Similar Trapezoids: Calculate the Perimeter of Polygon I with 8, and 18 Units

To solve the problem of calculating the perimeter of trapezoid $I$, which is similar to another trapezoid shown, follow these steps:

Firstly, identify the given dimensions for the smaller trapezoid (base lengths of 6 and 8) and then use the similarity property of trapezoids to establish scale.

Use the length on the smaller trapezoid: Smaller side = $6$, and Larger side = $8$, with their larger comparative sides corresponding to $9$ and $y$, the base of the larger trapezoid at $18$. Consider the similarity ratio from given lengths: $\frac{9+18}{6+8} = \frac{27}{14}$ implies scaling the length to the known marked length.

Determine the scale factor: Let the shorter sides $A = 6$ and $B = 9$, both primes give proportions, thus, solve the perimeter computation.

Perimeter of trapezoid $I$ involves using this ratio for conversion of each side:

• Shorter side $6$: Larger comparative side $= 9$ (same order)
• Longer side $8$: Base at $13.5$, matching similarly
• Total perimeter involves calculating: $5 \cdot (1) + 6 \cdot (1.5) + 7.5 \cdot (1) = 5$, cumulative correct adding.

Thus, summing the entire side runs yields $13.5 + 6 + 9 + 15$. Calculation of this comparable side perimeter sums up to $44$.

Given similar trapezoids and perimeter rule, we conclude that the perimeter of polygon $I$ equals $44$, affirming choice 3 is correct.

Therefore, the perimeter of polygon $I$ is $\boxed{44}$.