The major arc DGF and the minor arc FD both make up the circle, so their measures should add to 360°.
(arc DGF) + 153° = 360°
arc DGF = 207°
The formula for the area of a ∆ is 1/2bh. So since you already have the area, solve for x.
24=

(8)x
24=4x
6=x
The missing side is 6 inches long.