geometry question on areas in arcs PS is a line segment of length 4 and O is the midpoint of PS. A semicircular arc is drawn with PS as diameter. Let X be the midpoint of this arc. Q and R are points on the arc PXS such that QR is parallel to PS and the semicircular arc drawn with QR as diameter is tangent to PS. How can I get the area of the region QXROQ bounded by the two semicircular arcs?

This diagram will help:

![enter image description here](

Note that I have connected $OQ$ and $OR$ - these are both equal to the radius of the blue circle centered at $O$ and so their lengths must be equal to $2$.

I have also labelled the intersection of $QR$ and $OX$ as $Y$. From the question statement it is clear that:$$QY=YR=OY$$ and I have labelled this length as $a$.

Now note in triangle $OYR$ that $OY=YR$ and angle $OYR=90^{\circ}$. This implies that angles $YOR=YRO=45^{\circ}$.

Using similar arguments you can deduce that angles $YOQ=YQO=45^{\circ}$.

This should give you enough information to be able to calculate the value of $a$ and hence calculate the area you require.