Find the radius of the circle? Two Circle of an equal of an radii are drawn  , without any overlap , in a semicircle of radius 2 cm.
If these are the largest possible circles that the semicircle can accomodate , then 
what is the radius of each of the circles?
Thanks in advance.

Due to symmetry two circles in a semicircle is the same problem as one in a quatercircle or four in a full circle. If we look at a quatercircle originating at the origin, with radius $r$ and completely contained in the first quadrant, then the circle has to be centered at a point $(c,c)$, touching the x axis, the y axis and the quatercircle at $(r/\sqrt{2},r/\sqrt{2})$.
To find out how large the circle may be, without crossing the x and y axes, consider the distance from the center of this circle to one axis being equal to the distance to the point $(r/\sqrt{2},r/\sqrt{2})$
$$ c=\sqrt{2\left(\frac{r}{\sqrt{2}}-c\right)^2}=\sqrt{2} \left(\frac{r}{\sqrt{2}}-c\right)$$
$$ \Rightarrow c = \frac{r}{1+\sqrt{2}} $$
For $r=2\,\text{cm}$, $c\approx 0.83\,\text{cm} $.
