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Dilating shapes: expanding

Sal is given a rectangle on the coordinate plane and he draws the image of the rectangle under a dilation with scale factor 1 2/3 about an arbitrary point. Created by Sal Khan.

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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.