Q:

Geometry: Arcs of a Circle

I think this is solved using the Pythagorean theorem but cant figure out how to get the length of PQ. Any help?

A:

$PQ$ is the base of an isosceles triangle with height $\frac{1}{2}(XY) = 4$. The sides of the isosceles triangle are radii of the circles. The radius of each circle is $QR = PS = PQ + 1$, since we know $RP = QS = 1$.
Cutting the isosceles triangle along its height $XZ$, we have the right triangle $PXZ$. Its base is half of $PQ$, its height is $XZ = 4$, and its hypotenuse has length $PQ + 1$, so the Pythagorean theorem gives 
$$ \bigg(\frac{1}{2}PQ\bigg)^2 + 4^2 = (PQ + 1)^2.$$

