1951 AHSME Problems/Problem 48

Problem

The area of a square inscribed in a semicircle is to the area of the square inscribed in the entire circle as:

$\textbf{(A)}\ 1: 2\qquad\textbf{(B)}\ 2: 3\qquad\textbf{(C)}\ 2: 5\qquad\textbf{(D)}\ 3: 4\qquad\textbf{(E)}\ 3: 5$

Solution

Let the radius of the circle be $r$. Let $s_1$ be the side length of the square inscribed in the semicircle and $s_2$ be the side length of the square inscribed in the entire circle. For the square in the semicircle, we have $s_1^2 + (\frac{s_1}{2})^2 = r^2 \Rightarrow s_1^2 = \frac{4}{5}r^2$. For the square in the circle, we have $s_2 \sqrt{2} = 2r \Rightarrow s_2^2 = 2r^2$.

Therefore, $s_1^2 : s_2^2 = \frac{4}{5}r^2 : 2r^2 = \boxed{\textbf{(C)}\ 2: 5}$

See Also

1951 AHSC (ProblemsAnswer KeyResources)
Preceded by
Problem 47
Followed by
Problem 49
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