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## High school geometry

### Course: 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.

## Want to join the conversation?

- Why would it not be an isosceles trapezoid?(26 votes)
- I think he was just identifying it as a trapezoid in a general sense. I did the same thing when I worked it out on my own.(3 votes)

- Sometimes it's difficult to see the perpendicular to the line of reflection. Therefore, I've been using the following technique: plot the "transform" (I don't know the correct terminology) of the point [e.g. if the point is (0,9), then plot (-9,0) OR if the point is (3,4), then plot (-4,-3)] then move the point to the final, correct reflection in both the x & y directions using the x-intercept & y-intercept of the line of reflection as offsets. In the same example, (-9,0) will move +3 in the x-direction since the x-intercept of the line of reflection is +3 and also move +3 in the y-direction since the y-intercept is also +3 to the final reflection point of (-6,3) and for the point at (3,4), the final reflection point is (-4+3, -3+3) or (-1,0). Is this true in all cases? Even if the reflection isn't over a straight line but perhaps some other 2-dimensional shape such as a circle?(14 votes)
- Well the math is a little more complicated when the slope of the line isn't
`1`

or`-1`

, but yes, you can use the perpendicular line (which has a slope of`-1/m`

compared to the original line's slope of`m`

) to calculate the reflected point mathematically.(10 votes)

- At0:58Sal talks about "when x is 0, y is 3 - that's our y intercept" and then talks about how the slope goes down from there. I've been following everything I can on geometry but I seemed to have missed exactly how these slopes work. Is there another unit I can look at that describes how the whole y = 3 - x thing works?(5 votes)
- I would suggest looking up "equations of a line" and "slope-intercept form" on the KA search bar.(8 votes)

- I do not know how to solve Y=3-X. I did not find any explanation about it in previous videos in this section. Could you explain it to me please?

Thanks for your invaluable services

Monir(4 votes)- You don't really solve "y=3-x." He just put the line on the graph. You can rewrite it as "y=-x+3." the "x" is the slope which is almost like the distance between two points. 3 is the y-intercept which you plot on the y-axis.(3 votes)

- Is a trapezoid essentially the same as a trapezium?(5 votes)
- Yes, they are the same thing, trapezium in British, trapezoid in American.(1 vote)

- I realize this is a simple reflection line with a slope of -1, however if the coefficient was different so that perhaps it was a steeper slope, would we know what is perpendicular by taking the opposite reciprocal of the slope and then just shifting up or down and right or left so many units based on that calculation? For example, let's say the line we are reflecting over is y = -2/3x-2. A line perpendicular to this one would have a slope of +3/2. We would then use up 3 units and to the right 2 units as a measurement to determine a point's reflection across the original line (or work backwards if the point was above the line), counting how many units we have moved to get to the line of reflection? I'm fairly bad at visualizing reflections, etc. and I have to take an important math test where I cannot write anything down so I am attempting to see if this is a correct method that I can use in my head when I am challenged with this no-writing test. Thank you!(2 votes)
- Yes but you have to calculate the distance from the point to the line, and then double it to reflect across to the other side.(2 votes)

- Why are two of the lines parallel?(2 votes)
- If you're asking how we know that the lines are parallel, it's because we can see that their slopes are equal.

For example, the lines y=-7x+12 and y=-7x-8 are parallel, because their slopes (the slope is always a coefficient of x) are the same. The b-value, AKA the y-intercept, is irrelevant in the context of parallel lines, because regardless of a line's vertical translation, it cannot EVER intersect a line that has its same slope (parallel). If you play around with a graph and look at different lines, it might help you to visualize this better.

Also, to identify perpendicular lines (though I know this isn't what you asked), just remember that two lines are perpendicular if their slopes are the negative reciprocals of one another. For example, y=6x+4 is perpendicular to the line y=-1/6x+2 because the first line's slope of 6 can be turned into 1/6, and then made negative. This gives you -1/6, which - if we check again, is the exact slope of the other line, so they must be perpendicular.(2 votes)

- Do you need to know how to do slopes to do this kind of math?(2 votes)
- Absolutely. If you don't understand as of now, you can find more information under 7th grade math and algebra 1.(2 votes)

- It says a quadrilateral so why is it a triangle?(1 vote)
- At what point do you see it as a triangle? He even stated at the end that it ends up as a trapezoid which is a quadrilateral.(3 votes)

- At1:45is there any mathematical way you can find out the vertices of a quadrilateral for a reflection?(2 votes)

## 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.