Similar triangles & slope: proportion using coordinates | Grade 8 (TX) | Khan Academy

We're told that triangle DF and triangle DKL are similar right triangles. Complete the proportion to show that the slope of DF, so that's this segment right over here DF, equals the slope of DL. So pause this video and see if you can complete it. They started the proportion here, and so this is going to be equal to one of these choices.

All right, now let's do this together. So the first thing I want to do is I just want to figure out what they're trying to calculate here. We already know that we are dealing with slope. Let me get the right tool out, that we're dealing with slope. This looks like a slope calculation right over here, and we know slope is change in y over change in x, which you could view as y1 minus y2 over x1 minus x2.

When I look at the points that are used here, it looks like y1 is that point right over there, or it's -3. X1 is 8; well, I know it's equal to 8. It looks like y2—remember, you subtract y2—so y2 is 9, and it looks like x2 is -7. So it looks like they're trying to find the slope between the points (8, -3) and the point (-7, 9).

So let's see what points those are. (8, -3) is this one right over here; so (8, -3) is point F right over there. Then (-7, 9) is point D right over there. If you want to do the same thing for this smaller triangle, you would still use point D as the second point, but you would use point L as the first point.

So point L right over here is the point (3, 1). So essentially, let's do the same calculation, where this is x1 and y1, and we're going to use the same x2 and y2 as before. If we did that, we are going to get—so on the numerator, y1 is 1 minus y2, which is y2 for D; so that's -9. That's the same y2 for D, and then x1 here is 3 minus x2. x2 for D we've already seen is -7.

So let's see which of those choices are that. If I look at that, that's exactly what we wrote right over here in Choice D.