Main content

### Course: 8th grade (Illustrative Mathematics) > Unit 2

Lesson 1: Lesson 4: Dilations on a square grid# Dilating shapes: shrinking by 1/2

Dilations are transformations that change the size of a shape and its distance from the center of dilation. When the center is the origin, we can change the distance by multiplying the x- and y-coordinates by the scale factor. That's how we find the new positions of the points after dilation. Created by Sal Khan.

## Want to join the conversation?

- can you have a scale factor of a negative?(10 votes)
- Yep, it will basically flip it. Say the point is 2 inches to the left of a line. If you dilate it by -3 now the point will be 6 inches to the right of it.(11 votes)

- What would happen if the dilation is not centered at the origin but at another point?(14 votes)
- u would dilate at that point, the point could act like the origin(6 votes)

- why is this taught in eight grade(7 votes)
- idk know but it is to hard for me(4 votes)

- I don't get how to dilate by a number with unknown lengths. How do you dilate exactly by the points?(5 votes)
- You count the distance of x and y of the old points and then dilate the ordered pair. Do this for all the points of the shape.(8 votes)

- What does this mean at1:24??(5 votes)
- it means that you're basically shrinking said triangle by 1/20:22(4 votes)

- This makes sense, and I am able to make the 1/2 calculations. But what scale factor is 1/3? That is a lot of decimals... How do you map that?(7 votes)
- I am confused on the whole thing(5 votes)
- I still don't get how to move the shapes?(3 votes)
- In the video, Sal dilates the triangle under a scale of 1/2. You multiply each of the coordinates in the coordinate plane by 1/2. It's going to be a reduction because when you multiply something lower than one, you get a smaller number.(4 votes)

- i don't quit understand why do we change the point's coordinates by 1/2 Instead of length? the whole time, we were dealing with length(2 votes)
- It's easier to divide points by two than it is to do a length just like that. When we divide the points, then we also divide the length. For example, let's say we have a square with side length 2 and vertices at (2, 2), (0, 2), (2, 0), and (0, 0). If we shrink this square by 2, we divide each point by 2. This means that our new points would be (1, 1), (0, 1), (1, 0), and (0, 0). We can then measure the length. If a side goes from (0,0) to (0,1), it travels 0 units left/right and 1 unit up. Therefore, the length is one unit. This is exactly what we wanted to do when we shrunk the shape by 2, as 2/2 = 1.

Hope this helped!(6 votes)

- What if the scale factor was some other number, like 1/3?(4 votes)

## Video transcript

Plot the images of points
D, E, and F after a dilation centered at the origin
with a scale factor of 1/2. So we're going to center
around the origin. We want to scale this
thing down by 1/2. So one way to think
about it is the points that will correspond to
points D, E, and F are going to be half as far away from
the origin, because our scale factor is 1/2 in
either direction. So for example, let's
think about point D first. Point D is at negative 8. So if we have a scale factor of
1/2, what point D will map to is going to be at negative
4 on the x direction. And on the y direction,
D is at negative 9, so this is going to
be at negative 4.5. Half of that. So that is going to
be right over there. That's where point D is going
to be, or the image of point D after the scaling. Now let's think
about point E. E is 2 more than the origin
in the x direction. So it's only going to be 1
more once we scale it by 1/2. And it's 7 more in
the y direction, so it's going to
be at 3 and 1/2. 7 times 1/2 is 3 and 1/2. So we're going to stick
it right over there. And then finally F, its
x-coordinate is 6 more than the origin. Its y-coordinate is 6 less. So its image after
scaling is going to be 3 more in the x direction
and 3 less in the y direction. So it's going to be
right over there. So we've plotted the
images of the points. So if you were to
connect these points, you would essentially
have dilated down DEF, and your center of dilation
would be the origin. So let's just write
these coordinates. Point D-- and point D, remember,
was the point negative 8, negative 9. That's going to map to--
take 1/2 of each of those. So negative 4 and negative 4.5. Point E maps to--
well, E was at 2, 7. So it maps to 1, 3.5. And then finally, point
F was at 6, negative 6, so it maps to 3, negative 3. So the important
thing to recognize is the center of our
dilation was the origin. So in each dimension,
in the x direction or in the y direction, we
just halved the distance from the origin, because
the scale factor was 1/2. We got it right.