# Solution to the Square with Midpoints Puzzle +-- {.image} [[SquarewithMidpoints.png:pic]] > Dots are midpoints. What fraction of the square is shaded? =-- ## Solution by [[Similar Triangles]] +-- {.image} [[SquarewithMidpointsLabelled.png:pic]] =-- In the above diagram, the point labelled $G$ is directly below $E$. Triangle $A E G$ is [[similar]] to triangle $A D C$, while triangle $G B E$ is [[similar]] to $A B F$. So the length of $G B$ is half of $E G$ which is half of $A G$. So $G B$ is one fifth of $A B$, and so $E G$ is two fifths of $A B$ and so a fifth of $A C$. This means that triangle $A E B$ is one twentieth of the area of the full square. Triangles $A D C$ and $A B F$ are both one quarter of the area of the square. Therefore the shaded area is the following fraction of the square: $$ \frac{1}{4} - \frac{1}{10} + \frac{1}{4} - \frac{1}{10} = \frac{3}{10} $$