# Parallelogram is formed by joining midpoints of a quadrilateral

1. Sep 18, 2009

### zeion

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

P1(x1,y1), P2(x2,y2), P3(x3,y3), P4(x4,y4) are the vertices of a quadrilateral. Show that the quadrilateral formed by joining the midpoints of adjacent sides is a parallelogram.

2. Relevant equations

Midpoint: M = ((x0+x1)/2, (y0+y1)/2)

3. The attempt at a solution

I'm guessing I need to show that the line formed by the midpoints of say, P1P4 and P1P2, is parallel to the line formed by joining midpoints P2P3 and P3P4?

I've never proved anything in my life and I'm not sure where to start -_-;

2. Sep 18, 2009

### skeeterrr

"I'm guessing I need to show that the line formed by the midpoints of say, P1P4 and P1P2, is parallel to the line formed by joining midpoints P2P3 and P3P4?"

You already figured it out.

3. Sep 18, 2009

### zeion

But how do I say this in a mathematical way? Lol
Do I need to use the Pythagoras theorem somewhere?

4. Sep 18, 2009

### skeeterrr

You don't need Pythagorean theorem.

And the question says show, not prove.

5. Sep 18, 2009

### zeion

So that means I can just write it with words?
Or draw a picture?

6. Sep 18, 2009

### Dick

You don't need the Pythagorean theorem. skeeterrr just meant try to finish the strategy you originally set out with. Show slopes of opposite sides are parallel using exactly the midpoint formula you originally stated. Just DO it.

7. Sep 18, 2009

### PhaseShifter

Calculate the four midpoints, then calculate the slopes of the lines connecting them.

8. Sep 18, 2009

### skeeterrr

Yes, just do the strategy that you thought out.

As for explaining, read the guideline on the MAT137 site, I am working on this problem set as well.

9. Sep 19, 2009

### zeion

Ok, pretty sure I have this one down.
Thanks.

10. Nov 10, 2009

### Tim4674

How would you show this in a general sense? Using theory as opposed to actual measurements? Any hints would be appreciated. Thanks

11. Nov 11, 2009

### Dick

Same advice again. Just try it. Write out the expressions without using numbers and see what cancels. Just TRY it.

12. Nov 12, 2009

### Tim4674

I ended up figuring it out by connecting the vertices to make a convex quadrilateral and then applying the midline theorem. Thanks for the help though