Consider the centre of circle to be O, ends of chord as A, B and their mid-point as M.
We have, radius, r = 5√2 cm and chord length, l = 10 cm.
Now you can observe, ∆OAM & ∆OBM are isosceles & right angled at M.
Here, angle AOM = angle BOM = 45°, then area of ∆ AOB = 25 cm²
and the area of sector AOB = ¼(circle area) = (12.5)π cm², since angle AOB = 90°.
Then area of segments are 50(π-2)/4 and 50(3π+2)/4.