TRANSFORMATIONS:DILATIONS
OBJECTIVE
A few examples to ponder
So far we’ve only been working with triangles, but what about other shapes?
What if we would like to translate a square?
How exactly would we go about doing this?
So, let’s say we are given a square
So, from looking at our square, we have all four points at:
(0,0), (0,4), (4,4), (4,0)
Now let’s say we want to translate this square down 3 units, and to the left 6 units.
What would the square’s points be then?
Well, again, going down 3 units would mean we need to subtract 3 from all of the y coordinates of the square, which would leave us with: (0,-3),(0,1),(4,1),(4,-3)
Then, to go left 6 units would mean we need to subtract 6 from all of the x coordinates of the square, which would leave us with: (-6,-3), (-6, 1), (-2, 1), (-2, -3)
And here is our new square!
Last Example
Let’s say this time we’re working with a pentagon.
Now let’s say we want to translate this pentagon up 4 units, and to the right 5 units.
How do we do it?
Well let’s see out points:(0,0), (0,2), (1,3),(2,2), (2,0)
So how do we translate the figure, using the points, 4 units up and 5 units to the right?
Well, we know to go up we need to add, so to each y coordinate we add 4 points.
So now we have: (0,4), (0,6), (1,7), (2,6), (2,4)
And we know we need to add 5 to all of the x coordinates.
So now we have: (5,4), (5,6), (6,7), (7,6), (7,4)
And here is our new pentagon!
Quick review of Rotation examples
Remember, to rotate a shape around its own point, you need to pick a point on the shape, and then rotate it.
Sounds pretty straight forward right?
Here’s the example from yesterday
ROTATING A SHAPE FROM ITS OWN POINT
So let’s take a look at this triangle that we have here.
We have a point at the tip of the triangle (4, 4) and we would like to rotate the triangle from that point.
As we can see, this point is on the triangle, so it should be no problem right?
Let’s start off by rotating the triangle by 90 degrees.
So as you can see, the triangle has been rotated by 90 degrees by that same point.
So how did we do it?
Think of it like sticking a pin on the point, and rotating it 90 degrees.
Like this!
Now let’s try rotating that same triangle another 90 degrees.
Again, think about putting a pin in the point and spinning it 90 degrees again. �So, as we can see the blue triangle is 90 degrees from the red triangle, and the green triangle is 180 degrees from the red triangle.
ROTATING ABOUT A POINT AWAY FROM THE OBJECT
Again, you can always rotate the object from a point away from the object.
Here’s the example from yesterday:
Other types of rotation
You can also rotate an object by a point away from the object, however this type of rotation is a little more advanced.
Here is an example of this type or rotation.
Let’s say we have a triangle, and a point away from that triangle.
Now, let’s say we want to rotate that triangle around that point by say, 45 degrees?
Here’s what it would look like:
Now, let’s keep it going by rotating 45 degrees until...
Important note about rotations
If you rotate an object around the origin by:
90 degrees: You flip the coordinates.
Ex:
Let’s say we have a triangle
We can see the triangles points are:
(1,0), (4,0), (4,3)
And we rotated that triangle 90o about the origin
We can see the point coordinates are flipped!
The new points for the new triangle are:�(0,1), (0,4), (-3,4)�So why did 3 turn to -3?
We’re in quadrant II!
SO, WHAT ABOUT OTHER ROTATIONS?
Basically, if you rotate an object by a multiple of 90, you will keep switching the coordinates.
So, for 180, you’ll switch the coordinates twice,
Like this:
Now we can see that our new points went from:
(1,0), (4,0), (4,3)
To
(-1,0), (-4,0), (-4,-3)
So what about reflections?
Reflections are very similar to rotations, but a little bit different.
A reflection is exactly what you would think of when you look in a mirror, it’s a projected image of a shape that is symmetrical to its origin about the plane of reflection.
So what that means in plain English, is basically a line that causes the exact same , but mirrored, object to appear on each side of it.
See, now that is a perfect example of a reflection.
The mirror acts as the line of reflection, and the mirror image is the actual reflection.
This, however, is not a reflection. The image is a mirror image, however it doesn’t stay that way…..
(I know she’s supposed to be good and all, but that’s still creepy.)
You can also change how reflections work
It’s all about the line of reflection (it’s actually called a plane of reflection usually, but we’ll call it a line for right now).
For example, let’s say we have this figure:
And we toss in a line
And then, for fun, we reflect about that line.
Seems pretty easy right?
REFLECTION INSIDE THE OBJECT
But what if, say, we decide to reflect the object inside of itself?
So again, we have an object
But this time, the line is actually through the object
Now when we reflect about this line, it’s going to get a little weird.
See? I told you!�However, if you notice, the green star is a true mirror image of the blue star through the line of reflection.
THAT’S A GREAT REVIEW
So, now what does this have to do with dilations?
Well, translations, rotations, reflections, and dilations are all considered transformations; however, translations, rotations, and reflections are considered rigid transformations.
Dilations are considered non-rigid transformations, and there is a reason for that.
When we dilate a shape (in this case, a triangle) we are changing it’s size, either bigger or smaller.
We are not changing anything else, just the size of the object.
So, how do we dilate?
Well, the easiest way to do it by hand (when computer software is not available), is by drawing lines and increasing (or decreasing) the distance from our point to the new one.
This is one of those times when explaining the process is much harder than showing the process.
Example 1:
So, first, in order to dilate a shape, we need a shape.
Next we need a point away from the shape.
Now, we want to connect each point on the triangle, to our point.
Now, we want to duplicate those segments from the point on the triangle, away from our original point.
Now that we have all three points for our new triangles, let’s connect them to see what we have.
As we can see, each side of the triangle is doubled.
This is because we expanded our dilation two times.
So this triangle is twice as big as our first triangle.�
However, if you look closely, you’ll notice that both of the triangles share the same measurement of angles.
Just keep that in mind for later on…..�(Epic geometry foreshadowing)
Example 2:
So, let’s do another example for whole number dilation.
Again, First, we’re going to need our shape.
Next we need a point away from the shape.
Now, we want to connect each point on the triangle, to our point.
Now, we want to duplicate those segments from the point on the triangle, away from our original point.
Now that we have all three points for our new triangles, let’s connect them to see what we have.
As we can see, each side of the triangle is tripled.
This is because we expanded our dilation three times.
So this triangle is thrice as big as our first triangle.�
Again, if you look closely, you’ll notice that both of the triangles share the same measurement of angles.
Just an important note.
So, that’s whole number dilation, what about fractional dilation?
Fractional dilation
So when we dilate a shape by a whole number it makes it bigger.
So what do you think would happen if we broke a shape into a fractional portion?
It shrinks!
And of course, depending on how small the fraction is, that is how much smaller the shape will be.
For example:
EXAMPLE 2:
So, let’s do an example for fractional dilation.
Again, First, we’re going to need our shape.
Next we need a point away from the shape.
Now, we want to connect each point on the triangle, to our point.
Now, we want to find the points on those segments that are half way away from our triangle.
Now that we have all three points for our new triangles, let’s connect them to see what we have.
As we can see, each side of the triangle is halved.
This is because we shrank our dilation a half times.
So this triangle is half as big as our first triangle.�
Again, if you look closely, you’ll notice that both of the triangles share the same measurement of angles.
So, now we can get to our activity