Finding equivalent ratios in similar quadrilaterals | Grade 8 (TX) | Khan Academy

We are told Lucas dilated quadrilateral ABCD to create quadrilateral WXYZ. So it looks like he rotated and zoomed in or made it or expanded it to get this other quadrilateral. The fact that we used these types of transformations like a dilation and it looks like a quad rotation as well, it tells us that these are similar to each other. They are similar, similar quadrilaterals.

So based on that, which proportion must be true? Pause this video and see if you can work that through on your own before we do this together.

All right, now let's do this together. So for my brain, and given that I have access to a very nice palette of colors, what I want to do is color the corresponding sides the same. So let's think about side CD here. We know that this point, or this angle right over here with one arc corresponds to this angle, and then this other angle with the double arcs is right over there.

So this side YZ corresponds to side CD. Then we could say, all right, going from the right angle over here to the point C, that would correspond to going from the right angle to the point Y in this other quadrilateral. Maybe I'll use red for this one. Going from B to A would correspond to going from X to W.

These are corresponding sides, and then last but not least, side AD corresponds to side ZW. That'll help us keep track of what's going on here. So this first one has the length of segment CD. The length of segment CD. The ratio between that and BC, and BC is my blue one, or my teal color, I should say.

BC, they're saying that's the same as XY, which is in teal, to YZ. Well, this one isn't feeling right. In order for this to be true, you would have to flip one of these ratios because, once again, my pink one to blue one on this quadrilateral should be the same. It should be pink to blue on the other quadrilateral, not blue to pink. That is one way to think about it, so let's rule out that one.

Now, let's see. We have the ratio between CD and BC is the same as the ratio between XY and WX. Well, this isn't even using corresponding sides right over here, so let's rule that one out. All right, next we have the ratio between CD and YZ, so those are corresponding sides. Then they're saying that should be equal to BC over WZ. BC over WZ. Well, WZ is not corresponding to BC, so I'll rule that out.

So just deductive reasoning would tell us that this is likely our choice. But let's work through it. So they're saying the ratio of CD to YZ, CD to YZ, is the same as the ratio of BC, BC to XY. So yes, this is ratios of corresponding sides, so this proportion must be true.