Quadrilaterals Investigation

This is another investigation classes like to explore and can be proven rather easily using vectors.

The investigation

Choose points P, Q, R and S and connect them to form a quadrilateral.

Construct the midpoints of each side.

Connect those midpoints in order with straight lines.

What do you notice?

I encourage students to constuct accurate diagrams and measure side lengths or angles carefully to determine whether their claims seem correct. To keep things simple, I've asked students to find the midpoints by measuring side length and halving it. I've also challenged students with using perpendicular bisectors to find midpoints, but be warned that diagrams get cluttered pretty quickly!

The midpoints of any quadrilateral form a parallelogram, or a straight line.

The proof

Let $2\mathbf{a}=\vec{PQ}$, $2\mathbf{b}=\vec{QR}$ and $2\mathbf{c}=\vec{RS}$

Let $A$, $B$, $C$ and $D$ be midpoints of $PQ$, $QR$, $RS$ and $SP$, respectively.

First, $\vec{AB}=\mathbf{a}+\mathbf{b}$

Secondly, $\vec{DC}=\frac{1}{2}\vec{PS}-\mathbf{c}=\frac{1}{2}(2\mathbf{a}+2\mathbf{b}+2\mathbf{c})-\mathbf{c}=\mathbf{a}+\mathbf{b}$

Now we have that $\vec{AB}=\vec{DC}$. These vectors are parallel and equal in length, therefore the midpoints form a quadrilateral, unless they are collinear in which case they form a straight line.