Download MATH 310 ! Self-Test " Transformation Geometry

MATH 310 ! Self-Test " Transformation Geometry, Similarity and Symmetry $ rf7
The essential problems to solve are: 5, 6, 7, 10, A1 and A4
1. Explain isometry (rigid transformation or rigid motion) of the plane.
Name and describe* the four types of isometries of the plane.
(* what happens to the points of the plane under this transformation? How are they moved?
What happens to figures in the plane? In what way are they changed? Be specific for each
different type of transformation.)
2.
Any transformation of the plane can be accomplished by a series of reflections. Explain.
3.
Which transformations do not change the direction of a figure?
(I.e. if the figure is “facing” a particular direction in the plane, the image after the
transformation will face the same direction. )
4.
Which transformation(s) change the orientation (clockwise sense) of a figure?
5.
Identify the single transformation which moves figure 1 to each one of the other figures.
fig. A
fig. B
fig. C
fig. D
fig. E
fig. F
fig. 1
6.
V
7.
Sketch images of the figures under the isometries described at right. Follow directions.
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Translate the figure A
by vector V, label that A'.
Then rotate A'
label that A''.
(Not A) 90E about P ;
Then reflect A'' through line l .
Label the final result A'''
_
Briefly describe all symmetries of
a. a square
b. a regular pentagon. (Sketches are appropriate.)
d. The individual letters in “S Y M M E T R I C”.
e.
f.
g.
c.
_
8. Add squares (congruent to existing squares) to each of the following figures to make the
figure have a line symmetry, but no rotational symmetry. (What is the minimum needed?)
a.
b.
9. Add to each figure at right so the resulting figure
has rotational symmetry but no line symmetry.
c.
a.
b.
(Assume these are regular polygons.)
10. Given the figures at right are similar,
a. What is the scale factor from the left figure to the right?
b. Find x .
24
8
11. In a dilation of the plane with center C, the distances between
C and a figure and its image are as shown. What is the scale factor
20 cm
of the transformation?
30cm
C C
30
10
X
ST TG p 2
Some more detailed questions:
A1. How many rotational symmetries has each figure? How many line symmetries?
A2. Using any of the tools we have employed:
rotate line segment AB by +45o about center O.
@O
@A
B@
Now we get really esoteric:
A3. Is it possible to generate a given translation by using a series of reflections?
Is it possible to generate a given rotation by using a series of reflections?
Is it possible to generate a given glide-reflection by using a series of reflections?
(If yes to all the above, then we'd say the set of reflections generates all possible
isometries of the plane;
every type of isometry can be accomplished by a series of reflections.)
A4. Can translations be used to generate all possible isometries of the plane? Why or why not?
What about rotations? ...glide reflections?
A5. Match each description or attribute of a transformation with the most appropriate
transformation (T R M GR):
a. No point of the plane is fixed; a figure and its image have the same orientation.
b. Every point of the plane moves exactly the same distance.
c. Exactly one point of the plane is fixed ("fixed" means doesn’t move).
d. No point of the plane is fixed; a figure must be "flipped over" to match its image.
A6. Select the appropriate transformation(s) to fit each description below.
T = Translation, R = Rotation, M = Reflection, GR = Glide Reflection
a. The composition of two translations is (always/sometimes/never) a (T R M GR).
[How do points move if translated by vector U then V?]
b.
The composition of two reflections through parallel lines is a (T R M GR).
c.
d.
e.
The composition of two rotations about a common center is a (T R M GR).
It is possible for the composition of two reflections to be (T R M GR).
It is possible for the composition of two rotations to be (T R M GR).
[How do points move if reflected through line l, then through parallel line m? How far? In what direction?]
A7. Is the composition of rotations commutative? (You should know what that means:
given any two rotations, is the result the same regardless of the order in which they are
performed? ,,.or does order matter?)
A8. Match each description or attribute of a transformation with the most appropriate
transformation (T R M GR):
a. Moves every point of the plane the same distance and direction.
b. Determined by a vector, or arrow, that specifies the distance and direction each point
is moved.
c. Composition of two of these is always one of these. Composition is commutative (order
doesn’t matter).
d. Moves points on parallel straight paths. Figures are never reversed.
e. Moves points on concentric circular paths. Figures are never reversed.
f. Determined by a directed angle that specifies the center, amount and direction of
.
g. Points further from the center move a greater distance.
h. Determined by a line. Distances moved are proportional to distances from this line.
i. Points move on parallel paths, but not equal distances. Figures are reversed.
j. Points do not move on parallel paths, but figures are reversed.
k.
Determined by a line and a vector.
MATH 310 ¸SelfTest¹ Transformation Geometry SOLUTIONS & COMMENTS
1. From class notes!
rf7
A transformation is a one-t o-one mapping of the points of the plane to new points of the same plane.
An isometry, also called a " rigid motion" , is a transformation w hich preserves distances.
Preserving all distances preserves figures (think of triangles).
There are only four types of isometries of the plane:
Translation
Reflection
Rotation
Glide reflection
(“ Slide” )
(“ Flip” )
(“ Turn” )
(“ Flip’ nSlide” )
Translation
CDetermined by a vector (an arrow w ith specific length and direction)
CMoves all points of the plane in one direction, the same distance...
determined by the “ slide arrow ” or vector of the translation.
CSince all points move the same direction, points move on parallel paths.
Reflection
CExcept for those on the line of reflection, all points move across the line
of reflection (perpendicular to the line of reflection); points equally
m
distant from the line of reflection, but on opposite sides, essentially
sw ap places.
CThe reflection line is the 2 bisector of the segment joining a point
and its image.
CClockw ise vs counter-clockw ise sense/orientation reverses (ie figures
“ flip” ).
Rotation
O÷
CDetermined by a center and directed angle of rotation
CEvery point in the plane, except the center of rotation, moves on a
circular path around the center of rotation, through the same angle.
CThe center of rotation stays fixed. In example at right, angle is 180E.
C
The angle of rotation for the bunnies is about 42E.
Glide-Reflection
CDetermined by a line of reflection and vector parallel to the line.
CAll points of the plane flip across the line of reflection, then “ glide” .
CNo point stays fixed.
CThe reflection line contains the midpoints betw een points and their
images.
CClockw ise vs counter-clockw ise sense (orientation) reverses. (i.e.
figures “ flip” .)
What type of isometry is it?
°Find at least three pairs of matching points, and name them, e.g. A BC and A ’ B’ C ’ .
°Check the orientation of the figure & image. If path A BC is clockw ise & A ’ B’ C ’ is counter-clockw ise,
then orientation reversed, and the isometry must be a Reflection or Glide-Reflection. If the orientation
is not reversed, then the isometry is a translation or rotation.
Draw arrow s from A to A ’ , B to B’ .
If they are the same (length & direction), the isometry is a translation.
If they are one direction, but different lengths, the isometry must be a reflection.
If they differ in direction, the isometry is either a rotation or glide-reflection.
MATH 310 ¸SelfTest¹ Transformation Geometry SOLUTIONS & COMMENTS ( p2)
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2. See the exercises using multiple reflections on the TG-2 and TG-3 worksheets.
3. Translations leave a figure “facing the same direction”. All other transformations can change the direction a
figure is facing (although this may not be obvious in special circumstances, such as a 180E rotation of a
symmetric figure, e.g. a rectangle). A figure that is “facing NW” will remain “facing NW” after any number of
translations of the plane. The figure never gets to “turn”, as it does in a rotation. As for reflections, we have
seen how two reflections can cause a rotation, and just one reflection results in the figure “facing the opposite
direction”.
Q: Which of the four images is facing the
C
C
(2)
(4)
same direction as the original (shaded) ?
C
(3)
C
A: only the second one.
(1)
Q: What are these transformations;
C
Can you identify them completely?
A. The transformations are (1) Reflection (2) Translation (3) Rotation (4) Glide-reflection
After reflection, before the glide, the
C
C
(2)
(4)
¹ image is here.
C
(3)
C
Reflection
(1)
line for (1)¸
C
Translation üvector
¹Line of reflection for the glide reflection
(The rotation here is a 180E rotation. The center of a 180E rotation is always easy to find, because it is halfway
between a point and its image. (Find a point on the original figure, and its image on the new figure. The
midpoint is it. This works onl y for 180E rotations.) Other rotations require more effort to locate the center.
4. Only Reflections and Glide-reflections change the clockwise sense of a figure. That is because only these types
of transformations “flip” the plane over, and “flipping” is required to reverse the clockwise sense of the figure.
The effects on a figure, of translating and rotating the plane can be visualized by sliding and turning a figure on a
flat surface, and neither of these results in a “flipped over” figure.
5. fig. 1
Translation
fig. A
Reflection
fig. B
Glide-Reflection
fig. C
Rotation
fig. D
Dilation/Contraction
fig. E
fig. F
Figure E is the result of a dilation, a ”size transformation”, not an isometry (not rigid motion).
Figure F is not even similar to Figure 1. We do not study any such transformations in this course.
More details: In the following figures, dotted lines show how individual points moved.
5.
fig. 1
Translation
Notice that figure A faces the same direction
as the original figure (fig. 1). Also notice all points
fig. A
Showing the vector of translation.
5.
move the same distance and direction.
fig. 1
Notice that all points move on parallel paths,
but different distances....
ü
Showing the line of reflection.
fig. B
MATH 310 ¸SelfTest¹ Transformation Geometry SOLUTIONS & COMMENTS ( p2)
5.
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fig. 1
Showing the vector of translation,
and the line of reflection.
fig. C
Notice points move on non-parallel paths. ... and orientation is reversed (clockwise-> counterclockwise).
5.
fig. 1
Finding the center of
rotation is
trivial in the
case of 180E rotation.
1.
The angle of rotation is 180E
fig. D
Notice that points move on non-parallel paths,
but orientation is unchanged... what was clockwise, remains clockwise.
5.
fig. 1
fig. E
The scale factor is about b.
Figure E can be obtained from fig. 1 by a contraction of the plane (dilation in reverse), then
translating.
Don’t stress over the “contraction” terminology.
.
By the way, when we place figures 1 & E in a “perspective” arrangement, fig 1 is about 3/2 as far
away from the center as fig E. So the dilation factor from fig. E to fig. 1 is 3/2 (or 1.5 if you
prefer decimal form). Thus the contraction factor from fig. 1 to fig. E is the opposite ratio,
2/3.
6.
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V .
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. . .
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. . .
A“ü
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A''
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A'
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Translate the figure A (green)
by vector V, label that A'.
Then rotate A' 90E about P ;
label that A''.
Then reflect A'' through line l .
Label the final result A“.
P
Hints for getting this right:
For the translation, locate the new triangle vertices by counting squares. Since V points 3 “east”, 2 “north”, each vertiex
should move likewise... 3 squares right, 1 up.
For the rotation:
90E turns are easy to pinpoint on the grid.
Consider the lower left vertex of the triangle AN, which lies 1 square right and 3 squares above P.
When the plane rotates 90E around P, the new vertex will lie 1 square above and 3 squares left of P.
For the reflection:
Consider the top right vertex of triangle AO, which lies 3 squares right of reflection line l .
The new vertex is exactly opposite across the line l , and so lies 3 squares left of reflection line l .
A
7. 1
7a. .
Symmetries of a square– Four line symmetries
B
and Four rotations:
B
A
Rotates to...
C
C
D
D
D
B
A
C
90E
180E
D
A
270E
B
C
A
B
360E
D
C
7b. There are five line symmetries of the regular pentagon,
one from each vertex through the midpoint of the opposite side.
There are also five rotational symmetries of the regular pentagon;
the smallest is one-fifth of a rotation, 72E;
the other four are, of course, 144E, 216E, 288E and 360E.
The colored dots are not part of the pentagon
but have been placed so the rotation can be
seen. The 1st illustration is how it looks to start.
7c. The regular hexagon has six line, and six rotational, symmetries.
_
However, the figure given has no line symmetries and only 180E and 360E
rotational symmetries, because of the “flags” that have been added.
_
7d. About the individual letters in “S Y M M E T R I C” :
The letter S, as printed, is like the letters N and Z– although at first glance S may appear to have some line symmetry, it
has none. What we are noticing is the 180E rotational symmetry that all these letters (N, S, & Z) have. Of course there
is also the 360E rotation.
The letter I has two line symmetries (it can be flipped on the horizontal and vertical axes and land back on top of its
original footprint). This letter also has 180E and 360E rotational symmetries.
The letter R has no line nor rotational symmetry.
Y and M and T each have one line symmetry, about their vertical axes. No rotational symmetry.
E and C both have one line symmetry, about their horizontal axes. No rotational symmetry.
360E
72E
7e.
144E
216E
288E
f.
The regular octagon has eight line symmetries–
Four of type A, from midpoint to opposite midpoint.
No line symmetry
Four of type B from vertex to opposite vertex.
(As is the case with the square... one may generalize)
180E rotational symmetry
Type “A”
Type “B” In addition the regular octagon has eight rotational
(& 360E of course)
symmetries, rotating (about the center of course)
by 45E, 90E, 135E, 180E, 225E, 270E, 315E, 360E
7g.
Ribbon has one line symmetry,
About its vertical axis. .
No rotational symmetry
8. There are numerous ways to add squares (congruent to existing squares) to each of the following figures to make
the figure have a line symmetry, but no rotational symmetry.
a.
The given figure had line & rotational
symmetry. Squares added to destroy
the rotational symmetry.
b.
The given figure had rotational
symmetry, but not line symmetry.
Two added squares destroy the
Rotational sym. while adding line.
9. Add to the figure at right so the resulting figure
has rotational symmetry but no line symmetry.
Again, there are numerous ways to do this.
Two examples are shown for each figure.
These second alternatives assume the
figures were regular polygons.
c.
a.
The given figure already had the
required properties. The optional
added square preserves them.
b.
MATH 310 ¸SelfTest¹ Transformation Geometry SOLUTIONS & COMMENTS ( p2)
10. Given the figures at right are similar,
a. What is the scale factor from the left figure to the right?
b. Find x .
24
Rf7
30
10
8
X
a. Scale factor is 30/24 = 5/4 or 1.25
b. x/8 = 30/24
x = 8C 30/24 = 10
...Or you could just say x= 8C1.25
11. In a dilation of the plane with center C, the distances between
C and a figure and its image are as shown. What is the scale factor
of the transformation?
20 cm
30cm
C C
The distance between C and the top of fig. 2 is 50cm
The distance between C and the top of fig. 1 is 30cm
1
2
The ratio of any distance in fig. 2 to the corresponding distance in fig. 1 ...must be the same as the ratio of those
line segments: 5/3 (due to similar triangles). This constant ratio is the “scale factor” relating size of fig.2 to fig.1 .
A1. How many rotational symmetries has each figure? How many line symmetries?
Assuming the
first five figures
are regular
Rotational:
3
4
5
6
8
(smallest angle):
Lines :
120E
3
90E
72E
4
5
60E
6
see #7f
8
4
2
No smallest!
180E
4
A2. The rotated segment appears approximately as ANBN.
Use tracing paper to check that rotating around O really moves AB to A’B’.
You can draw angle AOAN and use a protractor to check the measure of the angle.
Notice AB was “leaning” on a 45E angle, and ANBN is vertical, but points are the same distance from O.
2
@O
AN
A @ BN
B@
A3. Yes. Yes. Yes. These questions merely restate question #2, but this time with hints as to the detail. Answers are
demonstrated by exercises C1 and M5 and C5 on pages TG-2 and TG-4.
A4. These questions are the logical extension of question #2, and the groundwork for justification of the answers is laid
in question #3.
Translations cannot be used to generate all rigid transformations, because no figure is ever turned (so no rotations) and
no figure is ever flipped (no clock-sense reversals) (so no reflections nor glide-reflections).
Rotations likewise cannot do the job, because, like translations, rotations never flip figures– never reverse the clocksense, thus it is impossible to generate any reflections or glide-reflections by any series of rotations.
Reflections were previously shown to do the job; glide-reflections can too.
A5. a. T
b. T
c. Rotation
d. GR
A6. a. Always a T (Never a R or M or GR) b. T perpendicular to the mirror lines, and twice the distance between.
c. R about the same center
d. T R M GR ( all !! ) e. T R (but never M or GR)
A7. Generally the order makes a difference (unless the centers are the same).
A8. a–d. T
e–g. R
h–i. M
j–k. GR