Dilating a triangle example

We are asked to draw the image of triangle ABC under a dilation whose center is P and scale factor is one fourth. So pause this video and at least think about how you would do this. You don't have access to the tool that I do, where I can move this around. This is actually from the Khan Academy exercise, but at least think about how you would do this. Draw the image of triangle ABC under a dilation whose center is P and scale factor is 1/4.

All right, now let's work through this together.

So one way to think about it is our center of dilation is P and our scale factor is 1/4. For each of these points, however far they were from P before, they're now going to be 1/4 as far. One way to think about it is to consider it in each of the x and y dimensions.

For example, point A right over here, to get from point P to point A, you have to go one, two, three, four to the right and one, two, three, four down. So 1/4 of that in the x and y direction. Instead of going 4 to the right and 4 down, 1/4 of that would be 1 to the right and 1 down, because one is one fourth of four.

So instead of four to the right, we'd go one to the right. Instead of four down, we would go one down. So this is what point A would get mapped to.

Now let's think about point B. In order to go from P to point B, you have to go in the y direction up one, two, three, four, five, six, seven, eight. Well, what's one fourth of eight? It's two. So now we're going to go up two.

To go from P to B, how much do we have to go in the x direction? Well, we have to go to the left one, two, three, four. So instead of going four to the left, we're now going to go one fourth of that, or just one to the left.

Instead of 8 up and 4 left, we're now going 2 up and 1 to the left, 1/4 in each direction.

Then last but not least, to go from P to C, it looks like we're going one, two, three, four, five, six, seven, eight to the left and one, two, three, four up. So if we take one fourth of each of those, instead of eight to the left, we're now going to go two to the left. Instead of 4 up, we're now going to go 1 up.

And we're done! We have just created the image of triangle ABC under dilation whose center is P and scale factor is 1/4.