Finding a quadrilateral from its symmetries
About this video
Given two vertices, (0, 9) and (3, 4), and the fact that the quadrilateral is unchanged by reflection over y = 3 - x, the video reflects each vertex over the line to find the other two vertices. The two new sides joining each point to its reflection are both perpendicular to the line, so they are parallel, and the figure is a trapezoid.
You'll be able to
- graph a line such as y = 3 - x from its intercept and slope.
- reflect a point over a line with slope -1 by moving diagonally across grid squares.
- use a reflection symmetry to find missing vertices and classify the quadrilateral.
Key words
- Reflection
- A transformation that flips a figure over a line so each point lands the same distance away on the other side.
- Perpendicular
- Meeting at a right angle; the path from a point to its reflection is perpendicular to the mirror line.
- Trapezoid
- A quadrilateral with at least one pair of parallel sides.
- Slope
- How much y changes when x increases by 1; y = 3 - x has slope -1.
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.
Video © Khan Academy, used under CC BY-NC-SA. Watch on Khan Academy. Puzzicle is not affiliated with Khan Academy.