ABC is an isosceles triangle with AB = AC = 2a and BC = a. If AD ⊥ BC, find the length of AD.

Given: AB = AC = 2a, BC = a and AD ⊥ BC. Since Δ ABC is isosceles with AB = AC, the perpendicular from A to BC will bisects BC, meaning: BD = DC = In right-angled triangle ADB, AB = 2a BD = Using the Pythagorean theorem, AB2 = AD2 + BD2 ⇒ (2a)2 = AD2 + 2 ⇒ 4a2 = AD2 + ⇒ AD2 = 4a2 - ⇒ AD2 = ⇒ AD2 = ⇒ AD = Hence, the length of the perpendicular from A to BC is units.