# Proving that each diagonal of a parallelogram bisects each other

• Jul 10th 2010, 05:28 AM
Glitch
Proving that each diagonal of a parallelogram bisects each other
Q: Prove that each diagonal of a parallelogram bisects each other

How do I attempt this?

Can I find the midpoints of the diagonals, then if they're the same, get the distance between this midpoint and the vertices? If they're the same, have I proved it?

Thanks.
• Jul 10th 2010, 06:51 AM
Quote:

Originally Posted by Glitch
Q: Prove that each diagonal of a parallelogram bisects each other

How do I attempt this?

Can I find the midpoints of the diagonals, then if they're the same, get the distance between this midpoint and the vertices? If they're the same, have I proved it?

Thanks.

It can be proved using vectors. Consider a parallelogram ABCD with a,b,c,d as the position vectors of points A,B,C,D respectively.

$\vec{AB}=\vec{DC}$

so b-a=c-d

Rearrange it to get a+c=b+d

Divide both sides by 2, (a+c)/2=(b+d)/2

We see that the midpoints of AC and BD are identical.

...
• Jul 10th 2010, 06:53 AM
Glitch
I don't think I'm supposed to solve this with vectors, since I haven't learnt about them yet.
• Jul 10th 2010, 08:29 AM
skeeter
use congruent triangles, then.
• Jul 12th 2010, 05:30 AM
HallsofIvy
Quote:

Originally Posted by Glitch
I don't think I'm supposed to solve this with vectors, since I haven't learnt about them yet.

There's the problem- if you don't show any work at all, we have no idea what you can use!

In addition to "congruent triangles" (basic geometry) and "vectors", I might use "coordinate geometry". We can always set up a coordinate system so that one vertex is at (0, 0), another at (a, 0), a third at (b, c) and then the fourth must be at (a+ b, c). Now, it is easy to find the midpoint of each diagonal- that is, if you can use coordinate geometry- and we don't know that.
• Jul 12th 2010, 05:33 AM
Glitch
Quote:

Originally Posted by HallsofIvy
There's the problem- if you don't show any work at all, we have no idea what you can use!

I just wanted to know if my reasoning was sound.