First, convert the mixed fractions into improper fractions: $3 \frac{1}{2} = \frac{7}{2}$ $8 \frac{3}{4} = \frac{35}{4}$

Now, find the greatest common divisor (GCD) of $7/2$ and $35/4$. The GCD of two fractions $a/b$ and $c/d$ is given by:

$\text{GCD}(a/b, c/d) = \frac{\text{GCD}(a, c)}{\text{LCM}(b, d)}$

1. Find the GCD of the numerators (7 and 35): $\text{GCD}(7, 35) = 7$
2. Find the LCM of the denominators (2 and 4): $\text{LCM}(2, 4) = 4$

Now, the GCD of the two fractions is: $7/4$

Convert $7/4$ back to a mixed fraction: $7/4 = 1 \frac{3}{4}$

The greatest length $x$ is $1 \frac{3}{4}$ m, which corresponds to Option D.