What is the largest rectangle that can be inscribed in an equilateral triangle with sides of 12?

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Answer 1 · 2015-12-11T15:48:54

#(3, 0), (9, 0), (9, 3 sqrt 3), (3, 3 sqrt 3)#

#Delta VAB; P, Q in AB ; R in VA; S in VB#

#A = (0, 0), B = (12, 0), V = (6, 6 sqrt 3)#

#P = (p, 0), Q = (q, 0), 0 < p < q < 12#

#VA: y = x sqrt 3 Rightarrow R = (p, p sqrt 3), 0 < p < 6#

#VB: y = (12 - x) sqrt 3 Rightarrow S = (q, (12 - q) sqrt 3), 6 < q < 12#

#y_R = y_S Rightarrow p sqrt 3 = (12 - q) sqrt 3 Rightarrow q = 12 - p#

#z(p) = #Area of #PQSR = (q - p) p sqrt 3 = 12p sqrt 3 - 2p^2 sqrt 3#

This is a parabola, and we want the Vertex #W#.

#z(p) = a p^2 + bp + c Rightarrow W = ((-b)/(2a), z(-b/(2a)))#

#x_W = (-12 sqrt 3)/(-4 sqrt 3) = 3#

#z(3) = 36 sqrt 3 - 18 sqrt 3#