Notes
triangles in a semi-circle solution

Triangles in a Semi-Circle

Solution by Similarity

Triangles $A B D$ and $D B C$ are similar because they are both right-angled and the angles at $D$ add up to $90^\circ$ since it is the angle in a semi-circle. The area scale factor is $4$, so the length scale factor is $2$. This means that $D B$ has twice the length of $A B$, and $B C$ twice that of $D B$. Let $a$ be the length of $A B$, then $D B$ has length $2 a$ and $B C$ length $4 a$.

The area of triangle $A B D$ is $\frac{1}{2} \times a \times 2 a = a^2$ so $a^2 = 8$. The area of the semi-circle is $\frac{1}{2} \pi \left( \frac{5}{2} a \right)^2 = \frac{25}{8} \pi a^2 = 25 \pi$.