Finding equivalent ratios in similar quadrilaterals
About this video
Lucas rotated and dilated quadrilateral ABCD to make WXYZ, so the two quadrilaterals are similar. The video colour-codes corresponding sides (CD with YZ, BC with XY, BA with XW, AD with ZW) and tests four proportions. Only CD/YZ = BC/XY must be true, because it compares corresponding sides in the same order; the others flip a ratio or use sides that don't correspond.
You'll be able to
- explain why a dilation and rotation produce similar shapes.
- match corresponding sides using the angle marks.
- decide whether a proportion between similar shapes must be true.
Key words
- dilation
- A transformation that enlarges or shrinks a shape by a scale factor.
- similar
- Same shape, possibly different size and orientation.
- corresponding sides
- Sides that match up between similar shapes.
- proportion
- An equation saying two ratios are equal.
Transcript
- [Instructor] We are told Lucas dilated quadrilateral ABCD, ABCD, to create quadrilateral WXYZ. So it looks like he rotated it and zoomed in or made it or expanded it to get this other quadrilateral. And so the fact that we used these types of transformations like a dilation, and it looks like a 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 wanna do is color the corresponding sides the same.
So let's think about side CD here. So 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 arc is right over there. So this side, YZ corresponds to side CD. And 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.
And then 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, and 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, 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, 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. And 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.
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