From the picture we see that the base is 8. The vertical line representing the length of x cuts the base into two equal parts. Knowing this fact allows us to use the Pythagorean Theorem.
(leg)^2 + (leg)^2 = (hypotenuse)^2
(4)^2 + (x)^2 = [sqrt{52}]^2
16 + x^2 = 52
x^2 = 52 - 16
x^2 = 36
sqrt{x^2} = sqrt{36}
x = -6 and x = 6.
We reject x = -6 because length is never negative.
Answer: x = 6