Parallelogram is formed by joining midpoints of a quadrilateral

[zeion]

Homework Statement

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.

Homework Equations

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

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 -_-;

 

[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.

 

[zeion]

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

 

[skeeterrr]

You don't need Pythagorean theorem.

And the question says show, not prove.

 

[zeion]

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

 

[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.

 

[PhaseShifter]

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

 

[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.

 

[zeion]

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

 

[Tim4674]

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

 

[Dick]

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

 

[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