The first two things to recognize regarding our tirangle are 1) it is a right triangle and 2) it is an isosceles triangle.  The two congruent sides tell us that the two non-right angles are also congruent, and a little quick math tells us that they each equal 45 degrees.  This means our right triangle is not just any right triangle but a 45-45-90 triangle.

This is important because the sides of every 45-45-90 triangle follow the same ratio.  The two legs are obviously always congruent to each other (being isosceles), but to find the hypotenuse, we simply have to multiply the length of a leg by .

Given this fact we would be in good shape if we had the length of a leg and needed the hypotenuse.  But we have the hypotenuse and need the leg, which we means we need to work backwards going this way, we need to divide the length of the hypotenuse by . Therefore,

However, general practice in mathematics doesn't allow us to leave a square root in the denominator.  We solve this problem by rationalizing the denominator, which is accomplished by multiplying the numerator and the denominator by .

This effectively eliminates the square root in the denominator and provides our answer.

A triangle with the sum of two angles equaling the third is a  triangle

We begin by realizing that the midpoints of the sides of our outer square divide each side in half.  Furthermore, the sides of our second square connecting these midpoints form four right triangles in each corner of our largest square.

But these right triangles are special right triangles.  They are 45-45-90 triangles, which means we can find the hypotenuse (and thus the side of our second square) by multiplying the length of the leg by .  Therefore the length of a side of our second square is .

We now repeat the process, beginning by forming four new 45-45-90 triangles 

To find the hypotenuse of each of these triangles (and thus the side length of our third square), we simply multiply by  again.

We then repeat the process one final time, multiplying by  again.

Our final hypotenuse and thus the side of our innermost square is .

One option is to use the Pythagorean Theorem.

Since we have an isosceles triangle, both legs must be congruent.

Plug in to get your answer.

Or, you can remember the 45-45-90 identity, which states that the hypotenuse is  times the leg. 

Write the Pythagorean Theorem.

In a 45-45-90 triangle, the length of the legs are equal, which indicates that:

Rewrite the formula and substitute the known sides.

The lengths of the triangle are:  

Sum the three lengths for the perimeter.

If 3 is one of the legs, then the hypotenuse is .

If 3 is the hypotenuse, then the legs are or equivalently

Divide the length of the hypotenuse by to get the length of the legs:

The perimeter of a square can be found using the formula , where P is the perimeter and s is the length of the side of the square.

The diagonal of a square forms the hypotenuse of a 45-45-90 triangle, where each leg is the side of the square. In a 45-45-90 triangle, the ratio of the hypotenuse to the leg is , so the diagonal of this square is .

Solving this problem begins with realizing that all three of our triangles are not only right triangles but isosceles and are therefore 45-45-90 triangles.  That means in each triangle to get from the length of a leg to the length of the hypotenuse, we simply multiply by .  Therefore, the hypotenuse of our bottom triangle is

However, the hypotenuse of the bottom triangle is also the leg of the middle triangle.  To find the hypotenuse of this triangle, we simply repeat the process.

However, again the hypotenuse of the middle triangle is also the leg of the upper triangle.  To find , the hypotenuse of the upper triangle, we simply repeat the process one last time.