# Cartesian vector question

1. Apr 20, 2010

### spoc21

1. The problem statement, all variables and given/known data

Use Cartesian vectors in two-space to prove that the line segments joining midpoints of the consecutive sides of a quadrilateral form a parallelogram.

2. Relevant equations

We could say vectors a and b

a = [a1, a2]
b = [b1, b2]

3. The attempt at a solution

Am really confused by this question..I would really appreciate it if anyone could offer tips to help me get started on this problem.

Thanks!

2. Apr 20, 2010

### tiny-tim

Hi spoc21!
So far so good!

Next would be

c = [c1, c2]
d = [d1, d2]

Now what are the midpoints of ab, bc, cd, and da ?

3. Apr 20, 2010

### spoc21

Hi tiny-tim,

so if we use the midpoint formula, we would get the following:

midpoint of ab = (a1+b1/2, a2 + b2/2)

midpoint of bc = (b1 + c1/2, b2c2/2)

midpoint of cd = ( c1+ d1/2, c2 + d2/2)

midpoint of da = (d1 + a1/2, d2 + a2/2)

Now I'm stuck, how would I show that joining these points would produce a parallelogram.

Thank you very much for your help!

4. Apr 20, 2010

### Staff: Mentor

You should identify the midpoints - give them names, such as by saying the E is the midpoint between A and B, F is the midpoint between B and C, G is the midpoint between C and D, H is the midpoint between C and D, and I is the midpoint between D and A.

Now if the segments EF and GH are parallel, and the segments FG and HI are parallel, then EFGH is a parallelogram.

One nit: In your midpoint formulas, add parentheses around the sums. E.g., ((a1 + b1)/2, (a2 + b2)/2)

5. Apr 20, 2010

### spoc21

ok, so I get the following results

EF = [(c1 - a1)/2, (c2 - a2)/2]
GH = [(a1 - c1)/2, a2 - c2)/2]

I'm a little lost here..(don't know how to proceed)

Also, you said "G is the midpoint between C and D, H is the midpoint between C and D, and I is the midpoint between D and A."

Are there two midpoints G, as well as H between C, and D?

6. Apr 20, 2010

### Staff: Mentor

I lost track of my midpoints. There should only be four of them E, F, G, and H - no I needed. I meant "H is the midpoint between D and A."

You can check that your vectors EF and GH are parallel by showing that their slopes are the same. The slope of a vector v = <a, b> is b/a.

7. Apr 21, 2010

### spoc21

ok thanks Mark..also, I was wondering if there is a way to solve this using the properties of cross products?

8. Apr 21, 2010

### Staff: Mentor

|u X v| = |u||v|sin(theta), where theta is the angle between u and v. If u and v are parallel, the angle between them will be 0 or pi, so the magnitude of the cross product will be zero.

9. Apr 25, 2010

### spoc21

So I have attached my working in pdf format, and would appreciate if you could just take a quick look at it, as I am unsure if this is correct...Thanks!!

#### Attached Files:

• ###### cross product proof 1.pdf
File size:
162 KB
Views:
87
10. Apr 26, 2010

### tiny-tim

Hi spoc21!

ok, but a little long-winded.

(and why are you using square brackets instead of the usual round ones? has your professor told you to? )

It would have saved time and space to choose an easier notation

let A = (xA,yA) etc

then the midpoints are MAB = 1/2 (xB + xA, yB + yA) etc

and the vectors joining consecutive midpoints are VAC = 1/2 (xC - xA, yC - yA) etc

so VAC = -VCA and VBD = -VDB

in other words, opposite sides of the quadrilateral are parallel, and so it must be a parallelogram ("equal in magnitude" is a bonus, but not necessary for the proof)

(note incidentally that there would have been no need to use Cartesian coordinates if the question hadn't required them … you can just use vector symbols, eg mAB = 1/2 (b + a) etc )

EDIT: just noticed another way, that throws slightly more light on the position of the parallelogram …

use the fact that if the midpoints of the two diagonals of a quadrilateral coincide, then it's a parallelogram …

and those midpoints, for the new quadrilateral, are (both) 1/4 (a + b + c + d) …

so it's a parallelogram, and "centred" in the same place as the original quadrilateral

Last edited: Apr 26, 2010