 In this video, we are doing a coordinate proof, and we are going to use rhombus band. It says right here, if a quadrilateral is a rhombus, then the diagonals are perpendicular. So what we're going to do is we're going to write out what's given to us, and that is that band is a rhombus. And what we're trying to prove is that the diagonals are perpendicular. So what I'm going to do before I write in my proof, I'm going to draw real quick BAND. I'm going to put BAND on my little graph here. So take a minute, pause the video, and put band onto your graph. Okay, so I'm going to say that this is band, and again, I'm just estimating where these points should go. You could actually use a real graph and have it exact. But because we're going to be using either the distance formula, the slope formula, or the midpoint formula, our proof can come from the numbers instead of looking at the picture. So the diagonals of this rhombus, if we look at our picture, the diagonals are AD and BN. So what we're trying to prove is that BN is perpendicular to AD. So what we're going to do in order to prove that these two segments are perpendicular is we are going to use the slope formula. And what we have to show is that the slopes are opposite reciprocals, because remember perpendicular segments have slopes that are opposite reciprocal. So using the slope formula, which remember is y2 minus y1 over x2 minus x1, I am going to first find the slope of BN. Now some of you might be saying, well, I see that BN is a horizontal line, and I already know that horizontal lines have a slope of 0. Well, in a coordinate plane, I'm sorry, coordinate proof, you're trying to show that even though you can see that it's horizontal, you're trying to actually prove it. So we're going to use the numbers. So B is the point 0, 4, and N is the point 8, 4. So when I put those numbers into the slope formula, 4 minus 4 over 8 minus 0, I get 0 over 8, which is, in fact, 0. OK, remember we're trying to prove that the slopes are opposite reciprocal. Well, what in the world is the opposite reciprocal of 0? Well, if you think about this fraction right here with 0 over 8, if we make that negative and the reciprocal, it's going to be, let's say, negative 8 over 0. But really, all we care about, the opposite reciprocal of 0, is that 0 will be on the bottom. We can see that AD looks like it's a vertical line. We're going to prove it. So again, we're going to use, this time we're going to use the points A and D. And when I plug those values into the slope formula, 0 minus 8 over 4 minus 4. Sorry, my dogs are barking. And we get negative 8 over 0, which, of course, is undefined. You can't have 0 on the bottom of a fraction. So what we've just shown is that these two segments are, in fact, perpendicular. And so the conclusion is, sorry. The conclusion is band is a rhombus, B, A, and D is a rhombus. So the diagonals are perpendicular. And we've proven that using the slope formula here to show that we have opposite reciprocal slopes, which means we have perpendicular segments.