and are not congruent XZ 0 5. m X m Q m Z m Y m X YZ QR 9. Lesson 7. Developing oncepts ctivit 7. (p. 395) Eploring the oncept. a. FH corresponds to KL. b. corresponds to JK. FG corresponds to KJ. corresponds to KL. GH corresponds to JL. D corresponds to LM. DE corresponds to MN. E corresponds to JN. c. WX corresponds to MN. d. NP corresponds to TU. XY corresponds to NP. PQ corresponds to UV. YZ corresponds to PQ. QR corresponds to VW. WZ corresponds to MQ. RS corresponds to WX. NS corresponds to TX.. a. Turn FGH to get LKJ. b. Flip figure DE to get figure JKLMN. c. Slide figure WXYZ to get figure MNPQ. d. Turn figure NPQRS to get figure TUVWX. 3. The three tpes of motion that preserve the congruence of a figure when it is moved in the plane are flip, slide, and turn. 4. Figure D is flipped over line n to get figure EFGH. 5. Yes, EFGH is congruent to JKLM. E D H m Z G F m Z QR RP PQ QR 0. QR QR K QR 7.0 L m J m Z 90 n M opright McDougal Littell Inc. Geometr 37 hapter 7 Worked-out Solutions Ke

2 6. D is turned to get JKLM. Yes, D is congruent to JKLM. 7. No; there is no line over which D can be flipped to give JKLM. 7. Guided Practice (p. 399). n operation that maps a preimage onto an image is called a transformation.. The preimage and image of a transformation are sometimes congruent. 3. transformation that is an isometr alwas preserves length. 4. n isometr never maps an acute triangle onto an obtuse triangle. 5. translation 6. reflection 7. rotation 8. ST 9. VW 0. Sample answer:. WXY QRS and VWX 7. Practice and pplications (pp ). Figure DE Figure JKLMN. 3. This transformation is a rotation about the origin. The figure DE is turned about the origin. 4. Sample answer: 5. Sample answer: E and JN L and 6., Sample answer: corresponds to JK JK So, JK. 8. true 9. false 0. true. reflection in the line ; flip over the line, 3, 4, 3,, and D 6,.. translation, slide 6 units to the right; L,, N 5,, and M 3, 4 3. Yes; the preimage and image appear to be congruent. 4. Yes; the preimage and image appear to be congruent. 5. No; the preimage and image are not congruent. 6. PQR 7. DEF LKJ 8. KJL EFD 9. PRQ 30. LJK DFE 3. RQP 3. FG RS So, FG RS. GH ST So, GH ST. FH RT So, FH RT ONTINUED 38 Geometr hapter 7 Worked-out Solutions Ke opright McDougal Littell Inc.

3 33. ONTINUED 34. XY So, XY. 5 YZ So, YZ. 4 XZ So, XZ. a 96 a 48 b 9 c 7 3d 6 d 36. translation 37. translation 38. reflection 39. rotation Yes, a point or a line segment can be its own preimage when it is rotated or when it is reflected. Points or line segments on a line of reflection are their own preimages. center of rotation is its own preimage. 4. From to, the stencil is reflected. From to, the stencil is reflected. From to D, the stencil is either rotated or reflected twice w 70 w The letters b, d, p, and q can be formed from each other b reflection or rotation; the letters n and u can be formed from each other b rotation or repeated reflection. 43. Sample answer: The lower right corner is the horizontal reflection of the upper right corner. Then reflect the lower right corner verticall to get the pattern for the lower left corner. From there, reflect horizontall to get the upper left corner D 46. Statements Reasons. PQR and. Given PQR XYZ are isometries.. PQ, QR,. Definition of isometr PR, PQ XY, QR YZ, and PR XZ 3. XY, YZ, XZ 3. Transitive propert of equalit 4. XY, YZ, 4. Definition of and XZ congruent segments 5. XYZ is an 5. Definition of isometr isometr. 7. Mied Review (p. 40) D EF GH opright McDougal Littell Inc. Geometr 39 hapter 7 Worked-out Solutions Ke

4 5. polgon 5. polgon 53. Not a polgon because one side is not a line segment. 54. Not a polgon because one side is not a line segment. 55. Not a polgon because two of the sides intersect onl one other side. 56. polgon 57. Sample answers: () slope of slope of RS slope of slope of PQ RS and QR PS So, PQRS is a parallelogram because both pairs of opposite sides are parallel. () PQ RS QR PS PQ QR PS PQ RS and QR PS PQRS is a parallelogram because opposite sides are congruent. 58. Sample answers: () slope of slope of YZ slope of XY slope of WZ WX YZ and XY WZ WXYZ is a parallelogram because opposite sides are parallel. () WX YZ XY WZ WX WX YZ and XY WZ WXYZ is a parallelogram because opposite sides are congruent. 40 Geometr hapter 7 Worked-out Solutions Ke opright McDougal Littell Inc.

5 Lesson 7. Developing oncepts ctivit 7. (p. 403) 7. Practice and pplications (pp ) Eploring the oncept Sample answer: X X' k k Z Y m Y' Z' 7. Investigate. Measurements will var, but X X, Z Z, and Y Y.. The measure of each angle is The line m is the perpendicular bisector of each segment. Eploring the oncept Sample answer: ' D m ' Make a onjecture 4. Line m is the perpendicular bisector of each segment; Sample answer: m D m D 90 and and D D. 5. The line of reflection is the perpendicular bisector of the segment connecting a point and its image. k 8. True; N is units above the line, so its image is units below the line. 9. True; M is 3 units to the right of the line 3, so its image is 3 units to the left of the line. 0. False; W is one unit below the line, so its image is one unit above the line. Its image should be W 6,.. True; U is 4 units to the right of the line, so its image is 4 units to the left of the line.. GH 3. D 4. FE 5. EF 6. S 0, 7. T 3, 8 T(3, 8) S(0, ) S'(0, ) T'(3, 8) 7. Guided Practice (p. 407). line of smmetr is a line in which a figure can be reflected onto itself.. When a point is reflected in the -ais, the -coordinates of the point and its image are the same and the -coordinates are opposites. 3. not a reflection 4. not a reflection 5. reflection 6. EF 7. D DEF 8. G 9. D D 0. GFE. D DG. 3 lines of smmetr 3. 4 lines of smmetr 4. 5 lines of smmetr 8. Q 3, 3 9. Q( 3, 3) Q'(3, 3) R 7, R'( 7, ) R(7, ) opright McDougal Littell Inc. Geometr 4 hapter 7 Worked-out Solutions Ke

6 30. Sample answer: '(9, ) '(3, 9) (, 9) (5, 7) (, 5) (7, ) (6, 0) '(, 5) '(, ) (5, 0) (, ) (, 4) (9, 3) '(7, 5) The coordinates of the vertices of the image of, are reversals of the coordinates of the vertices of the preimage,, , 0 (, 4) (6, 3) 40. 5, 0 ( 4, 6) (3.5, 9) 3. j 3. k '(, 4) 4 (3, 0) (, 0) '( 4, 6) PP QQ 33. Draw and intersecting line m at points S and T. PQ and P Q intersects line m at R. the definition of reflection, P S PS and and Q T QT and RT QQ. RT RT and SR SR b the Refleive Propert of ongruence. ngles P SR, RSP, RTQ and RTQ are right angles (definition of perpendicular) and are congruent (all right angles are congruent). It follows that P SR PSR and Q TR QTR b the SS ongruence Postulate. Since corresponding parts of s are, P R PR and Q R QR. So P R PR and Q R QR. Since and PQ PR QR b the Segment ddition Postulate, we get b substitution PQ P Q, or PQ P Q. P Q P R Q R 34. Since P is on m, then P P b definition of a reflection. definition of reflection, and QR Q R where R is the point where m and intersect. Since m QQ, PRQ and are right angles. PRQ PRQ because all right angles are congruent. PR PR b the Refleive Propert of ongruence. So PRQ PRQ b SS ongruence Postulate. Therefore PQ PQ or PQ P Q because corresponding parts of congruent triangles are congruent. Finall, PQ P Q b definition of congruent segments. PRQ RS PP m QQ QQ 35. definition of a reflection, m is the perpendicular bisector of PP, Q is on m, and Q Q. Then PQ P Q b the definition of reflection. ut Q Q, so PQ P Q. 36. Reflect H in line n to obtain its image H. Then draw a line H J. This will intersect n in a point K. Then the distance traveled, HK KJ, will be as small as possible. 4. The two molecules are reflections of each other. 4. Triangle is a reflection of triangle ; triangle 3 is a translation of triangle. 43. Triangles and 3 are reflections of triangle. Triangle 4 is a rotation of triangle Drawings will var. The distance between each verte of the preimage and line m is equal to the distance between the corresponding verte of the image and line m , 0 n m 80 4 m 80 n m 80 3 m 80 m 45 m z 5 z z , 0 n m 80 m 80 m 90 u 3 u u 6 5v 0 9 5v 9 v 9 5 3w 5 w 5 4 Geometr hapter 7 Worked-out Solutions Ke opright McDougal Littell Inc.

7 slope of 4 The slope of is So the slope of the line perpendicular to is because (5) (53) (54) slope of The slope of is so a line perpendicular to has slope because. (55) 7. Mied Review (p. 40) 57. P 58. PQ 59. QR 60. m m R m Q m 0 6. R 63. b a < c < a b 64. b a < c < a b 65. R(, ) Q(0, 0) R'(, ) RR 0 0 R(, 3) Q 3, 3 Q, RR RR RR RR 7 7 < c < < c < 4 b a < c < a b 33 < c < 33 < c < 45 Q(, ) 3 3 RR R'(3, ) 66. Q, Q 0, 0 Q 0, < c < 9 < c < 30 a b < c < a b 6 6 < c < < c < Lesson 7.3 Technolog ctivit 7.3 (p. 4)... a rotation about the point of intersection of the lines 3. nswers will var. 4. The measure of the acute angle is half the measure of P. 5. twice the measure of the acute angle formed b lines m and k 6. The measure of the angle of rotation is twice the measure of the acute angle formed b the two lines. Etension The conjecture is correct. ctivit 7.3 (p. 43) nswers ma var. Sample answer: ' ' m P m P ' a b < c < a b < c < m D 90 m 80 m m m 80 m 9 80 P 5.7 < c < 56.7 b a < c < a b.9 7. < c < < c < 9 m m D 6 m m 80 m D m m m D 80 m m 06 opright McDougal Littell Inc. Geometr 43 hapter 7 Worked-out Solutions Ke

8 7.3 Guided Practice (p. 46). center of rotation is the fied point about which a figure being rotated is turned.. counterclockwise 3. Yes, because a rotation is an isometr. 4. No, because the distance between an point and its image after a rotation is not fied. 5. The measure of the acute angle between k and m would be half the measure of the angle of rotation. So it would be clockwise rotation of 60 about P maps R onto S. 7. counterclockwise rotation of 60 about P maps R onto Q. 8. clockwise rotation of 0 about Q maps R onto W. 9. counterclockwise rotation of 80 about P maps V onto R. 0. The figure has rotational smmetr about its center with a rotation of 80, either clockwise or counterclockwise.. The figure has rotational smmetr about its center with a rotation of 80, either clockwise or counterclockwise.. The figure does not have rotational smmetr. 7.3 Practice and pplications (pp ) 3. D 4. LH 5. GE 6. M 7. M 8. FGL 9. P 0. definition of a rotation, PR PR and PQ PQ. the definition of congruent segments PR PR and PQ PQ. the Segment ddition Postulate, PR RQ, so RQ R Q b the subtraction propert of equalit. Finall RQ R Q b definition of congruent segments.. definition of rotation QP Q P. Since P and R are the same point and R and are the same point, then QR Q R.. 3. P PR R Q R T R Q P Q S S R T 4. X W 5. J,, K 4,, L 4, 3, M, 3 6. P, 3, Q 3, 5, R 4,, S, D 4,, E 0,, F, 5. 8.,, 4,,, 5 ; the -coordinate of the image is the -coordinate of the preimage. The -coordinate of the image is the opposite of the -coordinate of the preimage. 9. O 0, 0, X, 3, and Z 3, 4 ; the -coordinate of the image is the opposite of the -coordinate of the preimage. The -coordinate of the image is the opposite of the -coordinate of the preimage. 30. The measure of the angle of rotation from to is twice the measure of the acute angle of the intersecting lines, which is 35 or The measure of the angle of rotation from to is twice the measure of the acute angle of the intersecting lines, which is 5 or The measure of the angle of rotation about D is 36 or The measure of the acute angle between lines m and n is 6 or e q 60 Y 4e 7 e 7 4 3b b 4 d 0 d 8 c 7 c 4 a 0 a 55 Z P Z W 36. The wheel hub can be mapped onto itself b a clockwise or counterclockwise rotation of 45, 90, 35, or 80 about its center. Y X q 30 3t 3 t r 0 r 5 u 4 u s 44 Geometr hapter 7 Worked-out Solutions Ke opright McDougal Littell Inc.

9 37. The wheel hub can be mapped onto itself b a clockwise or counterclockwise rotation of or 54 (which is , 06 and , a. definition of a reflection, and Q Q where is the point of intersection of k and QQ. 7, 7, 7, respectivel) QP and Q P are right angles, and about its center. QP Q P because all right angles are congruent. 38. The wheel hub can be mapped onto itself b a clockwise or counterclockwise rotation of 7 and 44 (which is the Refleive Propert of ongruence, P P. So, QP Q P b the SS 360 and , respectivel) about its center. ongruence Postulate. corresponding parts of congruent triangles are congruent, PQ PQ. defini- 39. Yes, the image can be mapped onto itself b a clockwise or counterclockwise rotation of 80 about its center. tion of a reflection, and Q Q where is the point of intersection of m and Q Q. Q P 40. Yes; the answer would change to a clockwise or countercockwise rotation of 90 or 80 about its center. This is since all right angles are congruent. P P b the and Q P are right angles, and Q P Q P because the white figures can be mapped onto the black Refleive Propert of ongruence. So figures. Q P Q P b the SS ongruence Postulate. 4. The center of rotation is the point of intersection of the Since corresponding parts of congruent triangles are diagonals of the square. congruent, Since QP Q P then QP Q P b the Transitive Propert of ongruence. 4. Yes, it is possible for the piece to be hung upside down is a rotation of Q about point P. because the rotational smmetr has an angle of rotation of 80. This would make the picture the same right side up and upside down. b. QP Q P and Q P Q P because corresponding parts of s are. the definition 43. a. Graph for a c. of congruent angles, m Q P m Q P and m QP m Q P. the ngle ddition T'' Postulate, m P m Q P m Q P, T' S' S'' R'' R' R S T Q PQ PQ. m Q Q k QQ m QPQ m QP m QPQ m Q P m Q P m Q P m Q P. Then, m Q P m Q P m Q P b substitution. the Distributive propert, Finall b substitution, m QPQ m Q P m QPQ m Q P. m P. b. R,, S 4, 3, T 5, c. R,, S 3, 4, T, 5 d. single transformation that maps RST onto would be a counterclockwise rotation of 90 about the origin. e. n polgon can be rotated 90 counterclockwise about the origin b doing two reflections of the polgon. First, reflect the polgon in one of the ais. Then reflect the result of the first reflection in the line or. Then the measure of the acute angle between the two lines is 45 and the angle of rotation is 90. R S T 7.3 Mied Review (p. 49) 45. m 5 m m 7 m m 3 m 8 m 6 m m 4 m 80 m m m 6 98 m m 8 m Sample answer: 5. Sample answer: The circumcenter is outside the triangle when the triangle is obtuse. The circumcenter of a right triangle is alwas on the triangle. opright McDougal Littell Inc. Geometr 45 hapter 7 Worked-out Solutions Ke

10 hapter Sample answer:.. The circumcenter of an acute triangle is alwas inside the triangle. 54. Onl one pair of sides are given as parallel, which is not enough information to show that the figure is a parallelogram. Quiz (p. 40). Figure D Figure RSTQ.. The transformation is a reflection in line m. The figure is flipped over line m. 3. Yes, the reflection is an isometr because it preserves length. 4. L, 3 5. M, 4 6. N 4, 0 7. P 8., 3 8. The rotations that map the knot onto itself are rotations b multiples of clockwise or counterclockwise about the center of the knot where the rope starts to unravel. Math & Histor (p. 40). The design has lines of smmetr.. The design has rotational smmetr. It can be mapped onto itself b a rotation of 80 clockwise or counterclockwise about its center. Lesson Guided Practice (p. 45). vector is a quantit that has both direction and magnitude.. Sample answer: The direction is incorrect. PQ \ starts at P and ends at Q. So the vector from P to Q is 6,. 3., 6,. 4., 4, 3. 5., 7,. 6., 5, If 0, maps onto 0, 0, then 8, 5 maps 8, If 0, maps onto 5, 4, then 8, 5 maps onto 3, If 0, maps onto 3, 5, then 8, 5 maps onto 5,. 0. If 0, maps onto 8, 3, then 8, 5 maps onto 0, Sample figures are given Practice and pplications (pp.45 48) 5. a., 3, 4 b. 3, 4 6. a., 5, b. 5, 7. HJ \ ; 4, 8. LK \ ; 4, 4 9. MN \ ; 5, k and m. Sample answer: and inches 4. Yes, the distance from to m is the same as the distance from to m because is reflected in m onto and b the definition of reflection, the distances are equal. 5. The image of 5, 3 is 5, 3 7 or 7, The image of, is, 7 or, The preimage of, is, 7 or 4, The preimage of 0, 6 is 0, 6 7 or,. 9. The image of 0.5,.5 is 0.5,.5 7 or.5, The preimage of 5.5, 5.5 is 5.5, or 7.5, Q R P Q R S P S 46 Geometr hapter 7 Worked-out Solutions Ke opright McDougal Littell Inc.

11 3. Q R 4. P S Q R true 36. false 37. true. 38. true 39. P P 40. Q Q P S Q S P S P R Q R S S R R We are given P a, b and Q c, d. Suppose has coordinates a r, b s. Then or r s. The slope of s or If a r a r. and as given, then and the slope of s is So, the coordinates of r. are c r, d s. the Distance Formula, PQ a c b d PP a r a b s b PP QQ and QQ PP b s b PP QQ QQ r s P Q a r c r b s d s or a c b d. Thus, b the substitution propert of equalit, PQ P Q D es 49. no 50. es 5. Samples might include photographs of floor tiles or fabric patterns. 5. :,,, D:,, 6, E:, 6, 6. F:,, The two vectors are \ 6, 4 and \ 4, To arrive at D from, the vector is D \ 8,. 55. To go straight from town to town D, the vector would be D \ 8,. 56. The correct answer is because a translation preserves length. Q P opright McDougal Littell Inc. Geometr 47 hapter 7 Worked-out Solutions Ke

12 Mied Review (p. 48) m m m w 4 w z 3z 3 z r 3 3 r 3 r 8 6 s s t 3 5 t 5 t 4 t u 6 u 6 3u u m s m m If JK, then SR If QR 6, then JL. 70. If RL 6, then QS true 7. false 73. false Lesson 7.5 Developing oncepts ctivit 7.5 (p. 49) Eploring the oncept ' ' '' ' '' '' 48 Geometr hapter 7 Worked-out Solutions Ke opright McDougal Littell Inc.

13 Investigation.. 3. Yes, the order in which transformations are completed affects the final image. Investigate ,, 7, 5, 5, 4 H H, 3, J 8, 7, K, 8 H 0,,, 5, 3, 4 K K H' K' J J J' J' J'' J'' '' ' '' H, 3, J 8, 7, K, 8 ; the coordinates are the same; switching the order of the transformations did not affect the image in this eample. ' H' K' H'' K'' H'' K'' '' ' Etension So, So, So,. The transformation that maps onto is an isometr, b definition. 7.5 Guided Practice (p. 433). In a glide reflection, the direction of a the translation must be parallel to the line of reflection.. The order in which two transformations are performed sometimes affects the resulting image. 3. In a glide reflection, the order in which the two transformations are performed never matters. 4. composition of isometries is alwas an isometr The line of reflection is the -ais. 8.,, 3 opright McDougal Littell Inc. Geometr 49 hapter 7 Worked-out Solutions Ke

14 7.5 Practice and pplications (pp ) ( 3, 5) '( 3, ) ''(3, ) '( 9, 4) Q ( 7, 6) ( 3, 5) Q (0, ) P (, 7) Q (, 4) 0. F ( 4, 4) The order does affect the final image. G (, ) rotation, then reflection P (, ) P (, 7) reflection, then rotation ''(7, 4) R (6, 5) F (4, 4) R (, ) R (3, 4) G (, ) ( 3, 5) ( 3, 5) '( 6, ) R ( 8, ) P ( 4, ) Q ( 7, 6) ''(, 0) '(, 6) ''(, 6). G (4, 8) G (8, 8) F (, 7) F (5, 7) translation, then reflection reflection, then translation The order does affect the final image.. reflection in the line, followed b a clockwise rotation of 90 about the origin 3. reflection in the line, followed b a reflection in the line 4. clockwise rotation of 90 about the origin, followed b the translation,, 3 5. counterclockwise rotation of 90 about the point (0, ), followed b the translation,, 3 6. glide reflection is an isometr because it is a composition of a translation and a reflection, both of which are isometries. The composition of two isometries is an isometr. 7.,, and are preserved in a glide reflection. 8. nswers will var. 9., 7, 7, 7, 3 7, 6, So,, 6,. 30., 9, 4 9, 4 9 6, 4 4 8, 4 5, So,, 5, 3. fter each part was painted, the stencil was moved through a glide reflection. The translation moved it to the right and the reflection in a horizontal line through its center flipped the design , 4, 5, 6 34., The pattern can be made b a horizontal translation, 80 rotation, vertical line reflection, or horizontal glide reflection. 36. The pattern can be made b a vertical translation. 37. The pattern can be made b a translation or 80 rotation. 38. Sample answer: The X tile needs to be rotated 90 clockwise, then reflected in a horizontal line. The Y tile needs to be reflected in a vertical line, then rotated 90 counterclockwise. 50 Geometr hapter 7 Worked-out Solutions Ke opright McDougal Littell Inc.

15 39. a. nswers will var. 40. b. onjecture: The midpoint of the segment connecting the point and its image is on the -ais. c. Let, be the original point, since the translation must be parallel to the line of reflection, the coordinates of the image are a, for some number a. The coordinates of the midpoint are Then the midpoint is on the -ais d. Yes; the midpoint is the point where the segments and the line of reflection intersect. 7.5 Mied Review (p. 436) a, 3 c f f 4 3 5d 5d 4 3g 5 9 3g 3 g PQ QR c 0 c f a 4 a 0 5d d ONTINUED opright McDougal Littell Inc. a, 0. b 6 3 b 3 b 5 6 4e 6 4 e 3 e h 4 h 45. ONTINUED RS PS PQ QR RS PS so PQRS is a rhombus. PR QS Since PR QS, the diagonals of PQRS are congruent, so PQRS is a rectangle. the Square orollar, PQRS is a square. 46. PQ QR RS PS Since opposite sides are congruent, PQRS is a parallelogram. ONTINUED Geometr 5 hapter 7 Worked-out Solutions Ke

16 46. ONTINUED 48. PR QS Since PR QS, the diagonals are congruent, so PQRS is a rectangle PQ QR RS PS PR QS ll four sides are congruent, but the diagonals are not congruent, so PQRS is a rhombus but not a rectangle or a square. 8, 3,, 7. 6, 9, 6, 3,, 8 0,, 0, 4, 6, 3, 7, 3,,, 6 3, 3, 3, 3, 9, 9, 9.5, 9, 3.5, 5, Geometr hapter 7 Worked-out Solutions Ke opright McDougal Littell Inc.

17 54.. The pattern on the collar can be classified as TG. 3. The pattern on the collar can be classified as T. 4. nswers will var. 5. nswers will var Sample patterns are given ,.5, 7.5, 3.5, 3.5,.5 Lesson Guided Practice (p. 440). frieze pattern is a pattern that etends to the left and right in such a wa that the pattern can be mapped onto itself b a horizontal translation.. The pattern is an eample of TV because there is no 80 rotational smmetr. ut there is a vertical line smmetr because the triangle is equilateral. 3. translation, vertical line reflection 4. translation, rotation, horizontal line reflection, vertical line reflection, horizontal glide reflection 5. translation, rotation, vertical line reflection, horizontal glide reflection 6. translation, rotation, vertical line reflection, horizontal glide reflection 7. The five possible transformations that can be found in a freeze pattern are translation (T), 80 rotation (R), horizontal glide reflection (G), vertical line reflection (V), and horizontal line reflection (H) Practice and pplications (pp ) D 0... translation, horizontal line reflection, horizontal glide reflection 3. translation, 80 rotation 4. translation, 80 rotation 5. translation, 80 rotation, horizontal line reflection, vertical line reflection, horizontal glide reflection 6. Yes; there is a reflection in an vertical line that lies midwa between two figures. 7. Yes; there is a reflection in a the -ais. 8. The transformation that maps onto F is a reflection in the -ais, followed b a horizontal translation described b, 4,. 9. The transformation that maps D onto is a 80 rotation about 8, The frieze pattern is TRHVG.. The pattern on the collar can be classified as TRHVG. 3. T 33. TRHVG 34. TR 35. There are three bands of frieze patterns visible. 36. The patterns near the top and bottom of the jar are T. The pattern in the middle of the jar is TR. 37. nswers will var. 38. Sample answer: The band around the middle is an eample of THG. 39. d 9.5 in. d in. The circumference of the base is about 9.83 inches. If ou want 0 repetitions of the design, the design should be about or.98 inches wide. opright McDougal Littell Inc. Geometr 53 hapter 7 Worked-out Solutions Ke

18 ONTINUED b. c Sample answer: The design on the tiles limits what classifications of patterns can be made. For instance, in Eercise 40, the design on the tile would not allow the creation of THG in a single row because there is not a horizontal line of smmetr in the tile. The same would be true for the tile in Eercise If a pattern can be mapped onto itself b a horizontal glide reflection and b a vertical line reflection, it can be mapped onto itself b a 80 rotation about the point where the lines of reflection intersect. 45. If a pattern can be mapped onto itself b a horizontal line reflection and b a vertical line reflection, it can be mapped onto itself b a 80 rotation about the point where the lines intersect. It can also be mapped onto itself b a horizontal glide reflection involving the given horizontal line reflection and an translation. 46. If a pattern can be mapped onto itself b a 80 rotation about a point and a horizontal glide reflection, the center of rotation must be on the line of reflection for the glide reflection. Then the pattern can be mapped onto itself b reflection in a vertical line through the center of rotation. 47. Sample answer: 50. a. T b. TVG c. TRVG 5. Design does not have a rotational smmetr. Sample answer: 5. If it has 80 rotational smmetr, then its classification must be at least TR. The 80 smmetr means that the pattern can be mapped onto itself with a 80 rotation. Therefore, it must at least be TR. 53. Yes Yes Yes Is there a line of reflection? Is the reflection in a horizontal line? Is there a 80 rotation? No Yes Is the reflection in a horizontal line? No Is there a line of reflection? No Is there a glide reflection? No Yes No Yes No TRHVG TRVG TR THG TV TG T 48. Sample answer: D 49. Sample answers: a. E H F G ONTINUED 7.6 Mied Review (p. 444) girls girls girls bos girls girls girls bos girls girls girls bos 7 girls bos 3 8 girls bos 6 9 girls bos Geometr hapter 7 Worked-out Solutions Ke opright McDougal Littell Inc.

19 k k z 3 7z 3z 7z 3 0z 7z 0 3 z 4 3 7z 0 3 z 3 3 z 3 z d d square units h b b square units Quiz (p. 444) , 5, 5, 6,, 4 4, 6,, 7,, 5 3,,,,, 3 4, 4, 9, 5, 6, Q (, 0) P ( 3, 3) R (, 3) w 3 w w 4 w 3 w 4 w 4 3w w 8 3 w 8 d d square units Q (4, ) R (, ) P (5, 3) k 8. P 9. H L H G F 0... D 3. G F 4., 4, 5, followed b 90 clockwise rotation about the origin 5. reflection in the -ais, followed b a 90 counterclockwise rotation about the origin 6. The rainbow boa s snakeskin has a frieze pattern classified as TR. 7. The gra-banded kingsnake s snakeskin has a frieze pattern classified as TRHVG. hapter 7 Test (p. 449). The transformation that maps RST onto XYZ is a reflection in the -ais.. Yes, RT XZ because a reflection preserves length. 3. The image of T is Z. 4. The preimage of Y is S. 5. Sample answer: 6. Sample answer: K M L P M N P K N 7. Yes, the frieze pattern is TR. hapter 7 Review (pp ). Yes; it is an isometr because the figure and its image appear to be congruent.. No; it is not an isometr because the figure and its image are not congruent. 3. Yes; it is an isometr because the figure and its image appear to be congruent. 7. The transformation that maps figure T onto figure is a reflection in line m. 8. The transformation that maps figure T onto is a reflection in line m followed b a reflection in line n. Or it is a rotation about the point of intersection of lines m and n. 9. The measure of the angle of rotation is twice the measure of the acute angle formed b lines m and n. So, the measure of the angle of rotation is 85 or 70. T T opright McDougal Littell Inc. Geometr 55 hapter 7 Worked-out Solutions Ke

20 0. The transformation that maps figure R onto figure is a reflection in line k.. The transformation that maps figure R onto is a reflection in line k, followed b a reflection in line m, or it is a translation.. The distance between corresponding parts of figure R and figure is twice the distance between lines k and m. The distance is 5 or 0 units. 3. glide reflection is a composition of a translation followed b a reflection in a line parallel to the translation vector. 4. Sample answer: is the final image when is rotated 90 clockwise about the origin, then reflected in the -ais. is the final image when is reflected in the -ais, then rotated 90 clockwise about the origin. 5. Sample answer: R P(, ) Q(4, 6) P (6, ) PQ is reflected in the -ais, then translated, 5,. The same image results if the transformations are performed in reverse order. 6. The flag of Switzerland has a vertical line of smmetr, a horizontal line of smmetr, two diagonal lines of smmetr and rotational smmetr. It can be mapped onto itself b a clockwise or counterclockwise rotation of 80 about the center. 7. The flag of Jamaica has a vertical line of smmetr, a horizontal line of smmetr, and rotational smmetr. It can be mapped onto itself b a clockwise or counterclockwise rotation of 80 about the center. Q (9, 6) R R 8. The flag of the United Kingdom has a vertical line of smmetr, a horizontal line of smmetr, and rotational smmetr. It can be mapped onto itself b a clockwise or counterclockwise rotation of 80 about its center. 9. translation, 80 rotation, horizontal line reflection, vertical line reflection, glide reflection 0. translation, 80 rotation, horizontal line reflection, vertical line reflection, glide reflection. translation, vertical line rotation hapter 7 Standardized Test (pp ).. D 3. E z 8 4 z 6 5. W 3, 8 3 8, 8 0 or 5, X 7, 6 7 8, 6 0 or, 4 Y 5, 5 8, 0 or 3, 8 6. D counterclockwise rotation maps onto. reflection in the -ais maps onto. So the correct answer is D With the reflection in, S 6, is mapped onto S 6, 4 and T 3, 5 is mapped onto T 3, clockwise rotation about the point 3, maps S 6, 4 onto S, 5 and T 3, 7 onto T,. So the correct answer is E. 0. The letters that have a vertical line of smmetr are, H, I, M, O, T, W, X, and Y.. The letters that have a horizontal line of smmetr are, E, H, K, O, and X.. The letters with a rotational smmetr are H, N, O, S, X, and Z. 56 Geometr hapter 7 Worked-out Solutions Ke opright McDougal Littell Inc.

21 3. a.,, 7 b., 7 4. The coordinates of the vertices of figure WXYZ are W 7, 6 or W 9, 4, X 4 7, 6 or X, 4, Y 5 7, 3 or Y,, and Z 0 7, 3 or Z 7, clockwise rotation about the origin 6. Sample answer: Project hapters 6 7 (pp ) Investigation. No; the quadrilateral is not a regular polgon.. Some transformations that would map the pattern onto itself are translations and rotations. 3. The sum of the measures of the angles at an verte of a quadrilateral tessellation is 360. n quadrilateral will tessellate because the sum of the measures of its angles is 360. If all four angles are placed so the are adjacent, the full 360 rotation is covered. 4. Square and equilateral triangle tessellations are regular tessellations. 5. Tessellations will var. translation and a rotation map the pattern onto itself. 6. Sample answer: This is a semiregular tessellation. 7. These shapes cannot be used together to create a tessellation because there is no combination of multiples of 08 and multiples of 90 that add together to get These shapes can be used together to create a nonregular tessellation. Sample answer: opright McDougal Littell Inc. Geometr 57 hapter 7 Worked-out Solutions Ke