A semicircle is inscribed in a quarter of a circle of unit radius as shown in the figure. What fraction of the quarter circle is the semicircle?
+121
In the diagram above r is the radius of the inscribed half-circle and R is the radius of the quarter-circle. I will show that
\(\frac{r^2}{R^2}=\frac{1}{3}\)
so that the ratio of their areas would be
\(\frac{\frac{\pi\cdot r^2}{2} }{\frac{\pi\cdot R^2}{4}}=\frac{2\cdot r^2}{R^2}= \frac{2}{3}\).
(refer to the image above) we have
\(R^2=r^2+(r\cdot \sqrt{2})^2=r^2+2\cdot r^2=3r^2\)
so that
\(\frac{r^2}{R^2}=\frac{1}{3}\) as promised. Q.E.D
