Lesson 1 Transformations

1
Transformations Reflections, Translations and
Rotations
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 doesnt change the figure.

• Dilations This reduces or enlarges the figure
  to a similar figure.

3
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 .
l

• You could fold the picture along line l and the
  left figure would coincide with the corresponding
  parts of right figure.

4
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
l
n

• reflects across the y axis to line n
• (2, 1) ? (-2, 1) (5, 4) ? (-5, 4)

• reflects across the x axis to line m.
• (2, 1) ? (2, -1) (5, 4) ? (5, -4)

m
5
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)
6
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.

Four lines of symmetry
One line of symmetry
Two lines of symmetry
Infinite lines of symmetry
No lines of symmetry
7
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).

A
Example
Image A translates to image B by moving to the
right 3 units and down 8 units.
B
A (2, 5) ? B (23, 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
C
B
A
Image A reflects to image B, which then reflects
to image C. Image C is a translation of image A
9
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.
10
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.

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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.

13
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.
• 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.

15
Dilations examples

• Find the measure of the dilation image of segment
  AB, 6 units long, with a scale factor of
• 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.
• 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.
• 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.