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Esteban used a hinged mirror to create an equilateral triangle, as shown in the diagram below. If the area of the shaded region is 11.42 square inches, what is the area of the entire equilateral triangle? Justify your solution.

A triangle where from each vertex, a dotted line is drawn to the midpoint of the opposite side. The bottom left right triangle is shaded.

Each of the triangles is a reflection of another triangle, so all of the triangles are congruent.