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Dimensions of a rectangle from coordinates

How can we determine the length of a side of a rectangle given the coordinates of its vertices? In this example, by plotting the points on a graph, we can see that the length one side is five because the y-coordinate increases by five from one endpoint to the other.

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Video transcript

- [Instructor] Rectangle ABCD is graphed in the coordinate plane. The following are the vertices of the rectangle, and they give us the labels for the vertices as well. Given these coordinates, what is the length of side AD of this rectangle? So let's just plot it. That's one way that we could tackle it. So let's, let me see all of these are actually in the first coordinate. So I can focus on the first coordinate. So let's say that's my y-axis. That's my y-axis. And let me draw my x-axis, my x-axis. And actually I wanna really focus in on the points A and D because we just need to find the length of the side from point A to point D. So point A, lemme just use another color, so point A is at x equals 7, y is equal to 1. So 1, 2, 3, 4, 5, 6, 7, so that's x equals 7. y is equal to 1. y is equal to 1, so that is point A right over there, point A, let me label it, point A. And then where's point D? Point D has the exact same, has the exact same x-coordinate, but its y-coordinate is a little bit higher. Its y-coordinate is 6, it's actually 5 higher. So point D is that x equals 7, y is equal to 6. So y is equal to 1, 2, 3, 4, 5, and 6. So that's y is equal to 6. And so we can draw the point. It is going to be right over here. This is point D, and we can actually connect them with a line if we like to show that this is a side of a rectangle. So let me draw that. I can just draw this like this. And there you have it. I haven't even drawn the whole rectangle yet, but just by plotting these two points, we can think about how long side AD is. We can see, look, the x doesn't change going from A to D, but we do increase in the y direction by 5. We go from y is equal to 1 to y is equal to 6. So our change in y is equal to 5. So what's the length of this line? Well, it's gonna be 5. It's gonna be 5, whatever the units are. So that's the length of side AD. It's going to be equal to 5. We went from the point 7,1 to the point, to the point 7,6. Now they said that this was a rectangle, now just for satisfaction, we can draw the entire rectangle. We have the point B. That is at x equals 5, y is equal to 1. So let me draw that. And I'm just doing this just for fun now. At this point, we're done with the problem. So x equals 5, y equals 1. That's right over here. That's point B. Let me write B, that's the point 5,1. And then we have the point C. I'll use another color. The point C is at x equals 5, y equals 6. x equals 5, y equals 6. Point C is right over here. Point C is the point x equals 5, y equals 6. And then we can connect all the dots if we like and clearly see that it is a rectangle. So I could connect those, I could connect these two, and then I could connect these two. And then we see that we indeed have a rectangle. But we answered it a while ago that the length of side AD is equal to 5. If we cared about the other sides, the length of BA, well, this is 2, a difference of 2 along the x direction, the horizontal direction, this difference of 2 along the x direction, difference of 5 in the y direction, we go from y equals 1 to y equals 6. So there you have it. We are actually able to figure out all of the dimensions of this rectangle.