A square and an equilateral triangle have equal perimeters. The area of the triangle is 9√3 square inches. Expressed in inches the diagonal of the square is:

1962 AMC 12 Problem 6

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A square and an equilateral triangle have equal perimeters. The area of the triangle is

9\sqrt3

square inches. Expressed in inches the diagonal of the square is:

\dfrac92

2\sqrt5

4\sqrt2

\dfrac{9\sqrt2}{2}

none of these

Use

\frac{\sqrt3}{4}s^2

for the triangle’s area

Equate the two perimeters, then multiply the square’s side by

\sqrt2

s

is the triangle’s side, then

\frac{\sqrt3}{4}s^2=9\sqrt3,

s=6.

Its perimeter is

18,

making the square’s side

\frac{18}{4}=\frac{9}{2}.

The square’s diagonal is therefore

\frac{9\sqrt2}{2}.

Thus, the correct answer is D.