Let
x-------> the length side of the equilateral triangle
y-------> the length side of the square
we know that
The sum of the perimeters of an equilateral triangle and a square is
Perimeter of triangle is equal to
Perimeter of the square is equal to
so
------> equation
Find the area of equilateral triangle
Applying the law of sines
Find the area of the square
Fin the total area
----> equation
Substitute equation in equation
Using a graph tool
see the attached figure
we know that
the vertex of the graph is the point with the minimum total area
the vertex of the graph is the point
that means that
for the total area is equal to (is the minimum total area)
find the value of y
therefore
the answer is
the length side of the equilateral triangle is equal to
the length side of the square is equal to