# Solution to the Two Rectangles with Diagonals Puzzle +-- {.image} [[TwoRectangleswithDiagonals.png:pic]] > Two rectangles and their diagonals. What fraction of the total area is shaded? =-- ## Solution by [[Similar Triangles]] and [[Equilateral Triangles]] +-- {.image} [[TwoRectangleswithDiagonalsLabelled.png:pic]] =-- First, note that the rectangles are [[similar]] since the angle between a side and the diagonal is the same for each (angle $H \hat{B} G$). So the ratio of the side to the half-diagonal is the same for each, so the ratio of the lengths of $B G$ to $B H$ is the same as with them the other way round, which means that they must be equal in length. This is also the length of $G H$ so triangle $B G H$ is [[equilateral]] and [[quadrilateral]] $E G B H$ is a [[kite]]. Since triangle $B G H$ is equilateral, triangle $G I B$ is half an equilateral triangle of the same side length. Each rectangle therefore has area four times that of the triangle $B G H$. Triangle $J G H$ is then half of an equilateral triangle, so $H G$ has double the length of $G J$. The point $K$ is the [[midpoint]] of $G H$ so $H K$ is the same length as $J G$ and triangles $G J E$ and $E K H$ are [[congruent]]. Triangle $E G H$ is therefore two thirds of $G J H$ and so its area is one third of that of $B G H$. The area of the purple region is therefore $\frac{4}{3}$rds of the area of $B G H$. The total area is then $8 - \frac{4}{3} = \frac{20}{3}$ times the area of $B G H$, so the fraction that is shaded is $\frac{4}{20} = \frac{1}{5}$th of the total area.