square in a segment

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

square in a segment
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 $1:25$
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