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CCSS.Math:

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.