Notes
four squares and an isosceles triangle solution

Solution to the Four Squares and an Isosceles Triangle Puzzle

Solution by Similar Triangles

Consider the diagram labelled as above, where point $C$ is such that angle $B \hat{C} D$ is a right-angle.

Angles $C \hat{D} B$ and $G \hat{D} F$ are vertically opposite, so triangles $D C B$ and $D G F$ are similar. This means that the lengths of $C D$ and $B C$ are in the ratio $1 : 2$.

Angle $C \hat{A} B$ is $45^\circ$, so triangle $A C B$ is an isosceles right-angled triangle meaning that $A C$ and $C B$ have the same length. This means that the lengths of $C D$ to $C A$ are also in the ratio $1 : 2$. Since $A D$ has length $\frac{3}{2}$, this means that $C A$ has length $1$, and so also does $C B$.

Using the formula for the area of a triangle, triangle $A D F$ has area $\frac{1}{2} \times \frac{3}{2} \times 3$ and $A D B$ has area $\frac{1}{2} \times \frac{3}{2} \times 1$. So the total area of the blue triangle is: