Based on the triangles formed by the vertical lines on the figure, we can see that the ratio of the length of the longer segment to the shorter segment is 2:1. We already know that the shorter segment has a length of 2x, so the longer segment must have a length of 4x.
Next, we can apply the Pythagorean theorem to the right triangle on the left side of the figure. We have:
(2x)^2 + 6^2 = (4x)^2
Simplifying this equation gives:
4x^2 + 36 = 16x^2
Subtracting 4x^2 from both sides gives:
36 = 12x^2
Dividing both sides by 12 gives:
3 = x^2
Taking the square root of both sides gives:
x = ±?3
However, since we are dealing with a length, we can discard the negative solution. Therefore, the value of x is ?3.
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