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Transcript
Transformations
The three main Transformations are:
Rotation
Turn!
Reflection
Flip!
Translation
Slide!
After any of those transformations (turn, flip or slide), the shape still has the
same size, area, angles and line lengths.
If one shape can become another using Turns, Flips and/or Slides, then the two shapes are called
Congruent.
Dilation, contraction (scaling) (Resizing)
The other important Transformation is Resizing (also called dilation, contraction, compression,
enlargement or even expansion). The shape becomes bigger or smaller:
Resizing
If you have to Resize to make one shape become another then the shapes are not Congruent, but
they are Similar. These transformations keeps angles the same but distances change,
Congruent or Similar
So, if one shape can become another using transformation, the two shapes might be Congruent or
just Similar
If you ...
... only Rotate, Reflect and/or
Translate
... need to Resize
Then the shapes are ...
Congruent
Similar
Rotation
"Rotation" means turning around a
center:
The distance from the center to any point on the shape stays
the same.
Every point makes a circle around the center.
Here a triangle is rotated around
the point marked with a "+"
Try It Yourself
You can try rotating objects (point-by-point) by any angle, around any center point here:
Reflection
Reflections are everywhere ... in mirrors, glass, and here in a lake.
... what do you notice ?
Every point is the same distance from the central line !
... and ...
The reflection has the same size as the original image
The central line is called the Mirror Line ...
Can A Mirror Line Be Vertical?
Yes.
Here my dog "Flame" shows a
Vertical Mirror Line (with a bit of photo magic)
In fact Mirror Lines can be in any direction.
Imagine turning the photo at the top in different directions ...
... the reflected image is always the same size, it just faces the other way:
A reflection is a flip over a line
You can try reflecting some shapes about
different mirror lines here:
How Do I Do It Myself?
Just approach it step-by-step. For each corner of the shape:
1. Measure from the point to 2. Measure the same distance
the mirror line (must hit the again on the other side and
mirror line at a right angle) place a dot.
3. Then connect the
new dots up!
Labels
It is common to label each corner with letters,
and to use a little dash (called a Prime) to
mark each corner of the reflected image.
Here the original is ABC and the reflected
image is A'B'C'
Some Tricks
X-Axis
If the mirror line is the x-axis, just change each
(x,y) into (x,-y)
Y-Axis
If the mirror line is the y-axis, just
change each (x,y) into (-x,y)
Fold the Paper
And if all else fails, just fold your sheet of paper along the mirror line and
then hold it up to the light !
Translation
In Geometry, "Translation" simply means Moving ...
... without rotating, resizing or anything else, just moving.
To Translate a shape:
Every point of the shape must move:


the same distance
in the same direction.
Writing it Down
Sometimes we just want to write down the translation, without showing it on a graph.
Example: if we want to say that the shape gets
moved 30 Units in the "X" direction, and 40 Units in
the "Y" direction, we can write:
This says "all the x and y coordinates will become
x+30 and y+40"
Congruent
If one shape can become another using Turns, Flips and/or Slides, then the
two shapes are called Congruent:
Rotation
Turn!
Reflection
Flip!
Translation
Slide!
After any of those transformations (turn, flip or slide), the shape still has the
same size, area, angles and line lengths.
Examples
These shapes are all Congruent:
Rotated
Reflected and Moved
Reflected and Rotated
Congruent or Similar?
The two shapes need to be the same size to be congruent.
When you need to resize one shape to make it the same as the other, the shapes are called
Similar.
If you ...
Then the shapes are ...
... only Rotate, Reflect and/or Translate
Congruent
Similar
... also need to Resize
Congruent? Why such a funny word that basically means "equal"? Probably because they
would only be "equal" if laid on top of each other. Anyway it comes from Latin congruere, "to
agree". So the shapes "agree"
Resizing
When you resize a shape it gets bigger or smaller.
... but it still looks similar:


all angles stay the same
relative sizes are the same (for example
the face and body are still in proportion)
Note: here we call it resizing, but other people call it dilation, contraction,
compression, enlargement or even expansion! Same idea, just different names.)
To resize, just do this for every corner:



draw a line from the central point to the corner
increase (or decrease) the length of that line
put a dot at the new point
Then just connect the dots for the resized shape!
Similar
Two shapes are Similar if the only difference is size (and possibly the
need to turn or flip one around).
Resizing is the Key
If one shape can become another using Resizing (also called dilation, contraction, compression,
enlargement or even expansion), then the shapes are Similar:
These Shapes are Similar!
There may be Turns, Flips or Slides, Too!
Sometimes it can be hard to see if two shapes are Similar, because you may need to turn, flip or
slide one shape as well as resizing it.
Rotation
Turn!
Reflection
Flip!
Translation
Slide!
Examples
These shapes are all Similar:
Resized
Resized and Reflected
Why is it Useful?
When two shapes are similar, then:


corresponding angles are equal, and
the lines are in proportion.
Resized and Rotated
This can make life a lot easier when solving geometry puzzles, as in this example:
Example: What is the missing length here?
Notice that the red triangle has the
same angles as the main triangle ...
... they both have one right angle,
and a shared angle in the left corner
In fact you could flip over the red triangle, rotate it a little, then resize it and it would fit
exactly on top of the main triangle. So they are similar triangles.
So the line lengths will be in proportion, and we can calculate:
? = 80 × (130/127) = 81.9
(No fancy calculations, just common sense!)
Congruent or Similar?
But when you don't need to resize to make the shapes the same, they are called Congruent.
So, if the shapes become the same:
When you ...
... only Rotate, Reflect and/or
Translate
... also need to Resize
Then the shapes are ...
Congruent
Similar