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To solve for \( x \), the length of \( \overline{JK} \), we need to use the fact that the two pentagons \( ABCDE \) and \( JKLMN \) are similar. Similar polygons have corresponding sides that are proportional.

### Step-by-step solution:

1. **Identify corresponding sides:**
   - In \( ABCDE \), the side \( \overline{BC} = 4 \), and in \( JKLMN \), the corresponding side is \( \overline{KL} = 5.6 \).
   - In \( ABCDE \), the side \( \overline{DE} = 5 \), and in \( JKLMN \), the corresponding side is \( \overline{NM} = 7 \).

2. **Set up the proportion:**
   Since the polygons are similar, the ratio of corresponding sides should be equal. So, we can set up the proportion:
   \[
   \frac{JK}{AB} = \frac{KL}{BC} = \frac{NM}{DE}
   \]

3. **Use the known sides:**
   From the diagram, we know:
   \[
   \frac{KL}{BC} = \frac{5.6}{4} = 1.4
   \]
   This means that all the sides in \( JKLMN \) are 1.4 times the length of the corresponding sides in \( ABCDE \).

4. **Solve for \( JK \):**
   The side \( AB = 2 \), and since \( JK \) corresponds to \( AB \), we apply the ratio:
   \[
   JK = AB \times 1.4 = 2 \times 1.4 = 2.8
   \]

Thus, the length of \( \overline{JK} = 2.8 \).

Let me know if you'd like further details or have any questions! Here are some related questions you can explore:

1. What is the length of side \( \overline{JK} \) in relation to the other sides?
2. How do you determine the scale factor between two similar polygons?
3. What would happen to the shape of the pentagons if the scale factor were different?
4. How can you check if two polygons are similar by using angles and sides?
5. What would the length of side \( \overline{LM} \) be in a different pentagon with the same ratio?

**Tip:** When dealing with similar polygons, always remember to check both the corresponding angles and the corresponding side ratios to confirm similarity.

The pentagons ABCDE and JKLMN are similar. Find the length x of JK.

Learn how to find the length of side JK in similar pentagons ABCDE and JKLMN using proportional relationships between corresponding sides. This geometry problem is ideal for students in Grades 8-10.

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