# square in a segment

What is the ratio of the areas of the two squares?

Solution

Consider the right angled triangle $PQC$. Side $CP$ is the radius of the circle. Side $CQ$ has length $t+2s$ and side $QP$ has length $s$
It follows, by Pythagoras, that
$$
r^2 = (t + 2s)^2 + s^2
$$
The radius, is also given by $r = \sqrt{2}t$ (by considering the corner of the large square!)
So we get
$$
2t^2 = t^2 + 4st + 4s^2 + s^2
$$
This simplifies to
$$
5s^2 + 4st - t^2 = 0
$$
This quadratic factorises to give
$$
(5s - t)(s+t) = 0
$$
The factorised equation yields either $t = 5s$ or $t =-s$. The negative solution doesn't make sense in this context, so we get the ratio of $1^2 : 5^2$, that is $1:25$