A slightly more complicated way :-)
The arc can be replaced with a fillet, of same radius.
Trim as per Pilou.
Extrude of new arc and 2 edges, with cap ends, forms plane. Delete extrude except for plane.
Boolean Union .
Merge curved Edges.
Thanks for all the answers which helped me to understand that I cannot plane curves that do not end at the same point even if on the same plane. Interestingly there was a time or two that I tried a haphazard combination of joins/boolean merge/planars that it actually seemed to work but maybe I accidentally drew or created segments with booleans without realizing and this allowed for plane to work. In the end the suggestion from Brian worked very well after I unioned many rectangles into a single face.
