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Parallelogram on the coordinate plane

Remember our discussion of the coordinate plane? Sure you do! Let's graph the given coordinates of three of the polygon vertices, and find where the 4th vertex is. Created by Sal Khan.

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Video transcript

- [Instructor] You are graphing Polygon ABCD in the coordinate plane. The length of segment AB must be the same as the length of segment DC, and both segments are horizontal segments. The following are three of the vertices of the polygon. So vertex A is at the point one comma one. So one comma one, it puts us right over there. That is vertex A. Vertex C is at the point 4.5 comma four. So 4.5 along the horizontal axis, comma four, comma four. So we go all the way, we go all the way up to, all the way up to four. So that right over there is point C. Point D is that negative 1.5 comma four, so negative 1.5 along the horizontal, or I guess we know the x-axis, we could say. Negative 1.5 comma four, so four along the vertical or the y-axis. So we go right up, go right over there. That's close enough. So that of course is our y-axis. This is point D. And we need to figure out what are the coordinates of point B if B must be in Quadrant 1. And they tell us that the distance from A to B must be the same as the length of segment D to C, and both are horizontal. So let's draw what we know to draw. So DC, segment DC, segment DC is this segment right over here, is this segment right over here. And we see it's horizontal. Both of the vertical coordinates are four at both vertex D and vertex C. So both of the vertical coordinates are four. Now what is the length of this? Because we're gonna have to construct another segment that has the same length. Well, along the horizontal direction, we went from negative 1.5 to 4.5. So how far did we go? Well, to go from negative 1.5 to zero, you go 1.5 and then you have to go another 4.5. So this is going to be 4.5 plus 1.5, which is equal to four plus one is 5, .5 plus .5 is one. So five plus one is six. So this distance right over here is six of our units. Another way of thinking about it, and actually let me put the coordinates in here just so it becomes a little bit clearer. This was the point four point... Lemme do that in something easier to see. This right over here is the point 4.5 comma four. And this right over here is the point negative 1.5 comma four. And so another way of thinking about this distance is you could take the end point, and we're really thinking about the distance just along, it's a horizontal line, so the y value does not change. It doesn't change in the vertical direction, only the horizontal. So you really wanna say, well, how far is, if you start at negative 1.5 and you get to 4.5, how far have you gone? So you can just take your end point, your end value, your end horizontal value, your end x value, and from that you can subtract your starting x value. So you subtract negative 1.5. And this of course is equal to 4.5 plus positive 1.5, which once again is equal to six. So, fair enough, and lemme draw some of the rest of the polygon just so that we see it is indeed a polygon. So we have this side right over here, and it looks like it's gonna be a parallelogram. So we have this side right over here and we have to replace point B. And point B is gonna be someplace out here. It's going to have the same vertical value or the same y value is point A. So its y-coordinate is going to be one. So point B is going to be out here someplace, point, lemme just in a new color. So point point B. I haven't used this orange yet. Actually, I have used the orange yet. I haven't used the yellow. No, I've used the yellow. Let's see, I haven't used this green. Point B is going to be someplace out here. So point B is gonna be someplace out here. We already know what its y-coordinate is. It's a horizontal line, so it's gonna have to have the same exact y-coordinate as point A. Point A's y-coordinate was one. So this is going to have to have a y-coordinate of one. Now the big question is, what is its x-coordinate going to be? Well, it's going to have to be... Let me do that in a different color. It's going to have to be whatever A's x-coordinate was. It's gonna have to be whatever A's x-coordinate was. And we see that A's x-coordinate was one. It's gonna have to be that plus six 'cause we're gonna move the same distance in the horizontal direction. This thing has to be six. So if we start at one, we add six, we get to seven. So what are the coordinates of point B? And especially if point B must be in Quadrant 1. And notice we are definitely in Quadrant 1. This is Quadrant 1, this is Quadrant 2, this is Quadrant 3, and this is Quadrant 4. Coordinate for point B is seven comma one.