Unit 12 Transformations
•This unit addresses transformations in the
Coordinate Plane.
•It includes transformations, translations,
dilations, reflections, rotations, symmetry
(including line/plane and rotational),
compositions of reflections in parallel lines,
glide reflections, and the fundamental
theorem of Isometries.
•It also includes tessellations, symmetry in
tessellations, and proportions of dilations.
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Standards
SPI’s taught in this unit:
SPI 3108.1.1 Give precise mathematical descriptions or definitions of geometric shapes in the plane and space.
SPI 3108.1.3 Use geometric understanding and spatial visualization of geometric solids to solve problems and/or create drawings.
SPI 3108.2.2 Perform operations on vectors in various representations.
SPI 3108.3.1 Use algebra and coordinate geometry to analyze and solve problems about geometric figures (including circles).
SPI 3108.3.2 Use coordinate geometry to prove characteristics of polygonal figures.
SPI 3108.3.3 Describe algebraically the effect of a single transformation (reflections in the x- or y-axis, rotations, translations, and dilations) on twodimensional geometric shapes in the coordinate plane.
SPI 3108.4.10 Identify, describe, and/or apply transformations on two and three dimensional geometric shapes.
CLE (Course Level Expectations) found in Unit 12:
CLE 3108.1.1 Use mathematical language, symbols, definitions, proofs and counterexamples correctly and precisely in mathematical reasoning.
CLE 3108.1.4 Move flexibly between multiple representations (contextual, physical written, verbal, iconic/pictorial, graphical, tabular, and symbolic), to
solve problems, to model mathematical ideas, and to communicate solution strategies.
CLE 3108.1.5 Recognize and use mathematical ideas and processes that arise in different settings, with an emphasis on formulating a problem in
mathematical terms, interpreting the solutions, mathematical ideas, and communication of solution strategies.
CLE3108.2.2 Explore vectors as a numeric system, focusing on graphic representations and the properties of the operation.
CLE 3108.4.7 Apply the major concepts of transformation geometry to analyzing geometric objects and symmetry.
CFU (Checks for Understanding) applied to Unit 12:
3108.1.5 Use technology, hands-on activities, and manipulatives to develop the language and the concepts of geometry, including specialized vocabulary
(e.g. graphing calculators, interactive geometry software such as Geometer’s Sketchpad and Cabri, algebra tiles, pattern blocks, tessellation tiles, MIRAs,
mirrors, spinners, geoboards, conic section models, volume demonstration kits, Polyhedrons, measurement tools, compasses, PentaBlocks, pentominoes,
cubes, tangrams).
3108.1.10 Use visualization, spatial reasoning, and geometric modeling to solve problems.
3108.1.11 Identify and sketch solids formed by revolving two-dimensional figures around lines.
3108.2.4 Add vectors graphically and algebraically.
3108.2.5 Multiply a vector by a scalar graphically and algebraically.
3108.3.5 Use mapping notation to identify the image of a transformation given the coordinates of the pre-image.
3108.3.6 Identify a transformation given its mapping notation.
3108.4.13 Locate, describe, and draw a locus in a plane or space (e.g., fixed distance from a point on a plane, fixed distance from a point in space, fixed
distance from a line, equidistant from two points, equidistant from two parallel lines, and equidistant from two intersecting lines).
3108.4.29 Extend the effect of a scale factor k in similar objects to include the impact on volume calculations and transformations
3108.4.31 Use properties of single transformations and compositions of transformations to determine their effect on geometric figures (e.g. reflections
across lines of symmetry, rotations, translations, glide reflections, and dilations).
3108.4.32 Recognize, identify and apply types of symmetries (point, line, rotational) of two- and three- dimensional figures.
3108.4.33 Use transformations to create and analyze tessellations and investigate the use of tessellations in architecture, mosaics, and artwork.
3108.4.34 Create and analyze geometric designs using rigid motions (compositions of reflections, translations, and rotations).
Definitions
• Transformations -A change in a geometric
figure’s position, shape, or size.
• Preimage -This is the original figure before it
is transformed
• Image -This is the resulting image after it is
transformed
• Isometry -A transformation in which the
preimage and the image are congruent.
Examples of transformations which would
result in isometry would include: flips, slides,
and turns
Examples
• Flip -
• Slide • Turn -
Transformation Maps
• A transformation maps a figure onto its
image and may be described with an
arrow notation:
• Prime notation: (‘) is sometimes used to
identify image points. In this diagram, K’
is the image of K (K
K’)
K’
K
J
Q
Q’
J’
Reflections
• A reflection -or flip- is an isometry in which a
figure (preimage) and its image have
opposite orientations. Thus, a reflected image
in a mirror appears backwards.
• Here, triangle ABC is reflected along a line to
produce triangle A’B’C’. Since the reflection is
isometry, triangle ABC is congruent to triangle
A’B’C’
A
A’
B
C
C’
B’
Properties of Reflections
• Below is a reflection in line r, where the
following properties are true:
Line of reflection
B
r
A=A’
B’
• If a point A is on line r, then the image of A (remember,
the “after transformation” creates the image) is A itself that is, A = A’ In other words, the reflection of a point
is the same point
• If a point B is not on line r, then line r is the
perpendicular bisector of the line segment B/B’
Quiz (50 Points)
1.
2.
3.
4.
5.
6.
7.
8.
What is a Transformation?
What is a Pre-Image?
What is an Image?
What is Isometry?
Give an example of Isometry
What is the name for this symbology? __’__
What is a reflection?
What is one property concerning reflections and
points?
9. What is the second property concerning reflections and
points?
10. Are a Flip and a Slide the same type of
Transformation?
In Class Assignment
• Text, page 636-637 1-11
• You may hand draw # 10 and #11, as
long as the images are correct, and all
points are labeled properly
• Workbook, page 144 # 2-8
• 20 Points
Translation
• A translation (or slide) is an isometry
that maps all points of a figure the same
distance in the same direction. Thus
you can use a vector to describe a
translation.
• Mr Bass’ definition of a vector (for
translations): an ordered pair (x,y) that
tells me where to go from where I am
Translation Example
• What if you had a point A, at coordinate (-3,-1), and
you were told to use the vector (6,6) to find the image
of A (we’ll call it A’). What is the coordinate of A’?
•Use the vector in the same manner you use slope. So
you would go right 6, and up 6 (rise over run).
•The new coordinates would thus be (3,5)
A’
(3,5)
A(-3,-1)
Another way to work with
Translations
• You may see a pre-image, and an image, and be
asked to identify the vector. Again, this is no different
that determining slope.
• From A (-3,3) to A’ (3,5) you must go right 6, and up
2, so the vector would be (6,2)
A’
A
(3,5)
(-3,3)
Next Assignment
• Page 643, 1-21
Rotations
• A rotation occurs around a point -like the
center of a circle. These hold true:
– You need to know the center of the rotation -we’ll
call this a point such as r. After the rotation (the
image after the pre-image) this point gets
renamed -such as r’
– You need to know the angle of rotation -this will be
a positive number of degrees– You need to know whether the rotation is
clockwise or counterclockwise
– Unless stated otherwise, rotations in the book are
counterclockwise
Rotations Continued
• Having this information a rotation is
conducted by:
• Rotating x degrees about a point (such as r)
where a transformation occurs such that:
• The image of point r is itself (that is, r = r’)
• For any point (such as V), rV’ = rV (the distance
of each segment) and the measure of VrV’ = x
degrees
V’
r and r’
X degrees
V
Drawing a Rotation
• Example: Draw the image (after picture)
of triangle LOB for a 100 degree
rotation about C.
• Mark a point (i.e.) C. Draw a line from C
to a point on triangle LOB (i.e. O).
• Measure 100 degrees on the protractor
from that line, and draw another line
Drawing a Rotation Continued
• Use a compass to measure the distance
from C to the point on the line.
• Use the compass to mark the same
distance on the second line
• Do the same for each succeeding
corner (mark a 100 degree angle and
mark the distance)
• It looks like this….
Draw a Rotation :)
C
100 degrees
B’
O
O’
L
B
L’
Assignment
• Page 650, 10 - 16
• Page 651 35, 37, 38
Compositions of Reflections
• Theorem 12.1 - A translation or rotation
is a composition of two reflections
• Theorem 12.2 - A composition of
reflections in two parallel lines is a
translation
• Theorem 12.3 - A composition of
reflections in two intersecting lines is a
rotation
Composition of Reflections in
Parallel Lines
• This image is reflected on line l. Then it
is reflected on line m. The result is a a
translation (slide).
l
m
Composition of Reflections in
Intersecting Lines
• This image is reflected in line A, then
line B. The result is a rotation.
a
b
c
Fundamental Theorem of
Isometries
•
•
•
Theorem 12.4 - In a plane, one of two
congruent figures can be mapped onto the
other by a composition of at most three
reflections.
Isometry Classification Theorem:
Theorem 12.5 - There are only four
Isometries. They are:
1.
2.
3.
4.
Reflection
Translation (slide)
Rotation
Glide Reflection
Example of a Glide Reflection
• Remember -First you glide, then reflect
– Find the image of a triangle TEX for a glide
reflection where the glide vector is (0,-5)
and the reflection line is x = 0.
T
X
E
First, use the vector, and move (glide) down -5.
Then, create a reflection based on the x = 0 line.
(Numbers have been omitted for clarity of drawing).
T’
X’
E’
Homework
• Page 657 1-9 ( on 4-9, just draw the
images, don’t describe them).
Symmetry
• A figure has symmetry (is symmetrical) if
there is an isometry that maps the figure onto
itself.
• If the isometry is the reflection of a plane
figure, the figure has reflectional symmetry or
line symmetry.
• This is something most people already know
(the idea of symmetry), they just haven’t
applied it to geometry.
– For example, a face has symmetry, if you draw a
line of reflection down the middle -through the
nose, chin and so on
Symmetry
• A figure can have more than one line of
symmetry.
• Consider a hexagon:
How many lines of
symmetry are there in a
hexagon?
It looks like a lot, but
there are actually 6
Symmetry
• How many lines of symmetry are there
in this rectangle?
There are only 2.
Why aren’t there any
for the corners?
Rotational Symmetry
• Rotational Symmetry: Where a figure is it’s own
image (after transformation) for some rotation of 180
degrees or less. For example, this equilateral
triangle. The angle of rotation is 120 degrees. -if you
rotate the triangle 120 degrees, you get the exact
same image (isometry).
120 degrees
Rotational Symmetry
• Would a square have rotational symmetry? If so,
what would the angle of rotation be?
• Would a rectangle have rotational symmetry? If so,
what is the angle of rotation?
• Would a hexagon have rotational symmetry? What
would the angle of rotation be?
• Does this figure have rotational symmetry? If so,
what is the angle of rotation?
Point Symmetry
• A figure that has point symmetry has 180 degree
rotational symmetry.
• Thinking back to the examples given (Square,
Rectangle, Equilateral Triangle, unusual polygon),
which had Point symmetry?
• NOTE: 3 dimensional objects can have various types
of symmetry, including rotational symmetry about a
line, and reflectional symmetry in a plane.
• A pencil could have rotational symmetry about a line.
• A house reflected in a lake of water could have
reflectional symmetry in a plane.
3 Dimensional Examples
• Imagine a ping pong paddle.
– Does it have rotational symmetry?
– Does it have reflectional symmetry?
• How about a coffee mug?
• How about an umbrella?
Classwork
• Page 664 1- 19
Tessellations
• Tessellation (or Tiling) occurs when you
repeat a pattern of figures that
completely covers a plane -without gaps
or overlaps.
• These tessellations can be created with
translations, rotations, or reflections.
• You see it in art, nature (honeycombs),
and everyday life (tiled floors).
Example
• Identify the transformation, and the
repeating figure
• It is a translation (slide)
• The figure is a hexagon
Determining Figures that
Tessellate
• Because the figures in a tessellation do not
overlap or leave gaps, the sum of the
measures of the angles around any vertex
must be 360 degrees. If the angles around a
vertex are all congruent (they are) then the
measure of each angle must be a factor of
360.
• To determine whether a regular n-gon will
tessellate, you must calculate one angle
around it’s vertex.
• Then you determine whether the angle is a
factor of 360.
Determine the measure of one
angle to determine Tessellation
• A = (180(n-2))/n
• For example, determine one angle for an 18gon
• A = (180(18-2))/18
• A = 160
• Is 160 a factor of 360 degrees? No. Therefore
there is no pattern possible where you can
tessellate an 18-gon, and have no overlaps or
gaps
Examples
• What about an equilateral triangle?
• A = (180(3-2))/3 ---or 60. Thus, one
interior angle of an equilateral triangle is
60 degrees.
• 60 is a factor of 360, so equilateral
triangles do in fact tessellate.
Theorems and Ideas
• A figure does not have to be a regular
polygon to tessellate. It should be noted,
however, that the method we used to
determine tessellation only works on regular
polygons
• Theorem 12.6 - Every Triangle (whether it is
equilateral or not) tessellates
• Theorem 12.7 - Every quadrilateral
tessellates (whether they are regular or not -it
just has to be a 4 sided figure)
Symmetries in Tessellations
• You will often find symmetries in
Tessellations.
• These would include
– Line Symmetry
– Rotational Symmetry
– Glide Reflectional Symmetry
– Translational Symmetry
Classwork
• Page 670 1-16
• Page 672 40-44
Dilations
• A dilation is a transformation whose preimage and image are similar.
• This is the geometric definition of similar
– Angles must have the same measure
– Sides must be proportional
– Therefore a dilation is a similarity
transformation, but not, in general, an
isometry
Properties of Dilations
• Every dilation has a center, and a scale
factor n, where n > 0. This scale factor
describes the size change from the
original figure to the image.
• Generally, a dilation with a center (C for
example), and a scale factor n is a
transformation where the following hold
true:
Properties of Dilations
• The image (after picture) of C (the center) is itself.
That is C = C’
• For any point R, R’ is on ray CR and CR’ = n x CR.
n x CR
C = C’
R
CR’ = 25
R’
• In this example, if n = 2.5, and CR = 10, what is CR’?
• The dilation is an enlargement if the scale factor is
greater than 1. It is a reduction if the scale factor is
between 0 and 1.
• Is this example an enlargement or reduction?
Finding a Scale Factor
• Here, the larger triangle is a dilation of the
smaller one
• The center is X. Because the image is larger
than the pre-image, it is an enlargement
T’
• X’T’/XT = (4+8)/4 = 3
8
T
4
R’
R
• Therefore, the dilation has center X and a
scale factor of 3.
X=X’
Classwork
• Page 676 1-14