# Math Help - Proving parallelogram (vectors)

1. ## Proving parallelogram (vectors)

The midpoints of the adjacent sides of a quadrilateral are joined. Show that the resulting figure is a parallelogram.

I know that you suppose to somehow use vectors to prove this, but I dont know how you are suppose to model this?

2. Suppose that $A,~B,~C,~D$ are the consecutive vertices of the quadrilateral.
Let midpoints be $P \in \overrightarrow {AB},~ Q \in \overrightarrow {BC},~ R \in \overrightarrow {CD},~ S \in \overrightarrow {DA}$.
We know that $\overrightarrow {AB}+\overrightarrow {BC}+\overrightarrow {CD}+\overrightarrow {DA}=0$.
Moreover, $0.5(\overrightarrow {AB})=\overrightarrow {PB}~\&~ 0.5(\overrightarrow {BC})=\overrightarrow {BQ}$
So $0.5(\overrightarrow {AB}+\overrightarrow {BC})=\overrightarrow {PQ}$
Do the same thing for $\overrightarrow {RS}$
Prove that $\overrightarrow {PQ}=-\overrightarrow {RS}$
You will be done.

3. Originally Posted by Plato
We know that $\overrightarrow {AB}+\overrightarrow {BC}+\overrightarrow {CD}+\overrightarrow {DA}=0$.
What if the quadrilateral you were doing this proof in was a trapezium then how would this be true? (since all the sides are unequal). How would the vectors of the sides add up 0?

Originally Posted by Plato
So $0.5(\overrightarrow {AB}+\overrightarrow {BC})=\overrightarrow {PQ}$.
Sorry but I dont understand why the inner side of a parallelogram = 2 x 1/2 of the sides of a quadrilateral.

4. Originally Posted by SyNtHeSiS
What if the quadrilateral you were doing this proof in was a trapezium then how would this be true? (since all the sides are unequal). How would the vectors of the sides add up 0?
Sorry but I dont understand why the inner side of a parallelogram = 2 x 1/2 of the sides of a quadrilateral.
This is a standard question. It is done in any geometry course.
I have given you the standard vector proof.