Coordinates

Ramesh5492 · 2013-12-19 10:52:13

Hello everyone, I am shivam's friend, and I will open account soon. I had a math test and was stuck on this point:
I will make a general figure of the thing I don't understand.
We have a quadrilateral ABCD. I am given the coordinates for all four vertices. How do you know which edges (i.e. AB or CD) are opposite to each other? Lets say I have to find the centroid for which I need bimedians. How do I know which sides are opposite, from the coordinates, to find the midpoint and so on...

Second question
Lets say I have a parerellogram, ABCD.
I am given coordinates for ABC, how many possible coordinates can I have for D?
Is it related to that CD has to be parallel to AB or something like that?
Thanks...

bobbym · 2013-12-19 13:12:55

Hi Ramesh5492;

First question.

Try each pair of points, take the slope of the line between them. In a parallelogram the slopes of opposite sides are equal. When you find 2 slopes that are equal you have found the 2 opposing sides.

ramesh5492 · 2013-12-19 13:41:56

a quadrilateral is not necessarily a parerellogram

bobbym · 2013-12-19 13:48:07

Hi;

I would think that for a quadrilateral you are going to have to do a better job defining "opposite side." Which side is opposite AB?

And of course, complex or intersecting quadrilaterals are even more difficult to say what the opposite side is.

Bob · 2013-12-20 01:43:02

hi Ramesh5492

Welcome to the forum.

I wonder if I have understood your question. Let's check:

You know the coordinates of four points, A, B, C and D; but they are not necessarily lettered in order around the shape.

Now you want to know is AB opposite to CD or should it be AC opposite to BD or AD and BC ?

I have a method that seems to work but I haven't checked all cases yet to make sure it won't let you down in extreme cases.

(i) Out of the four x coordinates find the largest, xmax. Similarly find xmin, ymax and ymin.

(ii) This means you can enclose the quadrilateral in a rectangle. (see diagram below)

(iii) So find the equation of the line AC and also of BD.

(iv) If these lines cross inside the rectangle, then AB and CD are diagonals. That's the case in my diagram.

(v) But if I'd chosen my letters differently so that these lines cross outside the rectangle, then AB and CD are opposite.

Does that always work? Still thinking about that ................................

EDIT: Not if the quadrilateral has a reflex angle (arrowhead).

Bob

Last edited by Bob (2013-12-20 01:53:37)

Ramesh5492 · 2013-12-20 05:54:16

Bobbym I tried that, but that would be too long. We would have to find the opposite sides then the midpoint then the equations and then solve the system to get the point of intersection. But anyways, if this was the intended way then how many slopes would I have to test?

Bob bundy, thanks but the only tools we were given were the distance and mid point formulas. I doubt we were expected to do it in that method.

bobbym · 2013-12-20 06:02:01

My suggestion is one way if you have a parallelogram.

For a quadrilateral you are going to have define what you mean by opposite. In the quadrilateral below which side is opposite CEB or AB for that matter?

If it is difficult visually to find the opposite side of a given side how can we expect to do it mathematically? The word "opposite" is too ambiguous.

I think you will need to restrict the question to some particular type of quadrilateral.

Ramesh5492 · 2013-12-20 06:08:57

I guess opposite means non-consecutive sides. Lets say I give you points ABCD (with the coordinates) but no diagram. I tell you to find the centroid by finding the bimedian lines' intersection. How would you know which sides are opposite such that their midpoints make up the bimedian?

bobbym · 2013-12-20 06:15:53

In the above diagram in post #7, which is the opposite side to CEB ?

ramesh5492 · 2013-12-20 06:27:44

I don't know, but I meant simple quads like rectangles, squares, pararellograms etc. How can I get the opposite sides of that?

bobbym · 2013-12-20 06:30:31

With those 3 types the slope idea will work fine because all of them have the property that the opposite sides are parallel to the given side. But if you have a type like post #4 you will have a problem again.

ramesh5492 · 2013-12-20 06:33:32

Yeah, so for non-complex/non-intersecting quads the check for slopes method will work? And how many slopes do I need to test if I have ABCD?

bobbym · 2013-12-20 06:42:35

I am not following you?

ramesh5492 · 2013-12-20 06:44:08

I meant for quads like squares, rectangles etc the checking if slopes of equal method will work for finding opposite sides?

And secondly, how many slopes do I need to calculate for a quadrilateral ABCD to find the opposite sides?

bobbym · 2013-12-20 06:48:17

I meant for quads like squares, rectangles etc the checking if slopes of equal method will work for finding opposite sides?

I believe so.

If I were given 4 points, I would need only one test to find the line opposite a given line.

Do you have an example?

ramesh5492 · 2013-12-20 06:50:19

I don't get it. If I had ABCD, wouldn't I need slopes of AB, CD, AC, BD etc?

bobbym · 2013-12-20 06:53:53

Hi;

If you are given a side and 4 points then it is easy. 2 of your 4 points are the endpoints of the side you are given. The other 2 points are the endpoints of the other side. This will work provided you have a parallelogram ( square and rectangle are parallelograms) and are given a side.

ramesh5492 · 2013-12-20 06:55:59

I just have four points, ABCD. They form a quadrilateral. How many slopes do I need to test for equality?

bobbym · 2013-12-20 07:02:00

There are:

ways to pick 2 from the 4 points when order does not count.

If I pick A and B, I test the other 2, C and D. ( 1 test )

If I pick A and D, I test B and C. ( 1 test )

If I pick A and C then I have a small problem.

I have picked a diagonal and will have to choose again. Do you need to see a diagram?

Bob · 2013-12-21 04:17:41

hi ramesh5492

Method for finding the centroid of a quadrilateral (works for all convex quads.)

Split the quad into two triangles along one diagonal.

Find the centroid of each triangle by intersecting medians.

Join these centroids with a line.

Repeat for the other diagonal.

Where the two centroid joining lines cross is the centroid of the quad.

So for your four points:

(i) Choose any point and call it A

(ii) Temporarily call the other three points, P, Q and R in any order.

(iii) Calculate the gradients for AP, AQ and AR.

(iv) The gradient which lies between the other two will be the diagonal. Rename it AC

(v) Rename the other two points B and D either way round.

(vi) Find the centroid of ACB and also of ACD. Join these points with a line. The centroid of the quad will be somehwere on this line.

(vii) Repeat for triangles BDA and BDC. Make a second line by joining these centroids.

(viii) The centroid of the quadrilateral is where these two lines cross.

Bob

Math Is Fun Forum

#1 2013-12-19 10:52:13

Coordinates

#2 2013-12-19 13:12:55

Re: Coordinates

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Re: Coordinates

#4 2013-12-19 13:48:07

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#5 2013-12-20 01:43:02

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