Unit 1: Transformations, Congruence, and Similarity

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Unit 1Transformations, Congruence, and
Similarity
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Basic Types of Transformations

• Translations

Reflections
Rotations
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Quadrant I
Quadrant II
(x,y)
x-axis
Quadrant III
Quadrant IV
y-axis
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Object to Image(Before)
(After)

• Before Transformation

• After Transformation
• ( PRIME)

A
A
C
C
B
B
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Translations
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A translation "slides" an object a fixed
distancein a given direction.  The original
object and its translation have the same shape
and size, and they face in the same direction.
Objects that are translated are congruent.The
word "translate" in Latin means "carried across".
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Example 1 Translate the object down 2 and right
3 units.
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Example 1 Solution Translate the object down 2
and right 3 units.
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Example 2 Translate the object (-3, 4)
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Example 2 Solution Translate the object (-3, 4)
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• Remember
• Translations are SLIDING on a graph!!! The
  shape doesnt change at all.

Translations are SLIDES!!! Translations are SLIDES!!! Translations are SLIDES!!!
                                                                                                             
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• Reflections

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A reflection flips an object and can be seen in
water, in a mirror, in glass, or in a shiny
surface.  An object and its reflection have the
same shape and size, but the figures face in
opposite directions.  In a mirror, for example,
right and left are switched.
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The line (where a mirror may be placed) is called
the line of reflection.  The distance from a
point to the line of reflection is the same as
the distance from the point's image to the line
of reflection. A reflection can be thought of as
a "flipping" of an object over the line of
reflection.

The object ABCD is being reflected over the
x-axis.
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Example 3 Reflect the object over the y-axis.
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Example 3 Solution Reflect the object over the
y-axis.
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Example 4 Reflect the object over x 2.
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Example 4 Solution Reflect the object over x 2.
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• Rotations

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A rotation is a transformation that turns a
figure about a fixed point called the center of
rotation.  An object and its rotation are the
same shape and size, but the figures may be
turned in different directions.
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Rotations Graphically

• Physically rotate the graph paper and use the
  original points just graphed in the different
  quadrants.

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Rules

• When students complete rotating the original
  figure clockwise 90, 180, and 270, you can
  have them come up with the rules on their own.
• If you have time, you can have them use the same
  figure and rotate counter clockwise and come up
  with the rules of those too.

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Rotation Rules
Clockwise Counter Clockwise
90 (y, -x) (-y, x)
180 (-x, -y) (-x, -y)
270 (-y, x) (y, -x)
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Dilations
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What is a Dilation?

• Dilation is a transformation that produces a
  figure similar to the original by proportionally
  shrinking or stretching the figure.

Dilated PowerPoint Slide
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Proportionally
Lets take a look
And, of course, increasing the circle increases
the diameter.

• When a figure is dilated, it must be
  proportionally larger or smaller than the
  original.

So, we always have a circle with a certain
diameter. We are just changing the size or scale.
We have a circle with a certain diameter.
Decreasing the size of the circle decreases the
diameter.

• Same shape, Different scale.

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Which of these are dilations??
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Scale Factor and Center of Dilation

• When we describe dilations we use the terms scale
  factor and center of dilation.
• Scale factor
• Center of Dilation

Here we have Igor. He is 3 feet tall and the
greatest width across his body is 2 feet.
He wishes he were 6 feet tall with a width of 4
feet.
His center of dilation would be where the length
and greatest width of his body intersect.
He wishes he were larger by a scale factor of 2.
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The Object and the Image

• The original figure is called the object and the
  new figure is called the image.
• The object is labeled with letters.
• The image may be labeled with the same letters
  followed by the prime symbol.

Image
Object
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Determining Scale Factor

•  

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Scale Factor

• If the scale factor is larger than 1, the figure
  is enlarged.
• If the scale factor is between 1 and 0, the
  figure is reduced in size.

Scale factor gt 1
0 lt Scale Factor lt 1
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Ratio Fraction Decimal Percentage Reduction or Enlargement
 12 1/2  .5 50 Reduction
  3/4      
    0.9    
      400  
25        
  1/8      
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Are the following enlarged or reduced??
C
A
Scale factor of 1.5
D
Scale factor of 3
B
Scale factor of 0.75
Scale factor of 1/5
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Dilations Used Everyday
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Remember

• Dilations are enlargements or reductions.
• What are some things that you would not
• mind dilating to make larger or smaller?
• Practice Dilation Quiz

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Dilation

• A transformation that changes the size of an
  object, but not the shape.
• A Dilation will be a similar figure, but not a
  congruent figure.
• Example

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Dilate the object by a scale factor of ½
(2,2)
(-2,2)
(2,-2)
(-2,-2)
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Dilate the object by a scale factor of 3
(-6,6)
(6,6)
(2,2)
(-2,2)
(2,-2)
(-2,-2)
(6,-6)
(-6,-6)
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A spider has taken up residence in a small
cardboard box which measures 2 inches by 4 inches
by 4 inches. What is the length, in inches, of a
straight spider web that will carry the spider
from the lower right front corner of the box to
the upper left back corner of the box? A. 4.47
in. B. 5.66 in. C. 5 in. D. 6 in.
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