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

$(3, 0), (9, 0), (9, 3 \sqrt{3}), (3, 3 \sqrt{3})$ $(3, 0), (9, 0), (9, 3 sqrt 3), (3, 3 sqrt 3)$

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

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

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

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

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

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

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

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

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

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

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