Classifying figures with coordinates | Analytic geometry | High school geometry | Khan Academy

We're told that parallelogram A B C D has the following vertices, and they give us the coordinates of the different vertices. They say, "Is parallelogram A B C D 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 want 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.

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. 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.

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. Let me actually count by eights; let me count by twos here. So we have 2, 4, 6, and 8. Then we have negative 2, negative 4, negative 6, negative 8. We have 2, 4, 6, and 8, and then we'd have negative 2, negative 4, negative 6, and negative 8. So each hash mark is another 2. I'm counting by 2s here, and so let's plot these points, and I'll do it in different colors so we can keep track.

So A is (-6, -4). So negative 2, negative 4, negative 6, and then negative 4 would go right over here. That is point A. Then we have point B, which is (-2, 6). So negative 2, comma 6, so that's going to go up 2, 4, and 6. So that is point B right over there.

Then we have point C, which is at (8, 2). So 8, comma 2, right over there. That is point C. Lastly, but not least, we have point D, which is at (4, -8). Four comma negative 8, right over there. Point D.

Our quadrilateral, or we actually know it's a parallelogram, looks like this. So you have segment A B like that; you have segment B C that looks like that; segment C D looks like this; and segment A D looks like this. We know already that it's a parallelogram, so we know that segment A B is parallel to segment D C, and segment B C is parallel to segment A D.

But what we really need to do is figure out whether they are intersecting at right angles. To do that, using the coordinates to figure that out, we have to figure out the slopes of these different line segments. Let's figure out first the slope of A B. The slope of segment A B is going to be equal to our change in y over change in x.

So our change in y is going to be (6 - (-4)) over (-2 - (-6)). This is going to be equal to (6 + 4), which is 10, over (-2 - (-6)), which is the same as (-2 + 6). So that's going to be over 4, which is the same thing as 5/2. All right, that's interesting.

What is the slope of segment B C? The slope of segment B C is going to be equal to, once again, change in y over change in x. Our y coordinates change: (2 - 6) over (8 - (-2)). This is equal to (-4) over (8 + 2) over 10, which is equal to 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 are going to be the opposite reciprocals. 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. 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 actually perpendicular lines. This lets us know that A B is perpendicular; segment A B is perpendicular to segment B C. So we know that this is the case. We could keep on doing that, but in a parallelogram, if one set of segments intersects at a right angle, all of them are going to intersect at a right angle.

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; it would be that it is a rectangle because A B is equal. So the length of segment A B is equal to the length of segment A D, and the length of segment B C is equal to the length of segment C D. So that might be true; I haven't validated it, but just because this is true and because we do know that A B C D 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, so I would rule this top one out.

The second choice says yes, and it says because B C is perpendicular to A B. Yeah, we saw that by seeing that their slopes are the opposite reciprocals of each other, and of course, we know that A B C D 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.