Classifying figures with coordinates

- [Instructor] We're told that parallelogram ABCD has the following vertices, and they give us the coordinates of the different vertices. And they say is parallelogram ABCD a rectangle, and why? So pause this video, and try to think about this on your own before we work through it together. All right, now let's work through it together. So in general, if you know that something is already a parallelogram and you wanted to determine whether it's a rectangle, it's really a question of whether the adjacent sides intersect at a right angle. So for example, a parallelogram might look something like this. What we know about a parallelogram is that the opposite sides are parallel, so this side is parallel to that side and that this side is parallel to this side. And all rectangles are parallelograms, but not all parallelograms are rectangles. In order for a parallelogram to be a rectangle, these sides need to intersect at right angles. And clearly, the way I drew this one, it doesn't look like that. But let's see if we can figure that out based on the coordinates that they have given us. And to help us visualize, let me just put some coordinates. Let me draw some axes here. So that's my x-axis, and then this is my y-axis. Let's see, the coordinates, let's see, we have twos, fours, sixes, so let me actually count by, and eights, let me count by twos here. So we have two, four, six, and eight, and then we have negative two, negative four, negative six, negative eight. We have two, four, six, and eight, and then we'd have negative two, negative four, negative six, and negative eight. So each hash mark is another two. I'm counting by twos here. And so let's plot these points, and I'll do it in a different color, so we can keep track. So A is negative six comma negative four. So negative two, negative four, negative six, and then negative four would go right over here. That is point A. Then we have point B, which is negative two comma six. So negative two comma six, so that's going to go up two, four, and six. So that is point B right over there. Then we have point C, which is at eight comma two. So eight comma two right over there, that is point C. And then last but not least, we have point D, which is at four comma negative eight, four comma negative eight right over there, point D. And so our quadrilateral, or we actually know it's a parallelogram, looks like this. So you have segment AB like that. You have segment BC that looks like that. Segment CD looks like this. And segment AD looks like this. And we know already that it's a parallelogram, so we know that segment AB is parallel to segment DC and segment BC is parallel to segment AD. But what we really need to do is figure out whether they are intersecting at right angles. And to do that using the coordinates to figure that out, we have to figure out the slopes of these different line segments. And so let's figure out first the slope of AB. So the slope of segment AB is going to be equal to our change in y over change in x. So our change in y is going to be six minus negative four, six minus negative four over negative two minus negative six, negative two minus negative six. And so this is going to be equal to six plus four, which is 10, over negative two minus negative six. That's the same thing as negative two plus six. So that's going to be over four, which is the same thing as 5/2. All right, that's interesting. What is the slope of segment BC? The slope of segment BC is going to be equal to, once again, change in y over change in x. Our y-coordinates' change in y is two minus six, two minus six over eight minus negative two, eight minus negative two, which is equal to negative four over, and then eight minus negative two is the same thing as eight plus two over 10, which is the same thing as negative 2/5. Now in other videos in your algebra class, you might have learned that the slopes of lines that intersect at right angles or the slopes of lines that form a right angle at their point of intersection, that they are going to be the opposite reciprocals. And you can actually see that right over here. These are opposite reciprocals. If you take the reciprocal of this top slope, you'd get 2/5. And then you take the opposite of it or, in this case, the negative of it, you are going to get negative 2/5. So these are perpendicular lines. So this lets us know that AB is perpendicular, segment AB is perpendicular to segment BC. So we know that this is the case. And we could keep on doing that. But in a parallelogram, if one set of segments intersect at a right angle, all of them are going to intersect at a right angle. And we could show that more rigorously in other places, but this is enough evidence for me to know that this is indeed going to be a rectangle. If you want, you could continue to do this analysis, and you will see that this is perpendicular, this is perpendicular, and that is perpendicular as well. But let's see which of these choices match up to what we just deduced. So choice A says yes, and yes would be that it is a rectangle because AB is equal. So the length of segment AB is equal to the length of segment AD, and the length of segment BC is equal to the length of segment CD. So that might be true. I haven't validated it. But just because this is true and because we do know that ABCD is a parallelogram, that wouldn't let me know that we are actually dealing with a rectangle. For example, you can have a parallelogram where even all the sides are congruent. So you could have a parallelogram that looks like this. And obviously, if all of the sides are congruent, you're dealing with a rhombus, but a rhombus is still not necessarily going to be a rectangle. And so I would rule this top one out. This second choice says yes, and it says because BC is perpendicular to AB. Yeah, we saw that by seeing that their slopes are the opposite reciprocals of each other. And of course, we know that ABCD is a parallelogram. So I am liking this choice. And these other ones claim that this is not a rectangle, but we already deduced that it is a rectangle. So we could rule these out as well.