# Solution to the Overlapping Triangle and Rectangle Puzzle +-- {.image} [[OverlappingTriangleandRectangle.png:pic]] > The triangle and rectangle have the same area. What fraction of the total is shaded? =-- ## Solution by [[Similar Triangles]] +-- {.image} [[OverlappingTriangleandRectangleLabelled.png:pic]] =-- 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.