Math Help - How should the radius MP be drawn?

1. How should the radius MP be drawn?

I would really appreciate help with this one

The coloured parts space is limited by three semicircle archs of which the smaller has the radius r.
How should the radius MP be drawn, if you want it to divide the coloured space into two parts with
equally big areas?

2. Hello, Kevin!

The colored space is limited by three semicircle arcs of which the smaller has radius $r.$
How should the radius $MP$ be drawn, if you want it to divide the colored space
into two parts with equal areas?

Draw radius $MQ$ downward, perpendicular to $AB.$
Let $\theta = \angle PMQ.$

Ignore the two smaller semicircles for the moment.
If the shaded region is the semicircle $APB$, then $MQ$ bisects the area.

But a semicircle with area $\tfrac{1}{2}\pi r^2$ has been subtracted from quadrant $MQB$
. . and added to quadrant $MQA.$

For the new area is to be bisected by radius $MP$,
. . the area of sector $PMQ$ must equal $\tfrac{1}{2}\pi r^2$

The area of sector $PMQ \:=\:\tfrac{1}{2}(2r)^2\theta \:=\:2r^2\theta$

So we have: . $2r^2\theta \:=\:\tfrac{1}{2}\pi r^2 \quad\Rightarrow\quad \theta \:=\:\tfrac{\pi}{4}$

Hence: . $\angle PMQ \:=\:\angle AMP \:=\:\tfrac{\pi}{4}$

Therefore, we must draw radius $MP$ so that angle $AMP = 45^o.$

3. Thank U so much!