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# Parallelogram on the coordinate plane

Video transcript
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. Vertex A is at the point 1, 1. It puts us right over there. That is the vertex A. Vertex C is at the point 4.5 comma 4, so 4.5 along the horizontal axis comma 4. So we go all the way up to 4. That right over there is point C. Point D is at negative 1.5 comma 4, so negative 1.5 along the horizontal or the x-axis, we could say, negative 1.5 comma 4, so 4 along the vertical or the y-axis. We 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 I. And they tell us that the distance from A to B must be the same as the length of segment D to C in both our horizontals. So let's draw what we know to draw. So DC, segment DC is this segment right over here. And we see it's horizontal. Both of the vertical coordinates are 4 at both vertex D and vertex C. So both of the vertical coordinates are 4. Now what is the length of this? Because we're going to 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 0, 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 4 plus 1 is 5, 0.5 plus 0.5 is 1, so 5 plus 1 is 6. So this distance right over here is 6 of our units. Actually, let me put the coordinates in here, just so it becomes a little bit clearer. Let me do that in something easier to see. This right over here is the point 4.5, 4, and this right over here is the point negative 1.5, 4. Another way of thinking about this distance is you could take the end point-- and we're really thinking about the distance just along its horizontal line, so the y-value does not change. It doesn't change in the vertical direction, only the horizontal. So you really want to say, 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 or 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 6. Fair enough. And let me draw some of the rest of the polygon, just so that we see it is indeed a polygon. We have this side right over here. It looks like it's going to be a parallelogram. We have this side right over here. And we have to replace point B. Now, point B is going to be someplace out here. It's going to have the same vertical value or the same y-value as point A. So its y-coordinate is going to be 1. So point B is going to be out here someplace. Let me do this in a new color. I haven't used this orange yet. Actually, I have used the orange yet. I haven't use 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. We already know what its y-coordinate is. It's a horizontal line, so it's going to have to have the same exact y-coordinate as point A. Point A's y-coordinate was 1, so this is going to have to have a y-coordinate of 1. Now the big question is, what is its x-coordinate going to be? Let me do that in a different color. It's going to have to be whatever A's x-coordinate was. We see that A's x-coordinate was 1. And it's going to have to be that plus 6, because we're going to move the same distance in the horizontal direction. This thing has to be 6. So if we start at 1, we add 6, we get to 7. So what are the coordinates of point B, especially if point B must be in quadrant I? And notice we are definitely in quadrant I. This is quadrant I, this is quadrant II, this is quadrant III, and this is quadrant IV. The coordinate for point B is 7 comma 1.