Title: Lesson 1 Transformations
1Lesson 10-5
Transformations
2Types 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.
3Reflections
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.
4Reflections 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
5Reflections 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)
6Lines 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
7Translations (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)
8Composite 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
9Rotations
- 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.
10Rotations
- 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)
11Angles 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.
12Angles 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.
13Dilations
- 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.)
14Dilations 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.
15Dilations 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.