What percentage of the sector OAB is occupied by the inner circle: 50%, 69%, 75% or 81%
Could someone please show me a way to find it out?
1. Calculate the radius r of the inner circle: If R is the radius of the quarter circle then
$\displaystyle 2r^2 = (R-r)^2~\implies~r^2+2Rr-R^2=0~\implies~\boxed{r=-R+R\cdot \sqrt{2}}$ The negative solution isn't very plausible here.
2. The ratio you are looking for is:
$\displaystyle \dfrac{\pi (-R+R\cdot \sqrt{2})^2}{\dfrac14 \pi R^2} = \dfrac{\pi R^2(-1+ \sqrt{2})^2}{\dfrac14 \pi R^2} = 4(1-\sqrt{2})^2 \approx 0.68629...$