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Lesson 1 Transformations

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Title: Lesson 1 Transformations Author: Authorized User Created Date: 7/13/2006 2:55:48 PM Document presentation format: On-screen Show (4:3) Company – PowerPoint PPT presentation

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Title: Lesson 1 Transformations


1
Lesson 10-5
Transformations
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)
8
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)
11
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.

12
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.)

14
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.
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