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Answers to Exercises 7 CHAPTER 7 • CHAPTER CHAPTER 7 • CHAPTER LESSON 7.1
1.Rigid; reflected, but the size and shape do not change. 2.Nonrigid; the shape changes. 3.Nonrigid; the size changes. 4. 5.
6.
P
7.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.Extend the three horizontal segments onto the other side of the reflection line. Use your compass to measure lengths of segments and distances from the reflection line. 10b.
16.(Lesson 7.1)
Number of sides of regular polygon
Number of reflectional symmetries
Number of rotational symmetries (360°)
3
3
3
11.
P
12.reflectional symmetry 13.4-fold rotational and reflectional symmetry 14.reflectional symmetry 15.7-fold symmetry: possible answers are F or J. 9-fold symmetry: possible answers are E or H. Basket K has 3-fold rotational symmetry but not reflectional symmetry. 16.See table below;n, n 17. , or 18.
19.P(a,b),Q(a,b),R(a,b) 20.possible construction: P
21.50th figure: 154 (50 shaded, 104 unshaded); nth figure: 3n4 (nshaded, 2(n2) unshaded) 22.46 23.It is given that12, and23 because of the Vertical Angles Conjecture, so 13. SegmentDCis congruent to itself. DCEandDCBare both right angles, so they are congruent. Therefore,DCBDCEby ASA, andBCCEby CPCTC.
4
4
4
5
5
5
6
6
6
7
7
7
8
8
8
. . .
. . .
. . .
n
n
n
ANSWERS TO EXERCISES
85
y
–4
ytranslation
–8
6.Rules that involvexorychanging signs, or switching places, produce reflections. If bothxandychange signs, the rule produces a rotation. Rules that produce translations involve a constant being added to thexand/oryterms. 5, 0is the translation vector for Exercise 1. 7.(x, y)→(x,y) 8.(x, y)→(x,y) 9. N
–5
4.
H'
S
N
W
T
5
y 5
–5
yreflection
W
5
S
8 ball
E
Mason
Perry
13.
14.
3.
T H H' S 12.by the Minimal Path Conjecture Proposed freeway
N
5
2.reflection y 8
1.
86
5
5.rotation
H E
ANSWERS TO EXERCISES
E
Cue ball
10.There are two possible points, one on the N wall and one on the W wall.
5
reflection
LESSON 7.2
11.
x
7
5
15.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 18.
19.sample construction:
20.sample construction:
21.false; possible counterexample: trapezoid with two right angles 22.false; possible counterexample: isosceles trapezoid
ANSWERS TO EXERCISES
87
LESSON 7.3
1.10, 10 2.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 5.180° 6.3 cm 7.possible answer:
8.possible answer:
9.
Center of rotation
10.Two reflections across intersecting lines yield 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°.
88
ANSWERS TO EXERCISES
11.Answers may vary. Possible answer: reflection across the figure’s horizontal axis and 60° clockwise rotation. 12.
13. , 14.Sample answer: Draw a figure on an overhead transparency and then project the image onto a screen. 15.possible answers: rotational: playing card, ceiling fan, propeller blade; reflectional: human body, backpack 16.one: yes; two: no; three: yes
17.possible answer:
A O B 9 0 ? ?14 5 11 11 18a. 12 20 ? ? 0 13 127 18b. 2a3b4c ? ? ? a b c a2b3c d e f d d–e0 ? ? ? 0d f
LESSON 7.4
1.Answers will vary.2.Answers will vary. 3 2 4 3.3 .44.3 .6 2 2 5.3 .4.3.46.3.4.6.43.4 .6 3 2 2 6 2 7.3 .43 .4.3.48.33 .4.12 9a.The dual of a square tessellation is a square tessellation. 9b.The dual of a hexagon tessellation is a triangle tessellation. 9c.If tessellation A is the dual of tessellation B, then tessellation B is the dual of tessellation A. 4 8 10.The dual is a 33 tessellation of isosceles right triangles. 11.
12.
13.A ring of ten pentagons fits around a decagon, and another decagon can fit into any two of the pentagons. But another ring of pentagons around the second decagon doesn’t leave room for a third decagon. 14.
15.Answers will vary. 1 16.y x4 2 y
–3
4
– 6
5
17.possible answer: TOT 18.
W
8ball
N
Cue ball
S
E
ANSWERS TO EXERCISES
89
LESSON 7.5
1.Answers will vary. 3 4 2.The dual is a 55 tessellation.
3.
4.Yes. The four angles of the quadrilateral will be around each point of intersection in the tessellation. c a c a 5. b b b b b a c a c a c c a c a c a b b b b b a c a c
90
ANSWERS TO EXERCISES
By the Triangle Sum Conjecture,abc180°. Around each point, we have 2(abc) 2 180°360°. 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 ways
7.
8.y 2x3 y 8
–2
5
LESSON 7.6
1.Answers will vary. 2.Answers will vary. 3.Answers will vary. 4.regular hexagons 5.squares or parallelograms 6.squares or parallelograms 7.
8.
9.Answers will vary. 10.Answers will vary. 11. B
A
S
E
2 12.y x3; the slope is the opposite sign. 3 y
–10
5
10
13.3.4.6.44.6.12 440 rev228 ft1 min 14. 1290 ft/s 1 min1 rev60 s 15.Possible explanations: 15a.true; The kite diagonal between vertex angles is the perpendicular bisector of the other diagonal; in a square, diagonals would bisect each other 15b.False; it could be an isosceles trapezoid. 15c.False; it could be a rectangle. 15d.true; Parallel lines cut off congruent arcs of a circle, so inscribed angles (the base angles of the trapezoid) are congruent.
ANSWERS TO EXERCISES
91
LESSON 7.7
1.equilateral triangles. 2.regular hexagons. 3.
4.
5.Answers will vary. 6.Answers will vary. 7.sample design:
8.False; they must bisect each other in a parallelogram.
92
ANSWERS TO EXERCISES
9.true 10.true 11.False; it could be a kite or an isosceles trapezoid. 1 12.The path would be of Earth’s circumference, 4 approximately 6280 miles, which will take 1 126 hours, or around 5 days. 4 13a.Using the Reflection Line Conjecture, the line of reflection is the perpendicular bisector of AAandBB. Because these segments are both perpendicular to the reflection line, they are parallel to each other. Note that ifABis parallel to the reflection line, quadrilateralAABBwill be a rectangle instead of a trapezoid. 13b.Yes; it has reflectional symmetry, so legs and base angles are congruent. 13c.greatest: near each of the acute vertices; least: at the intersection of the diagonals (whereA, C,andBbecome collinear andA, C,andB become collinear) 108 9 8 7 ? ? 3 56 14a.6 0 41 0 ? ? 9 2 28 15 0 2 ? 8 3 13 30 14b. ? ? ? 12550 291 10
1.parallelograms 2.parallelograms 3.
4.
LESSON 7.8
5.Answers will vary. 6.Answers will vary. 7.Circumcenter is (3, 4); orthocenter is (10, 8). 8.
9.
10.
ANSWERS TO EXERCISES
93
USING YOUR ALGEBRA SKILLS 7 1 1.y x 6 2.y 2x2 3 2 3.5).Centroid is 2, ; orthocenter is (0,
94
ANSWERS TO EXERCISES
4.Centroid is (4, 0); orthocenter is (3, 0). 4 3 5.1, 6.(1,1) 7.(5,8)
CHAPTER 7 REVIEW
1.true2.true 3.true4.true 5.true6.true 7.False; a regular pentagon does not create a monohedral tessellation and a regular hexagon does. 8.true9.true 10.False; two counterexamples are given in Lesson 7.5. 11.False; any hexagon with one pair of opposite sides parallel and congruent will create a monohedral tessellation. 12.This statement can be both true and false. 13.6-fold rotational symmetry 14.translational symmetry 15.Reflectional; color arrangements will vary, 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 individually has bilateral symmetry. The center portion has bilateral symmetry. 17.Answers will vary. 18.Answers will vary. 6 2 19.32-uniform3 .4.3.4; 2 20.semiregular4.8 ; 1 21.yx 2
y
x
22.
T
H
23.Use a grid of squares. Tessellate by translation. 24.Use a grid of equilateral triangles. Tessellate by rotation. 25.Use a grid of parallelograms. Tessellate by 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 alternate the direction in which they lean (row 1 leans right, row 2 leans left, row 3 leans right, and so on). 28.
ANSWERS TO EXERCISES
95
