To find the length of $EF$, we can use the similarity of triangles in this problem.

Step-by-Step Solution

1. Identify Similar Triangles:

   • From the hint, we know that triangles $\triangle GHS$ and $\triangle FES$ are similar. This is because they have parallel sides and share a common angle at $S$, making the triangles similar by the Angle-Angle (AA) similarity criterion.

2. Set up a Proportion:

   • Since $\triangle GHS \sim \triangle FES$, the ratios of corresponding sides are equal: $\frac{GH}{FE} = \frac{HS}{ES} = \frac{GS}{FS}$

3. Plug in Known Values:

   • Given values:

     • $GH = 20 \, \text{m}$
     • $FE = x$ (the value we need to find)
     • $HS = 26 \, \text{m}$
     • $ES = 32.5 \, \text{m}$

   • We set up the proportion using $\frac{GH}{FE} = \frac{HS}{ES}$: $\frac{20}{x} = \frac{26}{32.5}$

4. Solve for $x$:

   • Cross-multiply to solve for $x$: $20 \times 32.5 = 26 \times x$ $650 = 26x$ $x = \frac{650}{26} = 25 \, \text{m}$

Conclusion

The length of $EF$ is $25 \, \text{m}$.

Would you like further details or have any questions?

Here are 5 related questions that might interest you:

1. How do you determine if two triangles are similar?
2. What are the properties of similar triangles?
3. Can this proportion method be used for all types of triangles?
4. How does the Angle-Angle (AA) criterion prove similarity?
5. What other methods can be used to solve for unknown sides in triangles?

Tip: When working with similar triangles, always double-check that corresponding sides are set up correctly in the ratio!