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## High school geometry

### Unit 2: Lesson 4

Symmetry# Finding a quadrilateral from its symmetries (example 2)

CCSS.Math:

Two of the points that define a certain quadrilateral are (-4,-2) and (0,5). The quadrilateral has a reflective symmetry over the lines y=x/2 and y=-2x+5. Draw and classify the quadrilateral. Created by Sal Khan.

## Video transcript

Two of the points that define
a certain quadrilateral are negative 4 comma negative 2. So let's plot that. So that's negative
4 comma negative 2. And 0 comma 5. So that's 0 comma
5 right over there. The quadrilateral
is left unchanged by a reflection over the
line y is equal to x over 2. So what does that
line look like? y is equal to x over 2. I'll do that in the blue. y is equal to x over 2. So when x is equal to 0, y is 0. The y-intercept is 0 here. And the slope is 1/2. Every time x increases by
1, y will increase by 1/2. Or when x increases by
2, y will increase by 1. So x increases by
2, y increases by 1. X increases by 2,
y increases by 1. Or another way to think about
it, y is always 1/2 of x. So when x is 4, y is 2. When x is 6, y is 3. When x is 8, y is 4. So we can connect these. Let me try my best
attempt to draw these in a relatively
straight line. And then I can keep going. When x is negative
2, y is negative 1. When x is negative
4, y is negative 2. So it actually goes through
that point right there. And it just keeps going
with a slope of 1/2. So this line, and I can draw
it a little bit thicker now, now that I've dotted it out. This is the line y
is equal to x over 2. And they also say that
the quadrilateral is left unchanged by reflection
over the line y is equal to
negative 2x plus 5. So the y-intercept here is 5. When x is 0, y is 5. So it actually goes
through that point. And the slope is negative 2. Every time we increase by 1-- or
every time we increase x by 1, we decrease y by 2. So that would go there. We go there. And we keep going at
a slope of negative 2. So it's going to look
something like this. It actually goes
through that point and just keeps going on and on. So this is my best attempt
at drawing that line. So that is y is equal
to negative 2x plus 5. Now let's think about it. Let's see if we can
draw this quadrilateral. So let's first reflect
the quadrilateral, or let's reflect the points
we have over the line y is equal to x over 2. So this is the line y
is equal to x over 2. This magenta point,
the point negative 4, 2 is already on that line. So it's its own reflection,
I guess you could say. It's on the mirror is one
way to think about it. But we can easily reflect
at this line over here. This line, if we were to drop
a perpendicular-- And actually, this line right over here,
y is equal to negative 2x plus 5 is perpendicular
to y is equal to x over 2. How do we know? Well if one line
has a slope of m, then the line
that's perpendicular would be the negative
reciprocal of this. It would be negative 1 over m. So this first line
has a slope of 1/2. Well what's the negative
reciprocal of 1/2? Well the reciprocal
of 1/2 is 2/1. and you make that negative. So it is equal to negative 2. So this slope is a negative
reciprocal of this slope. So these lines are
indeed-- I'm trying to erase that-- perpendicular. So we literally could
drop a perpendicular, literally go along this
line right over here, in our attempt to reflect. And we see that we're
going down 2, over 1 twice. So let's go down 2, over 1,
down 2, over 1, twice again. Let me do that in
that same color. The reflection of this point,
across y is equal to x/2 is this point right over there. So now we have three points
of our quadrilateral. Let's see if we
can get a fourth. So let's go to
the magenta point. The magenta point
we've already seen. It's sitting on top
of y equals x/2, so trying to reflect it
doesn't help us much, but we could try to
reflect it across y is equal to negative 2x plus 5. So once again these lines are
perpendicular to each other. Actually let me mark that off. These lines are perpendicular. So we can drop a perpendicular
and try to find its reflection. So we're going to
the right 2 and up 1. We're doing that once, twice,
three times on the left side. So let's do that once, twice,
three times on the right side. So the reflection
is right there. We essentially want
to go to that line. And however far we
were to the left of it, we want to go that same, that
bottom left direction, which you want to go in the same
direction, to the top right, the same distance to
get the reflection. So there you have it. There is our other point. So now we have the four
points of this quadrilateral. Four points of this
quadrilateral are-- or the four sides, let me actually just
draw the quadrilateral. We have our four points. So this is one side
right over here. This is one side,
right over here. This is another side,
right over here. And you can verify that
these are parallel. How would you verify
that they're parallel? Well they have the same slope. To get from this point to that
point, you have to go over, so your run has to
be 4, and you have to rise 1, 2, 3, 4, 5, 6, 7. So the slope here is 7/4. Rise over run, or change in
y over change in x, is 7/4. And over here,
you go 1, 2, 3, 4. So you run 4 and then you
rise 1, 2, 3, 4, 5, 6, 7. So the slope here is also 7/4. So these two lines are
going to be parallel. And then we could draw
these lines over here. So this one at the top. The one at the top
right over there. And what's its slope? Well, let's see. We go from x equals
0 to x equals 8. So we go down, our change in
y is negative 1 every time we increase x by 8. So this is the slope of--
slope is equal to negative 1/8. And that's the exact same slope
that we have right over here. Negative 1/8. So these two lines
are parallel as well. This line is parallel
to this line as well. So at minimum, we're dealing
with a parallelogram. But let's see if we can
get even more specific. Because this kind of
looks like a rhombus. It looks like a
parallelogram, where all four sides have
the same length. So there's a couple
of ways that you could verify that this
parallelogram is a rhombus. One way is you
could actually find the distance between the points. We know the coordinates, so you
could use the distance formula, which really comes straight
out of the Pythagorean theorem. Or even better, you could
look at the diagonals of this rhombus-- we could
look at the diagonals of this parallelogram. We are trying to figure
out if it's a rhombus. And if the diagonals
are perpendicular, then you're dealing
with a rhombus. And we've already shown
that these diagonals, that this diagonal, this
diagonal, and this diagonal are perpendicular. They intersect at right angles. And so this must be a rhombus.