Representing dilations algebraically, k less than 1 | Grade 8 (TX) | Khan Academy

- [Instructor] We are
told quadrilateral WXYZ

was dilated with the origin
as the center of dilation

to create quadrilateral W prime,
X prime, Y prime, Z prime.

So we started off with
this black quadrilateral,

and then it looks like
it was dilated down.

One way to think about it,
centered at the origin,

it was scaled down.

Write a rule to represent this dilation.

So like always, pause the video,
have a go at it on your own

before we do this together.

All right, so let's just remind ourselves

what a rule that represents
a dilation even looks like.

A rule would look something like this.

You take any x, y coordinate
on the original shape

and it's going to get mapped
to another x, y coordinate

which will now be on the new
shape, on the shape in green.

Actually, why don't I write that in green

just to make it clear what's going on.

So it's going to be scaled
in the x direction by K

and scaled in the y direction by K.

And so the key is we have to figure out

what this scaling factor actually is.

Now, there's a couple
of ways you could do it.

You could look at a corresponding side,

especially one that runs
horizontal or vertical

so that you can actually
just count how long it is

so you can know its dimensions
or you know its length.

For example, we could look at
that length right over there.

So we know that WZ is equal to

it looks like 1, 2, 3,
4, 5, 6, 7, 8, 9 units.

And now let's see what
W prime, Z prime is.

So W prime, Z prime looks
like it is 1, 2, 3, 4, 5, 6.

It is six units.

So it looks like when we went
from WZ to W prime, Z prime,

it looks like we scaled,

we multiplied by two over three.

So that gives us a pretty good clue

that the scaling factor is two over three,

and that makes sense.

If the scaling factor is less than one,

the shape that we are mapping to

after the dilation is going to be smaller.

If the scale factor is greater than one,

then we're going to enlarge it.

But let's see if we can find
other confirmation of that.

Well, there aren't any other sides

that are horizontal or vertical,

but we could actually also confirm that

by looking at a point

where we can clearly get the
coordinates of that point.

So for example, we see that point Z,

right over here, it has the coordinates,

this looks negative nine, negative three,

negative nine, negative three.

And now Z prime.

If we believe this scaling factor,

if we believe that it is 2/3
for both the x and the y,

then if we multiply this by 2/3,

it should be negative six.

And if we multiply this by 2/3,

it should be negative two.

Let's see, Z prime is indeed,

the coordinates are negative six

or negative six, negative two.

So once again, we have multiplied by 2/3

in either of these situations.

So we feel very comfortable

that the rule to
represent this dilation is

for any x, y on the original shape,

it is going to get mapped
to, instead of a K,

we now know that the K is 2/3,

2/3 of the original x

and 2/3 for the new y
coordinate of the original y.

And we are done.