Notes
overlapping triangle and rectangle solution

Solution to the Overlapping Triangle and Rectangle Puzzle

Solution by Similar Triangles

As the triangle and rectangle have the same area, the parts that don’t overlap must also be of the same area as each other. In the above diagram, the vertical line down from the apex of the triangle divides the shaded regions into four triangles. Each pair, $A$ with $E$ and $F$ with $J$, consists of similar triangles as their angles at their meeting points are vertically opposite angles so are equal and the sides that do not meet are parallel. By considering those parallel sides, the scale factors from $A$ to $E$ and from $J$ to $F$ must be the same. But since the areas of $A$ and $J$ must be equal to those of $E$ and $F$, these triangles must in fact be congruent.

The other regions in the division are then all either congruent to $A$ or to $J$, showing that the shaded regions comprise $\frac{2}{5}$ths of the total.