(a)

∠ QHF = 90°

QF is the hypothenuse of the right-angled triangle QHF

QH is the adjacent side to ∠ EQH

QE = QH = Radius of the circle = 8$\sqrt{3}$

QF = QE + EF

EF = QF - QE

CAH

cos(θ) = $\frac{\mathrm{Adjacent}}{\mathrm{Hypothenuse}}$

cos 60° = $\frac{8\sqrt{3}}{\mathrm{|QF|}}$

Multiply both sides by the denominator QF

QF x cos 60° = 8$\sqrt{3}$

Divide both sides by cos 60°

QF = $\frac{8\sqrt{3}}{\mathrm{cos\; 60°}}$

QF = 27.71 cm

EF = QF - QE

EF = 27.71 cm - 8$\sqrt{3}$ cm

EF = 13.856 cm ≈ 13.9 cm

(b)

Shaded portion = Area of ∆QHF - Area of the sector of the circular shape

Area of ∆ QHF = $\frac{\mathrm{|QH|\; x\; |HF|}}{2}$

TOA

tan(θ) = $\frac{\mathrm{Opposite}}{\mathrm{Adjacent}}$

tan 60° = $\frac{\mathrm{|HF|}}{8\sqrt{3}}$

Multiply both sides by the denominator 8$\sqrt{3}$

|HF| = tan 60° x 8$\sqrt{3}$

|HF| = 24 cm

Area of ∆ QHF = $\frac{8\sqrt{3}\mathrm{x\; 24}}{2}$

Area of ∆ QHF = 166.28 cm²

Area of the sector of the circular shape = $\frac{60}{360}$ x π x QH x QH

Area of the sector of the circular shape = $\frac{60}{360}$ x $\frac{22}{7}$ x 8$\sqrt{3}$ x 8$\sqrt{3}$

Area of the sector of the circular shape = 100.57 cm²

Shaded portion = 166.28 cm² - 100.57 cm²

Shaded portion = 65.71 cm² ≈ 65.7 cm² (three significant figures)