ABC is an equilateral triangle of side 2a. Find each of its altitudes.

In the equilateral ΔABC, we see that AB = BC = CA = 2a [From the figure shown above]

AD ⊥ BC [Construction]

⇒ BD = CD = 1/2 BC = a [Since the perpendicular drawn from a vertex to the opposite side bisects the opposite side in an equilateral triangle]

In ΔADB,

AB² = AD² + BD²

AD² = AB² - BD² 

AD² = (2a)² - a²

AD² = 4a² - a²

AD² = 3a²

AD = 3a

⇒ AD = √3a units

Similarly, we can prove that, BE = CF = √3a units