Quadrilateral relationships: Rectangles and rhombuses

0:00- [Instructor] We are told to identify a rhombus that is also a rectangle. So pause this video and have a go at choosing which of these meets those constraints, and then we'll do this together. All right, I do this, let's do this together. So let's think about what it means to have a rhombus.

0:15So there's a few ways to approach it, but my brain always thinks four equal sides. You might also hear people say two pairs of parallel sides, but it turns out if you have four equal sides, you're definitely going to have two pairs of parallel sides. So let's think about which of these choices at least meet the condition to be a thrombus.

0:35Well, choice A is definitely a rhombus. All four of these sides are the same. So that one looks good. Choice B is clearly not a rhombus. These two sides are the same, but then these other two sides are potentially different. And so I will rule choice B out. Now, choice C is a rhombus, all four sides are the same.

0:57Now, what's the difference between choice A and choice C? Well, choice C, all of my angles are right angles, where in choice A, they at least don't make it very clear that it is, and they're probably not right angles. And so now let's look at the choice for a rectangle.

1:12Well, a rectangle is a quadrilateral where you have four, four right angles. Now you could also say you have two pairs of parallel sides, but it turns out if you have four right angles, you're going to have two pairs of parallel sides. So choice A right over here does not meet the constraint for a rectangle.

1:31So we need to meet both of 'em. So we're gonna rule this one out. So A is a rhombus, but not a rectangle. B actually is a rectangle, but not a rhombus, and C is both. You have four equal sides and you have four right angles. So I'm liking choice C. Another way you could think about it, if you think about all quadrilaterals, so all things with four sides.

1:55So this is quadrilaterals, I'll just write quad for short there. And that might include things that are, have no parallel sides, something like this. But then you could say, "All right, well, what if I want to have at least one pair of parallel sides?" Well, then you could have this category of trapezoids.

2:16Trapezoids. Trapezoids. Where you have at least one pair of parallel sides, and so you could have a trapezoid like this that has one pair of parallel sides, but then you could also have a trapezoid that has two pairs of parallel sides. This is the inclusive definition of the trapezoids, where even two pairs of parallel sides meets the conditions.

2:36So two pairs of parallel sides might look something like that. And we call this right over here, we call this a parallelogram. Parallelogram. Now, within parallelograms, there's some parallelograms where all of the sides are equal. So if all of the sides are equal, then we're talking about a rhombus.

3:01Rhombus over here. So this right here is a rhombus, and I could, let me actually make it clear that these sides are parallel to each other. Just to make it very, very, very clear. This side over here is parallel to that side. Now here, they're parallel to each other, and we have four equal sides.

3:23But then you can also imagine that, well, what about a situation where it's four equal sides and four right angles? Well, that's a subset. Then you're talking about a square, essentially. And a square is a subset of rectangles. But we just saw that some rhombuses are rectangles, some rhombuses aren't, and some rectangles are rhombuses and some rectangles aren't.

3:44So if we make another set right over here of rectangles, rectangles, where rectangles and rhombuses intersect, you get squares. And then right over here, you would have things like choice B where you have four right angles, four right angles, but you have pairs of equal sides, but all of them aren't the same.

4:07So hopefully this helps you out, or you could do it the way we did it initially with a little bit of deductive reasoning.