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To find the length of \( EF \), we can use the fact that triangles \( \triangle GHS \) and \( \triangle FES \) are similar, as noted in the hint. This similarity implies that the ratios of corresponding sides are equal. Here’s the step-by-step approach:

1. **Identify Corresponding Sides in Similar Triangles**:
   - Since \( \triangle GHS \sim \triangle FES \), we know:
     \[
     \frac{GH}{FE} = \frac{HS}{ES} = \frac{GS}{FS}
     \]

2. **Set Up the Proportion**:
   - From the diagram:
     - \( GH = 20 \) m
     - \( HS = 26 \) m
     - \( ES = 32.5 \) m
   - We want to find \( FE \), so we’ll use the ratio:
     \[
     \frac{GH}{FE} = \frac{HS}{ES}
     \]

3. **Plug in the Known Values and Solve for \( FE \)**:
   \[
   \frac{20}{FE} = \frac{26}{32.5}
   \]
   Cross-multiply to solve for \( FE \):
   \[
   20 \times 32.5 = FE \times 26
   \]
   \[
   650 = 26 \times FE
   \]
   \[
   FE = \frac{650}{26}
   \]
   \[
   FE = 25 \text{ m}
   \]

So, the length of \( EF \) is **25 meters**.

Would you like further explanation on any part of this process?

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Here are 5 related questions for further practice:
1. What is the definition of similar triangles, and why are corresponding side ratios equal?
2. How can you determine if two triangles are similar based on their angles and sides?
3. What is the relationship between the scale factor and corresponding side lengths in similar triangles?
4. If \( GH \) was measured to be different, say 22 m, how would you solve for \( EF \) with the new value?
5. Can similar triangles be used in three-dimensional space, and if so, how?

**Tip**: When working with similar triangles, always ensure that you correctly match corresponding sides to maintain the proportion.

To measure the distance EF across a lake, a surveyor at S locates points E, F, G, and H as shown. What is the length of EF?

Learn how to calculate the distance EF across a lake by setting up a proportion between similar triangles GHS and FES. This problem applies the triangle similarity theorem and proportional relationships, ideal for students in Grades 8-10.

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