Equilateral triangle with perpendicular height h

Question

I don't know how to go about this question :

An Equilateral triangle has perpendicular height h. A) find the length of each side, in terms of h B) without using your calculator, show that the area of the triangle is h^2/√3

 

Do I use Pythagoras? But that would be if the sides are length l : l²=(1/2l)²+h². I don't see a way to do that in terms of h... 

Answer 1

Since an equilateral triangle has (by definition) 3 equal sides, if I draw a perpendicular line from the base to the opposite vertex, I have not only drawn an angle bisector, but I have bisected the the base.  This means that I have created two new triangles which are a 30/60/90 (degree) triangles.  If my original equilateral triangle has sides x, then each 30/60/90 triangle have sides x, ¹/₂ x and h (in terms of x, h = ^√3/₂ x). using the Pythagorean Theorem, we can find x in terms of h {h² + (¹/₂ x)² = x²}. Solving for x we would get x = ^h/_√3 (or rationalizing the denominator, x = ^h√3/₃). Now using the formula for the area of a triangle (A = ¹/₂ bh) we get A = ¹/₂ (^h/_√3)(h) = h²/_√3 (or h²√3/₃)