A transformation is a change that you make to a shape or an object in a specific way.
You specify very precisely exactly what the change will be.
We use the word object to refer to the original shape that you started with, and the word
image to refer to the new shape you get by applying the transformation.
The object is the original shape, the image is the result.
There are two basic kinds of transformations that we’ll look at in this video: Congruent
(or isometric) transformations that preserve the angles and lengths of the shape, and size-changing
transformations that preserve angles but change all the lengths in the same ratio.
Remember, the word congruent means exactly the same.
Isometric just means “same size”.
The congruent transformations are translation, which is a move or a slide, rotation and reflection.
We’ll look at each of these in turn in a moment.
And changing the size by a specified factor is called enlargement or reduction.
Translation means to move or slide an object.
You specify the move either by a distance up or down and a distance left or right, or
you can specify the distance to go in a particular direction.
For example, suppose I asked you to translate this shape down two squares and right three.
You would simply do exactly that: Take the shape down two squares, and then across three
to the right.
The image would look exactly the same as the original (remember this is an isometric transformation,
the image will be congruent to the original object), but every point of the image will
be two down and three right of where it was before.
If instead I ask you to translate the shape a distance of seven squares in the direction
east-northeast, then it’s a little bit trickier.
First, find where the distance of seven takes you.
That’ll put it somewhere on a circle of radius seven.
Now all we have to do is get the direction correct: it’s half-way between east and
north-east, at twenty-two-and-a-half degrees.
The image is congruent to the original object, but every point of the image is now seven
squares distance away east-northeast of where it started.
A rotation is a turn of a specified angle about a specific point called the centre of
rotation.
The centre of rotation must always be specified.
Every point on the shape will remain at the same fixed distance from the centre of rotation.
For example, let’s rotate this shape around the marked centre of rotation at the bottom
left of the shape.
We’ll rotate it ninety degrees clockwise.
(Sorry, even if you prefer digital watches, you still need to know which way clockwise
or anticlockwise is.)
The easiest way to do this is actually to cut the shape out of another piece of paper.
Then you can literally hold the bottom left corner still and rotate the cut-out shape
around it.
We want a quarter turn clockwise.
Did you notice as we rotated it that each part of the shape moved in a circle around
the centre of rotation?
Every point in the image is the same distance from the centre as it was in the object.
But the centre of rotation doesn’t have to be part of the object.
Here’s an example where the centre of rotation is off to the right.
We have to rotate it anticlockwise ninety degrees.
This is a little trickier, but again let’s get our cut-out shape and rotate it around.
Can you see the circles each point on the shape is following?
Where are they centred?
It may help you, if you’re cutting the shape out, to include the centre of rotation in
the piece you cut.
Then you can hold it still there as you rotate.
And of course the angle of rotation need not be ninety degrees.
This one says to rotate forty-five degrees clockwise.
Getting the distances and positions right can be a bit harder.
I really recommend you cut the shape out and literally pull it around.
It really will help you.
A reflection is a flip across a specified line.
Every point in the shape will remain the same distance from that line, but on the other
side.
When the line is vertical or horizontal, it’s usually fairly easy to do.
The image is inverted, but we still say that it’s congruent to the object: It’s flipped
over, but the angles and lengths are all still the same and are still in the same relationships
with each other.
A flipped-over shape, an inverted shape, is still said to be congruent.
But if the reflection line is at a different angle, it can be a bit trickier.
What you have to do is use the definition of the reflection to work out where the image
is.
Each point is the same distance from the line, but on the other side.
Any part of the shape on the line doesn’t move, but every other point gets moved directly
across the line, to the same distance it was on its original side.
Even points from the other side get moved across the line, to the same distance they
were.
See?
And the resultant image will be congruent to the object, but inverted.
Again, if you prefer, you can cut the shape out and literally flip it over.
Finally, transformations that change the size of the shape.
This is variously called enlargement, reduction, dilation or magnification.
Generally enlargement is for making the shape bigger with a scale factor greater than one,
and reduction is for making it smaller with a scale factor less than one.
However the terms are more-or-less interchangeable.
Each point in the shape has its distance from the centre of enlargement changed by the same
factor, called the scale factor.
The word “factor” should remind you to multiply the length by the scale.
The angles in the shape will stay the same, but the size will change: every side’s length
will be bigger or smaller by the same scale factor.
This one is difficult to do with a cut-out, because you can’t really stretch the paper.
So instead, you’ll have to work out the points separately and then join them up.
Take each point, and in this case double each distance away from the centre of enlargement.
See how it works?
The trickier ones just do carefully: double the vertical distance, and double the horizontal
distance.
And there’s the image.
It’s twice the size, but the angles and relationships haven’t changed.
We say the image is similar to the object: the same angles, but not the same lengths.
It’s similar, but not congruent.
Let’s try a reduction, where the scale factor is smaller than one.
Here, every point’s distance from the centre has to be reduced to one-third of what it
was.
Lightly draw in the lines from the centre to each point and measure one-third of the
distance.
Then connect them up, and there’s the image.
Exactly one-third the size of the original.
And of course the centre of enlargement doesn’t have to be part of the shape, just like when
we did rotation.
But the process is exactly the same.
Move each point one-and-a-half times the distance it currently is from the centre of enlargement.
And there’s the image.