Lesson 1 Transformations

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Lesson 10-5
Transformations
Lesson 10-5: Transformations
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Types of Transformations
Reflections: These are like mirror images as seen across a line or
a point.
Translations ( or slides): This moves the figure to a new location
with no change to the looks of the figure.
Rotations: This turns the figure clockwise or
counter-clockwise but doesn’t change the
figure.
Dilations: This reduces or enlarges the figure to a
similar figure.
Lesson 10-5: Transformations
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Reflections
You can reflect a figure using a line or a point. All measures (lines
and angles) are preserved but in a mirror image.
Example:
The figure is reflected across line l .
You could fold the picture along line l and the
left figure would coincide with the
corresponding parts of right figure.
Lesson 10-5: Transformations
l
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Reflections – continued…
Reflection across the x-axis: the x values stay the same and the
y values change sign.
(x , y)  (x, -y)
Reflection across the y-axis: the y values stay the same and the
x values change sign.
(x , y)  (-x, y)
Example: In this figure, line l :
 reflects across the y axis to line n
(2, 1)  (-2, 1) & (5, 4)  (-5, 4)

n
l
reflects across the x axis to line m.
(2, 1)  (2, -1) & (5, 4)  (5, -4)
Lesson 10-5: Transformations
m
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Reflections across specific lines:
To reflect a figure across the line y = a or x = a, mark the
corresponding points equidistant from the line.
i.e. If a point is 2 units above the line its corresponding image point
must be 2 points below the line.
Example:
Reflect the fig. across the line y = 1.
(2, 3)  (2, -1).
(-3, 6)  (-3, -4)
(-6, 2)  (-6, 0)
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Lines of Symmetry

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If a line can be drawn through a figure so the one side of the
figure is a reflection of the other side, the line is called a “line of
symmetry.”
Some figures have 1 or more lines of symmetry.
Some have no lines of symmetry.
Four lines of symmetry
One line of symmetry
Two lines of symmetry
Infinite lines of symmetry
No lines of symmetry
Lesson 10-5: Transformations
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Translations (slides)
If a figure is simply moved to another location without change to
its shape or direction, it is called a translation (or slide).
 If a point is moved “a” units to the right and “b” units up, then
the translated point will be at (x + a, y + b).
 If a point is moved “a” units to the left and “b” units down, then
the translated point will be at (x - a, y - b).
Example:
A

Image A translates to image B by
moving to the right 3 units and down 8
units.
B
A (2, 5)  B (2+3, 5-8)  B (5, -3)
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Composite Reflections

If an image is reflected over a line and then that image is
reflected over a parallel line (called a composite reflection), it
results in a translation.
Example:
A
C
B
Image A reflects to image B, which then reflects to image C.
Image C is a translation of image A
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Rotations
 An image can be rotated about a fixed point.
 The blades of a fan rotate about a fixed point.
 An image can be rotated over two intersecting lines by
using composite reflections.
Image A reflects over line m to B, image B reflects over line n to
C. Image C is a rotation of image A.
A
C
m
B
Lesson 10-5: Transformations
n
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Rotations
It is a type of transformation where the object is rotated around a fixed
point called the point of rotation.
When a figure is rotated 90° counterclockwise about the origin,
switch each coordinate and multiply the first coordinate by -1.
(x, y) (-y, x)
Ex: (1,2) (-2,1) & (6,2)  (-2, 6)
When a figure is rotated 180° about the
origin, multiply both coordinates by -1.
(x, y) (-x, -y)
Ex: (1,2) (-1,-2) & (6,2)  (-6, -2)
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Angles of rotation

In a given rotation, where A is the figure and B is the resulting
figure after rotation, and X is the center of the rotation, the
measure of the angle of rotation AXB is twice the measure of
the angle formed by the intersecting lines of reflection.
Example: Given segment AB to be rotated over lines l and m,
which intersect to form a 35° angle. Find the rotation image
segment KR.
B
A
35 °
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Angles of Rotation . .
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Since the angle formed by the lines is 35°, the angle of rotation is
70°.
1. Draw AXK so that its measure is 70° and AX = XK.
2. Draw BXR to measure 70° and BX = XR.
3. Connect K to R to form the rotation image of segment AB.
B
K
A
R
35 °
X
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Dilations


A dilation is a transformation which changes the size of a figure
but not its shape. This is called a similarity transformation.
Since a dilation changes figures proportionately, it has a scale
factor k.
• If the absolute value of k is greater than 1, the dilation is an
enlargement.
• If the absolute value of k is between 0 and 1, the dilation is a
reduction.
• If the absolute value of k is equal to 0, the dilation is
congruence transformation. (No size change occurs.)
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Dilations – continued…

In the figure, the center is C. The distance from C to E is three times the distance
from C to A. The distance from C to F is three times the distance from C to B.
This shows a transformation of segment AB with center C and a scale factor of 3
to the enlarged segment EF.
E
A
R
A
C

B
W
C
F
B
In this figure, the distance from C to R is ½ the distance from C to A. The
distance from C to W is ½ the distance from C to B. This is a transformation of
segment AB with center C and a scale factor of ½ to the reduced segment RW.
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Dilations – examples…
Find the measure of the dilation image of segment AB, 6 units long,
with a scale factor of
1. S.F. = -4: the dilation image will be an enlargment since the
absolute value of the scale factor is greater than 1. The image will
be 24 units long.
2.
S.F. = 2/3: since the scale factor is between 0 and 1, the image
will be a reduction. The image will be 2/3 times 6 or 4 units long.
3.
S.F. = 1: since the scale factor is 1, this will be a congruence
transformation. The image will be the same length as the original
segment, 1 unit long.
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