Solve for x in the given triangle.

Let's analyze the given triangle ABC where angle A is 30 degrees, side BC is 8 and we need to find the length of a segment (let's call it CD, where D lies on AB) which is 'x', and angle ACB is 90 degrees and AD = DC. Since AD = DC, triangle ADC is an isosceles triangle. Therefore, angle DAC = angle DCA = 30 degrees. Then, angle CDB is an exterior angle to triangle ADC, so angle CDB = angle DAC + angle DCA = 30 + 30 = 60 degrees. Now, in triangle BCD, angle BCD = 90 - 30 = 60 degrees. So, triangle BCD is a triangle with angle BCD = 60 degrees and angle CDB = 60 degrees, which means the remaining angle CBD = 180 - 60 - 60 = 60 degrees. Hence, triangle BCD is an equilateral triangle, so BC = CD = BD = 8. Therefore, x = 8. **Final Answer: The final answer is $$\boxed{8}$$**