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## High school geometry

### Unit 2: Lesson 4

Symmetry# Finding a quadrilateral from its symmetries

CCSS.Math:

Two of the points that define a certain quadrilateral are (0,9) and (3,4). The quadrilateral has reflective symmetry over the line y=3-x. Draw and classify the quadrilateral. Created by Sal Khan.

## Video transcript

Two of the points that define
a certain quadrilateral are 0 comma 9 and 3 comma 4. The quadrilateral is left
unchanged by a reflection over the line y is
equal to 3 minus x. Draw and classify
the quadrilateral. Now, I encourage you
to pause this video and try to draw and
classify it on your own before I'm about to explain it. So let's at least plot the
information they give us. So the point 0
comma 9, that's one of the vertices of
the quadrilateral. So 0 comma 9. That's that point
right over there. And another one of the
vertices is 3 comma 4. That's that right over there. And then they tell us
that the quadrilateral is left unchanged by reflection
over the line y is equal to 3 minus x. So when x is 0, y is 3--
that's our y-intercept-- and it has a slope
of negative 1. You could view
this as 3 minus 1x. So it has a slope of negative 1. So the line looks like this. So every time we increase our
x by 1, we decrease our y by 1. So the line looks something like
this. y is equal to 3 minus x. Try to draw it relatively,
pretty carefully. So that's what it looks like. y is equal to 3 minus x. So that's my best attempt
at drawing it. y is equal to 3 minus x. So the quadrilateral is
left unchanged by reflection over this. So that means if I were to
reflect each of these vertices, I would, essentially,
end up with one of the other vertices
on it, and if those get reflected you're going to
end up with one of these so the thing is not
going to be different. So let's think about
where these other two vertices of this
quadrilateral need to be. So this point, let's just
reflect it over this line, over y is equal to 3 minus x. So if we were to try to drop
a perpendicular to this line-- notice, we have gone
diagonally across one, two, three of
these squares so we need to go diagonally
across three of them on the left-hand side. So one, two, three gets
us right over there. This is the reflection of
this point across that line. Now, let's do the same
thing for this blue point. To drop a perpendicular
to this line, we have to go diagonally
across two of these squares. So let's go diagonally across
two more of these squares just like that to get to that
point right over there. And now we've defined
our quadrilateral. Our quadrilateral
looks like this. Both of these lines
are perpendicular to that original
line, so they're going to have the same slope. So that line is parallel
to that line over there. And then we have this line
and then we have this line. So what type of
quadrilateral is this? Well, I have one pair
of parallel sides, so this is a trapezoid.