Name ________________________________________ Date __________________ Class__________________
LESSON
17-3
Practice C
Tessellations
1. A tiling can be made using these three regular polygons: a decagon,
a triangle, and one other polygon. Find the other polygon.
___________________
2. Describe a regular or semiregular tiling that uses a nonagon. Explain how you
know this is the only possible tiling using a nonagon.
3. There are 12 theoretically possible regular or semiregular tessellations using the
regular polygons having between 3 and 12 vertices. Describe each tessellation by
giving the number and types of the polygons that meet at each vertex.
4. Draw the tessellation you listed in
Exercise 3 that includes a pentagon.
5. Tell what is remarkable about the drawing.
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6. Choose which of these two figures will tessellate. Draw the tessellation.
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5.
6.
7.
8.
2. yes
3. yes; 120°; 3
4. yes; 180°; 2
17-3 TESSELLATIONS
Practice A
1. tessellation
Review for Mastery
2. regular polygons
3. glide reflection symmetry
1. yes; one line of symmetry
4. semiregular
5. both
6. glide reflection symmetry
7. translation symmetry
8.
2. no
3. yes; 180°; order: 2
4. yes; 90°; order: 4
5. both
9. neither
6. plane symmetry
7. neither
8. both
Challenge
10. regular
11. semiregular
12. no
13. yes
14. yes
Practice B
1. TVRG
2. THVRG
1. translation symmetry
3. T
4. TV
2. both
5. THG
6. TR
3. glide reflection symmetry
7. Patterns will vary.
4.
8. Answers will vary.
9. For all integers n,
­ x 12n, where 12n 2 d x d 12n 2
°2, where 12n 2 d x d 12n 4
°
f(x) ®
° x 12n 6, where 12n 4 d x d 12n 8
°¯2, where 12n 8 d x d 12n 10
5.
Problem Solving
1. yes
2. yes; 180°; order: 2
3. both
4. rotational symmetry of order 2
5. D
6. H
7. D
8. H
6. regular
7. semiregular
8. neither
9. no
10. yes
11. no
Practice C
Reading Strategies
1. 15-gon
1. no
2. The tiling uses a nonagon, a triangle, and
an 18-gon; possible answer: Each angle
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Holt McDougal Coordinate Algebra
in a regular nonagon measures 140°,
which does not divide evenly into 360°.
So the nonagon cannot be used in a
regular tessellation. If two nonagons are
used in a semiregular tessellation, then
the measures of the angles of the
polygons that are not nonagons must be
360° 280° 80°. There is no polygon or
combination of polygons with angles that
measure 80°. If one nonagon is used in a
semiregular tessellation, then the
measures of the angles of the polygon or
combination of polygons that are not
nonagons must be 360° 140° 220°.
No polygon can have an angle measure
greater than 180°, so a combination of
polygons must make up 220°. A triangle
has 60° angles, so that leaves 160°. The
formula for interior angle measure of a
regular polygon shows that an 18-gon
has 160° angles. This tiling works, but
there could be more. A square would
leave 130°, a pentagon would leave
112°, a hexagon would leave 100°, and
an octagon would leave 85°. No polygon
or combination of polygons makes up
any of those angle measures. So there is
only one possible tiling using a nonagon.
3. six triangles; four triangles and one
hexagon; three triangles and two
squares; two triangles and two hexagons;
two triangles, one square, and one
dodecagon; one triangle and two
dodecagons; one triangle, two squares,
and one hexagon; four squares; one
square and two octagons; one square,
one hexagon, and one dodecagon; two
pentagons and one decagon; three
hexagons
6.
Review for Mastery
1. translation symmetry
2. translation symmetry and glide reflection
symmetry
3.
4.
5. regular
7. no
8. yes
Challenge
1. 60°
3. 6 u 60°
2. 6
360°
4. Explanations will vary
5. 60° 60° 60° 90° 90°
360°
6. Although both codes indicate that 3
equilateral triangles and 2 squares meet
at a vertex point of the tessellation, each
code specifies a different arrangement.
4.
5. Although the angles at each vertex add
to 360°, the polygons do not tessellate.
They overlap.
6. neither
7. There are only three regular tessellations.
There are exactly eight semiregular
tessellations.
Problem Solving
1. translation, reflection, rotation
2. translation
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