20 -90 The ratio of the area of a square inscribed in a semicircle to the area of a square inscribed in a full circle is
(A) $1: 2$.
(B) $2: 3$.
(C) $2: 5$.
(D) $3: 4$.
(E) $3: 5$.
(2nd American High School Mathematics Examination, 1951)
[Solution] Let the side length of the square inscribed in the semicircle be $x$, the side length of the square inscribed in the circle be $y$, and the radius of the circle be $R$.
From $x^{2}+\left(\frac{x}{2}\right)^{2}=R^{2}$, we get $x^{2}=\frac{4}{5} R^{2}$.
Also, $y=\sqrt{2} R$, so $y^{2}=2 R^{2}$.
Therefore, $x^{2}: y^{2}=\frac{4}{5} R^{2}: 2 R^{2}=2: 5$.
Hence, the answer is $(C)$.