2
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.
3.
3
Reflections
You could foldthe picture along line l and the
left figure would coincide with the
corresponding parts of right figure.
l
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 .
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Reflections – continued…
reflects across the y axis to line n
(2, 1) (-2, 1) & (5, 4) (-5, 4)
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)
-8 -6 -4 -2 2 4 6 8
-8
-6
-4
-2
2
4
6
8
x
y
Example: In this figure, line l :
reflects across the x axis to line m.
(2, 1) (2, -1) & (5, 4) (5, -4)
l
n
m
5.
5
Reflections across specificlines:
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.
(-3, 6) (-3, -4)
(-6, 2) (-6, 0)
(2, 3) (2, -1).
Example:
Reflect the fig. across the line y = 1.
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Lines of Symmetry
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.
One line of symmetry
Infinite lines of symmetry
Four lines of symmetry
Two lines of symmetry
No lines of symmetry
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Translations (slides)
Ifa 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).
A
B
Image A translates to image B by
moving to the right 3 units and down 8
units.
Example:
A (2, 5) B (2+3, 5-8) B (5, -3)
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Composite Reflections
Ifan 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.
A B C
Image A reflects to image B, which then reflects to image C.
Image C is a translation of image A
Example:
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9
Rotations
An image canbe 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
B
C
m
n
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Rotations
It is atype 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)
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)
Ex: (1,2) (-2,1) & (6,2) (-2, 6)
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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.
A
B
35 °
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Angles of Rotation. .
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.
A
B
35 °
X
K
R
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Dilations
A dilationis 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.
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.
C
E
A
F
B
C
R
A
B
W
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Dilations – examples…
Findthe 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.
