As a complementary information, here is a way it can be done with MetaPost (code inserted in a LuaLaTeX program for typesetting convenience):
u := 2cm;
path horizontal, bisector;
horizontal = (left -- right) scaled u;
bisector1 = horizontal rotated 45;
bisector2 = bisector1 rotated (-90);
fill buildcycle(bisector1, halfcircle scaled (2u) rotated -90, bisector2)
draw fullcircle scaled (2u) withpen pencircle scaled 1.25;
draw bisector1; draw bisector2;
draw horizontal; draw (down--up) scaled u;
To fill the circular sector, the
buildcycle macro of MetaPost has been used here. It takes two or more paths as arguments and tries to return the path bordering the area they delimit together (if it exists).
It works best when each path has only one intersection point with the following in the list, though. This is the reason I chose the right half of the given circle (
halfcircle scaled (2u) rotated -90) to be given as argument to
buildcyclebesides the two bisectors, and not the full circle:
buildcycle could have been utterly confused by the two intersection points the full circle has with each bisector.
It would have also worked if I had given
fullcircle scaled (2u) rotated 180 as argument, but to understand why, it is necessary to read this discussion and/or the manual, p. 30-32.