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## High school geometry

### Course: High school geometry > Unit 6

Lesson 4: Parallel & perpendicular lines on the coordinate plane# Classifying quadrilaterals on the coordinate plane

Watch Sal classify a quadrilateral as a square, rhombus, rectangle, parallelogram, or trapezoid given its four vertices. Created by Sal Khan.

## Want to join the conversation?

- Is there anyway of verifying the angles on the coordinate plane? For example, you draw something that looks like a rectangle, but you can't determine if it is a rectangle or a parallelogram. You determine the sides are parallel by checking the slopes. You verify the distances between the sides are the same by using the distance formula. How do you verify if the angles between are 90 degrees or some other measurement?(7 votes)
- Well if two of the lines are perpendicular to each other then they meet at a right angle. if a line is perpendicular two another then their slopes will be the negative inverse of each other.

https://www.khanacademy.org/math/algebra/linear-equations-and-inequalitie/more-analytic-geometry/v/perpendicular-lines

So if you find the slope of both lines and then and they are the negative inverse of each other then those lines meet at a right angle.

Hope that helps.(14 votes)

- Why did he use delta for "change" in the formula for the slope of a line?(5 votes)
- The capital delta is a standard notation for the change in a variable, and it is used primarily in calculus (and sometimes in those disciplines leading up to it).(11 votes)

- When proving that a Quadrilateral is a parallelogram, can I just prove that one pair of sides are parallel and one distinct pair of sides are congruent? or do I have to prove that both pairs of sides are parallel and congruent?(4 votes)
- It all depends on what you really mean by distinct pairs of sides, IF you have two sides that are parallel and congruent, then that is enough to prove congruency, the other two sides must also be parallel and congruent. But if the "top" and "bottom" sides are parallel, and the left and right sides are congruent, it could either be a parallelogram or a isosceles trapezoid.(5 votes)

- What is the best way to do an exercise like this without the use of a visual coordinate plane to draw on? Is there an "All-Numbers" way of solving this problem?(4 votes)
- Sketching the drawing also helps avoid errors such as finding distances and slope of pairs of points that are diagonals as opposed to sides of the quadrilateral.(2 votes)

- At2:00, why would I need to see if the sides are parallel to each other?(3 votes)
- To know if any of the "shapes" are the right answer. And to know for sure that what you think is right(3 votes)

- what kind of quadrilateral would it be than?(3 votes)
- It's none of the above. It is just a quadrilateral and not one of the special ones mentioned above.

Quadrilateral -> 4 sided figure

Parallelogram -> 2 pairs of parallel sides

Trapezoid -> Exactly 1 pair of parallel sides

Rectangle -> 4 right angles

Rhombus -> 4 equal sides

Square -> 4 equal sides and 4 right angles

Check out this lesson plan to learn more about the different types of quadrilaterals: https://www.khanacademy.org/math/basic-geo/basic-geometry-shapes/basic-geo-quadrilaterals/a/quadrilaterals-review(2 votes)

- At4:50, why would it not be a parallelogram or trapezoid?(3 votes)
- It's none of the above. It is just a quadrilateral and not one of the special ones mentioned above, because it does not fulfill any of their constraints:

Quadrilateral -> 4 sided figure

Parallelogram -> 2 pairs of parallel sides

Trapezoid -> Exactly 1 pair of parallel sides

Rectangle -> 4 right angles

Rhombus -> 4 equal sides

Square -> 4 equal sides and 4 right angles

Check out this lesson plan to learn more about the different types of quadrilaterals: https://www.khanacademy.org/math/basic-geo/basic-geometry-shapes/basic-geo-quadrilaterals/a/quadrilaterals-review(2 votes)

- Sal used the slope formula to help identify the type of quadrilateral in this video. How could you use other coordinate methods--such as the distance and midpoint formulas--to classify other quadrilaterals? Thank you!(2 votes)
- It was obvious from the diagram that it was none of the quadrilaterals listed. I am not sure how midpoint formula would have any meaning, but it you found the length of the lines by distance formula (or Pythagorean Theorem), the only thing you could know for sure is that if both pairs of opposite sides are equal length, then it must be a parallelogram (or rhombus if all sides are equal). Side lengths would not say anything about a trapezoid. Then you would have to check slopes to see if you could get a more precise answer. By doing slopes, you get a little more information such as a trapezoid if only one set of opposite slopes are the same, at least a parallelogram if both sets of opposite slopes are the same, or at least a rectangle if adjacent slopes are perpendicular (negative reciprocal). So you get more info with slopes, but if you find it is at least a parallelogram or rectangle, you have to go to distance formulas to check and see if there are more precise quad answers.(4 votes)

- At the end of the video, Sal said that it is "none of the above". Wouldn't trapezoid be the answer? (Side AB is parallel to side CD)(1 vote)
- Unfortunately, sides AB and CD are not parallel. You can confirm that if you find the slope from points AB and the slope from points CD. They are not equal which tells you the line segements are not parallel. Thus, Sal's selection of "none of the above" is correct.

If the line segments had been parallel, then he would have selected the option for Trapezoid.(5 votes)

- How do you draw the conclusion with the slopes you found one the first figure(3 votes)

## Video transcript

Classify quadrilateral ABCD. Choose the option that best
suits the quadrilateral. We're going to pick whether it's
a square, rhombus, rectangle, parallelogram, trapezoid,
none of the above. And I'm assuming we're
going to pick the most specific one possible, because
obviously all squares are rhombuses, or rhombi,
I guess you'd say. Not all rhombi are squares. All squares are also rectangles. All squares, rhombi, and
rectangles are parallelograms. So we want to be as specific
as possible in picking this. So let's see, point
A is at 1 comma 6. I encourage you to
pause this video and actually try this on your
own before seeing how I do it, but I'll just proceed. So that's point A
right over there. Point B is at
negative 5 comma 2. That's point B. Point C is at-- just had
some carbonated water, so some air is coming up. Point C is at
negative 7 comma 8. So that is point C
right over there. And then finally, point
D is at 2 comma 11. And actually that kind
of goes off the screen. This is 10, 11 would
be right like this. So that would be 2 comma 11. If we were to extend
this, this is 10 and this is 11 right up here. 2 comma 11. So let's see what this
quadrilateral looks like. You have this line right
over here, this line right over there, that line
right over there, and then you have
this line like this, and then you have
this like this. So right off the bat, well it's
definitely a quadrilateral. I have four sides. But the key question, are
any of these sides parallel to any of the other sides? So just looking at it, side CB
is clearly not parallel to AD. You just can look at it. Now it also looks like
CD is not parallel to BA, but maybe I just drew it badly. Maybe they actually
are parallel. So let's see if we
can verify that. So the way to tell whether
two things are parallel is to actually figure
out their slopes. So let's first figure out
the slope of AB, or BA. So let's figure
out our slope here. Your slope is going
to be the change in y over your change in x. And in this case,
you could think of it as we're starting at the
point negative 5 comma 2, and we're ending at
the point 1 comma 6. So what's our change in y? Our change in y, we
go from 2-- we're going from 2 all the way to 6. Or you could say it's 6 minus
2, our change in y is 4. What's our change in x? Well we go from negative 5
to 1, so we increased by 6. Or another way of thinking
about it, 1 minus negative 5, well that's going
to be equal to 6. So our slope here is 2/3. It's 2/3. Every time we move 3
in the x direction, we go up 2 in the y direction. Move 3 in the x direction,
go up 2 in the y direction. Now let's think about
line CD up here. What is our slope? Our change in y over change
in x is going to be equal to, let's see. Our change in-- let's figure
out our change in x first. Our change in x, we're
going from negative 7 comma 8 all the
way to 2 comma 11. So our change in x, we're
going from negative 7 to 2. Or we could say 2
minus negative 7. So we are increasing by 9. That's our change in x. So that's going
to be equal to 9. And then our change in y, well
it looks like we've gone up. We've gone from 8 to
11, so we've gone up 3. Or we could say 11 minus 8. Notice, endpoint
minus start point. Endpoint minus start point. You have to do that on the
top and the bottom, otherwise you're not going to actually
be calculating your change in y over change in x. But you notice our change,
when our x increases by 9, our y increased by 3. So the slope here
is equal to 1/3. So these actually
have different slopes. So none of these lines are
parallel to each other. So this isn't even
a parallelogram. This isn't even a trapezoid. Parallelogram, you have to have
two pairs of parallel sides. Trapezoid, you have to have
one pair of parallel sides. This isn't the case for any of
these, or none of these sides are parallel. So we would go with
none of the above.