Find x in this image

Question

The two of 1/4 circles with radius r are fitted in a rectangle with one of the sides a. Find another side x.

Answer 1

Consider the point where both quadrants touch each other, join the corners. Now use Pythagoras Theorum to get $X^2 + A^2 = 4R^2$

Hope it helps.

Answer 2

The intersection point of these circles is the center $M$ of the rectangle :

Indeed, the image of the first circle under the central symetry of center $M$ is the second circle. Hence, the intersection $I$ of both circles must be invariant under the central symetry of center $M$. Hence, $I=M$.

Now, that implies that the length of a diagonal of the rectangle is $2r$. And you can apply Pythagoras theorem as suggested by Chief VS, and get $x = \sqrt{(2r)^2 - a^2}$.