Notes
parallelogram in a rectangle solution

Parallelogram in a Rectangle

Solution by Similar Triangles and Area of a Triangle

With the points labelled as above, triangles $A D C$ and $G D E$ are similar: angles $D \hat{A} C$ and $D \hat{G} E$ are equal as they are alternate angles, the same applies to angles $A \hat{C} D$ and $G \hat{E} D$. Since $E G$ has half the length of $A C$, the length of $D F$ is half of that of $B D$. Therefore the length of $B D$ is $\frac{2}{3}$rds of the length of $B F$. The area of triangle $A C D$ is then $\frac{1}{3}$ of the area of the lower half of the square. Hence the area of the parallelogram is $\frac{1}{3}$rd of the area of the square.

Solution by Tessellation or Dissection

The above diagram shows either by tessellation or dissection that the square has the same area as three parallelograms.