Answer: C
Since E and F are the midpoints of AB and AC respectively, ∴ \(BC = 2EF = 2 \times 4\) cm = 8 cm. Also, AD ⊥ BC and AB = AC ∴ \(BD = \frac{1}{2}BC = 4\) cm. Now, from right-angled triangle ABD, \(AB = \sqrt{AD^2 + BD^2}\) \(= \sqrt{(2\sqrt{5})^2 + 4^2}\) \(= \sqrt{20 + 16} = \sqrt{36} = 6\) ∴ The length of AB is 6 cm.
Since E and F are the midpoints of AB and AC respectively, ∴ \(BC = 2EF = 2 \times 4\) cm = 8 cm. Also, AD ⊥ BC and AB = AC ∴ \(BD = \frac{1}{2}BC = 4\) cm. Now, from right-angled triangle ABD, \(AB = \sqrt{AD^2 + BD^2}\) \(= \sqrt{(2\sqrt{5})^2 + 4^2}\) \(= \sqrt{20 + 16} = \sqrt{36} = 6\) ∴ The length of AB is 6 cm.