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## Geometry (all content)

### Course: Geometry (all content)>Unit 10

Lesson 6: Dilations

# 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.

## 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.