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In the given diagram, we need to find the length of segment \( HI \), where:

- \( JFGI \) forms a trapezoid, and
- \( F \) and \( H \) are points on the sides of the trapezoid.

The lengths provided are:
- \( JF = 20 \),
- \( FG = 10 \),
- \( GH = 13 \),
- And the segments \( FH \) and \( JI \) are parallel.

This is a typical case where we use properties of similar triangles or the trapezoid rule to solve for unknown lengths, but the geometry and arrangement suggest similarity-based reasoning.

### Step-by-step approach:
1. **Identify similar triangles**: 
   Triangles \( JFI \) and \( HGI \) are similar due to the parallel segments \( JI \parallel GH \).

2. **Use the ratio of sides**:
   Since the triangles are similar, we can set up a proportion between the corresponding sides of the triangles.

   The ratio between \( JF \) and \( FG \) (base lengths) is:
   \[
   \frac{ JF }{ FG } = \frac{ HI }{ GH }
   \]
   Plugging in the known values:
   \[
   \frac{ 20 }{ 10 } = \frac{ HI }{ 13 }
   \]
   Simplifying the ratio:
   \[
   2 = \frac{ HI }{ 13 }
   \]

3. **Solve for \( HI \)**:
   Now, solve for \( HI \) by multiplying both sides of the equation by 13:
   \[
   HI = 2 \times 13 = 26
   \]

Thus, the length of segment \( HI \) is \( \boxed{26} \).

---

Would you like further details or have any questions?

### 5 Related Questions:
1. How can you identify similar triangles in other types of problems?
2. What are the key properties of trapezoids used in geometric problems?
3. How does the parallel lines rule apply to triangle similarity?
4. Can you solve this using the midpoint theorem or another geometric concept?
5. How does the concept of proportionality apply in different geometric shapes?

### Tip:
When working with similar triangles, always focus on the corresponding sides and their proportional relationships to set up equations.

Find the length of segment HI in the given trapezoid JFGI, where the lengths JF = 20, FG = 10, GH = 13, and JI is parallel to GH.

This geometry problem focuses on finding the length of segment HI in a trapezoid using the properties of similar triangles. Given values include JF = 20, FG = 10, GH = 13. The solution involves setting up a ratio based on the similarity of triangles JFI and HGI. Suitable for students in Grades 8-10.

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