5. What is the area of a square inscribed in a semicircle of radius 1 , with one of its sides flush with the diameter of the semicircle?
Solution: Call the center of the semicircle $O$, a point of contact of the square and the circular part of the semicircle $A$, the closer vertex of the square on the diameter $B$, and the side length of the square $x$. We know $O A=1, A B=x, O B=\frac{x}{2}$, and $\angle A B O$ is right. By the Pythagorean theorem, $x^{2}=\frac{4}{5}$.