Answers to Exercises 11.

Transcription

1 CHAPTER 7 CHAPTER LESSON 7.1 CHAPTER 7 CHAPTER 1. Rigid; reflected, but the size and shape do not change. 2. Nonrigid; the shape changes. 3. Nonrigid; the size changes possible answer: a boat moving across the water 8. possible answer: a Ferris wheel 9a. Sample answer: Fold the paper so that the images coincide, and crease. 9b. Construct a segment that connects two corresponding points. Construct the perpendicular bisector of that segment. 10a. Etend the three horizontal segments onto the other side of the reflection line. Use our compass to measure lengths of segments and distances from the reflection line. 10b. P reflectional smmetr fold rotational and reflectional smmetr 14. reflectional smmetr 1. 7-fold smmetr: possible answers are F or J. 9-fold smmetr: possible answers are E or H. Basket K has 3-fold rotational smmetr but not reflectional smmetr. 16. See table below; n, n , or 19. P( a, b), Q( a, b), R(a, b) 20. possible construction: P 21. 0th figure: 14 (0 shaded, 104 unshaded); nth figure: 3n 4 (n shaded, 2(n 2) unshaded) It is given that 1 2, and 2 3 because of the Vertical Angles Conjecture, so 1 3. Segment DC is congruent to itself. DCE and DCB are both right angles, so the are congruent. Therefore, DCB DCE b ASA, and BC CE b CPCTC. P 16. (Lesson 7.1) Number of sides of n regular polgon Number of reflectional n smmetries Number of rotational n smmetries ( 360 ) ANSWERS TO EXERCISES 8

2 1. translation LESSON Rules that involve or changing signs, or switching places, produce reflections. If both and change signs, the rule produces a rotation. Rules that produce translations involve a constant being added to the and/or terms., 0 is the translation vector for Eercise (, ) (, ) 8. (, ) (, ) 9. N 2. reflection 8 W Cue ball E 8 ball S reflection reflection There are two possible points, one on the N wall and one on the W wall. 11. W N T N S H'' H E W E T H H' S 12. b the Minimal Path Conjecture. rotation Proposed freewa Perr Mason , 86 ANSWERS TO EXERCISES

3 1. possible answer: HIKED 16. one, unless it is equilateral, in which case it has three 17. two, unless it is a square, in which case it has four sample construction: 19. sample construction: 21. false; possible countereample: trapezoid with two right angles 22. false; possible countereample: isosceles trapezoid ANSWERS TO EXERCISES 87

4 O LESSON , A 180 rotation. If the centers of rotation differ, rotate 180 and add a translation. 3a. 20 cm 3b. 20 cm, but in the opposite direction 4a. 80 counterclockwise 4b. 80 clockwise cm 7. possible answer: 11. Answers ma var. Possible answer: reflection across the figure s horizontal ais and 60 clockwise rotation , 14. Sample answer: Draw a figure on an overhead transparenc and then project the image onto a screen. 1. possible answers: rotational: plaing card, ceiling fan, propeller blade; reflectional: human bod, backpack 16. one: es; two: no; three: es 8. possible answer: N N 9. O A H A H H Center of rotation 10. Two reflections across intersecting lines ield a rotation. The measure of the angle of rotation is twice the measure of the angle between the lines of reflection, or twice 90, or 180. A O N 17. possible answer: A O B 11 18a.?? b. a b d e 12 c a 2b 3c??? f 20 d 9 0?? 12 7 d e a 3b 4c??? 0 d f 88 ANSWERS TO EXERCISES

5 LESSON Answers will var. 2. Answers will var a. The dual of a square tessellation is a square tessellation. 9b. The dual of a heagon tessellation is a triangle tessellation. 9c. If tessellation A is the dual of tessellation B, then tessellation B is the dual of tessellation A. 10. The dual is a tessellation of isosceles right triangles Answers will var possible answer: TOT A ring of ten pentagons fits around a decagon, and another decagon can fit into an two of the pentagons. But another ring of pentagons around the second decagon doesn t leave room for a third decagon. 14. W 8-ball N Cue ball S E ANSWERS TO EXERCISES 89

6 LESSON Answers will var. 2. The dual is a 3 4 tessellation. B the Triangle Sum Conjecture, a b c 180. Around each point, we have 2(a b c) Therefore, a triangle will fill the plane edge to edge without gaps or overlaps. Thus, a triangle can be used to create a monohedral tiling. 6. three was Yes. The four angles of the quadrilateral will be around each point of intersection in the tessellation.. c a c a c b a b b c b a c b a c a c a c a b b b a c b a c b 2 90 ANSWERS TO EXERCISES

7 LESSON Answers will var. 2. Answers will var. 3. Answers will var. 4. regular heagons. squares or parallelograms 6. squares or parallelograms Answers will var. 10. Answers will var. 11. A S B E ; the slope is the opposite sign r 1 m ev in 2 28 ft 1 min ft/s 1 rev 0 s 1. Possible eplanations: 1a. true; The kite diagonal between verte angles is the perpendicular bisector of the other diagonal; in a square, diagonals would bisect each other 1b. False; it could be an isosceles trapezoid. 1c. False; it could be a rectangle. 1d. true; Parallel lines cut off congruent arcs of a circle, so inscribed angles (the base angles of the trapezoid) are congruent. 10 ANSWERS TO EXERCISES 91

8 1. equilateral triangles. 2. regular heagons Answers will var. 6. Answers will var. 7. sample design: LESSON False; the must bisect each other in a parallelogram. 9. true 10. true 11. False; it could be a kite or an isosceles trapezoid. 12. The path would be 1 4 of Earth s circumference, approimatel 6280 miles, which will take 126 hours, or around 1 4 das. 13a. Using the Reflection Line Conjecture, the line of reflection is the perpendicular bisector of AA and BB. Because these segments are both perpendicular to the reflection line, the are parallel to each other. Note that if AB is parallel to the reflection line, quadrilateral AA B B will be a rectangle instead of a trapezoid. 13b. Yes; it has reflectional smmetr, so legs and base angles are congruent. 13c. greatest: near each of the acute vertices; least: at the intersection of the diagonals (where A, C, and B become collinear and A,C,and B become collinear) 14a ?? b ? ?? ?? 28 1? ANSWERS TO EXERCISES

9 1. parallelograms 2. parallelograms 3. LESSON 7.8. Answers will var. 6. Answers will var. 7. Circumcenter is (3, 4); orthocenter is (10, 8) ANSWERS TO EXERCISES 93

10 USING YOUR ALGEBRA SKILLS Centroid is 2, 2 3 ;orthocenter is (0,). 4. Centroid is (4, 0); orthocenter is (3, 0).. 1, ( 1, 1) 7. (, 8) 94 ANSWERS TO EXERCISES

11 CHAPTER 7 REVIEW 1. true 2. true 3. true 4. true. true 6. true 7. False; a regular pentagon does not create a monohedral tessellation and a regular heagon does. 8. true 9. true 10. False; two countereamples are given in Lesson False; an heagon with one pair of opposite sides parallel and congruent will create a monohedral tessellation. 12. This statement can be both true and false fold rotational smmetr 14. translational smmetr 1. Reflectional; color arrangements will var, but the white candle must be in the middle. 16. The two towers are not the reflection (or even the translation) of each other. Each tower individuall has bilateral smmetr. The center portion has bilateral smmetr. 17. Answers will var. 18. Answers will var ; 2-uniform ; semiregular Use a grid of squares. Tessellate b translation. 24. Use a grid of equilateral triangles. Tessellate b rotation. 2. Use a grid of parallelograms. Tessellate b glide reflection. 26. Yes. It is a glide reflection for one pair of sides and midpoint rotation for the other two sides. 27. No.Because the shape is suitable for glide reflection, the rows of parallelograms should alternatethedirectioninwhichthelean(row1 leans right,row 2 leans left,row 3 leans right,and so on). 28. T H ANSWERS TO EXERCISES 9