Geometric Transformations Notes
Advanced Math
A translation is nothing more than a geometric transformation that slides each point in a figure the same distance in the same direction
In this translation, ∆CDE is being translated to the right by the same length as segment AB. What do you think is true about ∆CDE and ∆C'D'E'?
∆CDE ∆C'D'E'
Mar 1­9:23 AM
A figure can be translated in any direction and any distance. A car traveling down the road is a good example of a translation in action. The shape of the car is not being altered in any way, it is simply being moved from one point to another.
Translation Example.gsp
Mar 1­9:34 AM
1
Geometric Transformations Notes
Advanced Math
Translation means SLIDE
Aug 8­7:45 AM
Translation Rules
*To translate a figure a units to the right, increase the x­coordinate of each point a units.
*To translate a figure a units to the left, decrease the x­coordinate of each point a units.
*To translate a figure a units up, increase the y­
coordinate of each point a units.
*To translate a figure a units down, decrease the y­coordinate of each point a units.
Aug 13­8:58 AM
2
Geometric Transformations Notes
Advanced Math
A reflection is a transformation that flips a figure across a line to create it's image. Another way to think about it is that each point in a reflected image is the same distance away from the line of reflection as the original point was.
What is true about a figure and it's reflected image?
Mar 1­10:27 AM
Reflection Rule
* The reflection of the point (a, b) across the x­axis is the point (a, ­b)
*The reflection of the point (a, b) across the y­axis is the point (­a, b)
Aug 13­9:03 AM
3
Geometric Transformations Notes
Advanced Math
Rotations are probably the most difficult type of geometric transformation to understand.
A rotation is a transformation that turns, or spins, a figure around a point.
Mar 1­10:32 AM
A figure can be rotated around a point on the figure itself.
In this case the triangle was rotated multiple times around a center point that was also one vertex of the triangle.
Or a figure can be rotated around a point that is completely separate from the figure itself
In this case, the same triangle was rotated around a point separate from the triangle.
Mar 1­11:43 AM
4
Geometric Transformations Notes
Advanced Math
Rotations are generally measured by the angle of rotation.
This figure would represent a 90o rotation because the angle created by the corresponding vertices and the center of rotation is a 90o angle. 90o
This figure represents a 135o rotation for the same reason.
135o
Rotation Example.gsp
Mar 1­11:48 AM
Rotation Rules
1) The rotation of the point (x, y) 90 degrees clockwise about the origin, is the point (y, ­x).
2) The rotation of the point (x, y) 180 degrees clockwise about the origin is the point (­x, ­y).
3) The rotation of the point (x, y) 90 degrees counterclockwise about the origin is the point (­y, x)
Aug 13­8:50 AM
5
Geometric Transformations Notes
Advanced Math
Rotate Figure ABCD 90 degrees and 180 degrees
A
D
B
C
Aug 13­11:31 AM
Rotate the triangle 90 degrees and 180 degrees using the rule
Aug 13­11:47 AM
6
Geometric Transformations Notes
Advanced Math
Dilations are a type of transformation during which we grow or shrink a figure. We've actually already studied this type of transformation quite in depth? Does anyone remember when?
The majority of our Stretching and Shrinking unit was devoted to similar figures created by dilating a plane figure, so I'm not going to spend a lot of time talking about them.
Mar 1­11:54 AM
Dilations are usually measured by their scale factor.
Reminder: A scale factor that is greater than one creates a larger image.
Scale Factor = 3
A scale factor between 0 and 1 creates a smaller figure.
Scale Factor = 1/2
Mar 1­8:02 PM
7
Geometric Transformations Notes
Advanced Math
Dilation Rule
To dilate a figure with respect to the origin, multiply the coordinates of each of its points by the percent of dilation.
Aug 13­8:55 AM
There are four basic transformations that we will be concerned with this year. Talk to the people at your able and try to define each of these in your own words
Reflection
Translation
Rotation
Dilation
Mar 1­9:17 AM
8
Geometric Transformations Notes
Advanced Math
Recap
Reflection
Translation
A shift or a slide that creates a congruent image
Rotation
A flip that creates a congruent image.
Dilation
A spin or turn that creates a congruent image
Growing or shrinking a figure to create a similar figure. Mar 1­8:24 PM
In addition to recognizing and describing these 4 basic tranformations, there are also some rules that govern how translations behave on a coordinate grid. Consider quadrilateral CDEF below. What are the coordinates of the 4 vertices?
(3, 6)
What would happen to those coordinates if I translated the figure 7 units to the left?
(2, 4)
(4, 2)
(2, 1)
Mar 1­8:07 PM
9
Geometric Transformations Notes
Advanced Math
(3, 6)
(­4, 6)
(2, 4)
(­5, 4)
(­3, 2)
(­5, 1)
(4, 2)
(2, 1)
I would have to subtract 7 from the x­value of each coordinate pair!
Mar 1­8:17 PM
What kind of transformation are we dealing with?
­Translation, Reflection, Rotation, or Dilation
A
B
D
C
D'
A'
C'
B'
Aug 13­8:42 AM
10
Geometric Transformations Notes
Advanced Math
What kind of transformation are we dealing with?
­Translation, Reflection, Rotation, or Dilation
A
B
D
C
D'
C'
A'
B'
Aug 13­8:42 AM
What kind of transformation are we dealing with?
­Translation, Reflection, Rotation, or Dilation
A
B
D
C
A'
B'
D'
C'
Aug 13­8:42 AM
11
Geometric Transformations Notes
Advanced Math
What kind of transformation are we dealing with?
­Translation, Reflection, Rotation, or Dilation
A
A'
B
B'
D
D'
C
C'
Aug 13­8:42 AM
Do you know the rules?
Horizontal translation: (x, y) = (x ± a, y)
Vertical translation: (x, y) = (x, y ± a)
Diagonal translation: (x, y) = (x ± a, y ± a)
Reflection about y­axis: (x, y) = (­x, y)
Reflection about x­axis: (x, y) = (x, ­y)
Rotation 90 degrees clockwise: (x, y) = (y, ­x)
Rotation 180 degrees: (x, y) = (­x, ­y)
Rotation 90 degrees counterclockwise: (x, y) = (­y, x)
Dilation by .5: (x, y) = (x/2, y/2)
Dilation by 2: (x, y) = (x ∙ 2, y ∙ 2)
Aug 14­12:08 PM
12
Attachments
Translation Example.gsp
Rotation Example.gsp