Let's look at the given picture. As we can see, there are four 45^(∘)-45^(∘)-90^(∘) triangles. We will name the vertices and the missing inner sides in our figure.
First of all let's analyze △ ABF. The length of the leg is a, and the length of the hypotenuse is 6. By the 45^(∘)-45 ^(∘)-90 ^(∘) Triangle Theorem, the length of the hypotenuse is sqrt(2) times the length of a leg, a.
asqrt(2) = 6
Let's solve the above equation for a.
The value of a is 3sqrt(2). Let's add this information to our picture.
Now let's evaluate the value of b, which is the leg of an isosceles right triangle BCF. In this triangle the length of the hypotenuse is 3sqrt(2). We will use the fact that the length of the hypotenuse in 45^(∘)-45 ^(∘)-90 ^(∘) triangles is sqrt(2) times the length of its leg, b.
bsqrt(2) = 3sqrt(2) ⇒ b= 3
Let's add this information to our picture.
Moving to △ CDF, we can evaluate the value of c. Again we will use the fact that in an isosceles right triangle the length of the hypotenuse, which is 3, is sqrt(2) times the length of its leg, c.
csqrt(2) = 3
Let's solve the above equation for c.
The value of c is 32sqrt(2). For the last time, let's add the information we just found to our picture.
Finally we can evaluate the value of x using the fact that △ FDE is also a 45^(∘)-45 ^(∘)-90 ^(∘) triangle. Therefore, the length of the hypotenuse of this triangle, which is 32sqrt(2) is sqrt(2) times the length of its leg, x.
xsqrt(2) = 3/2sqrt(2) ⇒ x=3/2
The value of x is 32.
