Main content

# Dilating shapes: expanding

Let's dilate a rectangle around a point P using a scale factor 1 2/3. This changes the size of the rectangle while keeping its shape. Because 1 2/3 is bigger than 1, most of the points get farther from P. Point P stays in place, because a distance of 0 times any scale factor is still 0. Created by Sal Khan.

## Want to join the conversation?

- I get almost all the video, but this one part confuses me. At1:19, where did Sal get the 3 to minus the 10 with?(57 votes)
- Point P is located at (-7,3) and the dilation needs to be 10 away from that. The 3 is the y-coordinate of the original point, so Sal subtracted the 10 from the positive 3 to get -7, the y-coordinate of the new point.(58 votes)

- How do you do a dilation when the scale factor is a percentage?(15 votes)
- you could make it into a decimal by dividing by 100 and then use that instead of the percentage. From that, it is basically the same as the video.

Hope it helps!(17 votes)

- nothing in the comments, or "questions" is relevant to the video xD(13 votes)
- I agree wholeheartedly with your comment. Thank you for pointing this out! Hopefully others will see the importance of only posting necessary and relevant math questions.(11 votes)

- What about dilating triangles?(14 votes)
- It's not hard. You just find the original line segment and then you multiply it by the scale factor. Just do that for all 3 sides. Then you get your new triangle!(7 votes)

- Is there a formula for this?(14 votes)
- if you're dilating about the origin then multiply all points' coordinates by however much you're dilating by yes?(1 vote)

- the problems on the practice are all triangles so this example that is being shown is not very helpful because triangles have a center that you have to account for and most of the triangles I had to do were off-centered/crooked which made it much harder..... just saying this video would be way more helpful if it included more shapes(11 votes)
- Rather than thinking of it as a triangle, think about having to move three points from the center. Each point has a horizontal and vertical distance from the dilation point, so whatever the scale factor is, multiply each part by the scale factor to find where the prime point should go. Do this three times, and you have the right answer. The prime points should be co-linear with their vertex point and the center of dilation. The shape does not matter as long as you do all the vertices the same way,(8 votes)

- Does anyone know a easier way to do this?(10 votes)
- Well, if you have a square, its very easy, just find the difference between the line segments, and because all the sides are the same so you just have to plot out the shape. But if you do this on a rectangle it's the same steps it just takes longer.(3 votes)

- At0:17, if the base shape is a triangle in which its sides and points are not on the marks of the graph, how do you expand the shape? Please help!(10 votes)
- So can I understand Dilation as this: The changing in the scale of a shape. ?(6 votes)
- Yeah, essentially. You increase or decrease the scale of a shape in a certain direction 👍(5 votes)

- When you reduce one, do you divide or multiply?(5 votes)
- ? If you mean reducing the side lengths of the shape by 1, you are dividing.(6 votes)

## Video transcript

The graph below contains
the rectangle ABCP. Draw the image of ABCP under a
dilation whose center is at P and a scale factor is 1 and 2/3. What are the lengths of
the side AB and its image? So we're going to do a
dilation centered at P. So if we're centering
a dilation at P and its scale
factor is 1 and 2/3, that means once we
perform the dilation, every point is going to be 1 and
2/3 times as far away from P. Well P is 0 away from
P, so its image is still going to be at P. So let's put
that point right over there. Now point C is going
to be 1 and 2/3 times as far as it is right now. So let's see, right
now it is 6 away. It's at negative 3. And P, its x-coordinate is the
same, but in the y direction, P is at 3. C is at negative 3. So it's 6 less. We want to be 1 and
2/3 times as far away. So what's 1 and 2/3 of 6? Well, 2/3 of 6 is 4, so
it's going to be 6 plus 4. You're going to be 10 away. So 3 minus 10, that
gets us to negative 7. So that gets us
right over there. Now point A, right
now it is 3 more in the horizontal direction
than point P's x-coordinate. So we want to go
1 and 2/3 as far. So what is 1 and 2/3 times 3? Well that's going to be 3 plus
2/3 of 3, which is another 2. So that's going to be 5. So we're going to
get right over there. Then we could complete
the rectangle. And notice point B is
now 1 and 2/3 times as far in the
horizontal direction. It was 3 away in the
horizontal direction, now it is 5 away from
P's x-coordinate. And in the vertical
direction, in the y direction, it was 6 below P's y-coordinate. Now it is 1 and
2/3 times as far. It is 10 below P's y-coordinate. So then let's answer
these questions. The length of segment AB--
well, we already saw that. That is, we're going
from 3 to negative 3. That is 6 units long. And its image, well
it's 1 and 2/3 as long. We see it over here. We're going from
3 to negative 7. 3 minus negative 7 is 10. It is 10 units long. We got it right.