EB=(r−8)+r=2r−8
From the Theorem of Intersecting Chords:
AE×EB=CE×DE
8(2r−8)=12(20)
r=19 cm
r−8=11 cm
Apply Cosine Law to Triangle EOC
122=r2+(r−8)2−2r(r−8)cosθ
122=192+112−2(19)(11)cosθ
cosθ=192+112−1222(19)(11)
θ=36.04∘
Area of AEC = Area of Sector AOC – Area of Triangle EOC
AAEC=12r2θrad−12r(r−8)sinθdeg
AAEC=12(192)(36.04∘)(π180∘)−12(19)(11)sin36.04∘
AAEC=52.05 cm2 ← Answer: [ B ]