Answer:
x = 64√13 / 9
Explanation:
Starting at the top triangle, the hypotenuse is found with Pythagorean theorem:
c² = a² + b²
c² = 4² + 6²
c = √52
Moving on to the 30-60-90 triangle, the hypotenuse if found with trigonometry:
sin 60 = √52 / c
c = √52 / sin 60
c = √52 / (√3 / 2)
c = 2√(52/3)
Next is the 45-45-90 triangle. The hypotenuse is again found with trig:
sin 45 = 2√(52/3) / c
c = 2√(52/3) / sin 45
c = 2√(52/3) / (√2 / 2)
c = 4√(26/3)
The next 30-60-90 triangle:
cos 30 = 4√(26/3) / c
c = 8√(26/9)
Another 30-60-90 triangle:
sin 60 = 8√(26/9) / c
c = 16√(26/27)
A 45-45-90 triangle:
sin 45 = 16√(26/27) / c
c = 32√(13/27)
The final 30-60-90 triangle:
sin 60 = 32√(13/27) / x
x = 64√(13/81)
x = 64√13 / 9