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The problem presents two similar pentagons \( JKLMN \) and \( PQRST \), and asks to find the length \( x \) of the side \( ST \) in the pentagon \( PQRST \).

Since the pentagons are similar, the ratio of corresponding sides will be equal. Let's identify the corresponding sides and set up a proportion.

From the image:
- The side \( MN = 4 \) in pentagon \( JKLMN \) corresponds to the side \( ST = x \) in pentagon \( PQRST \).
- The side \( LJ = 5 \) in pentagon \( JKLMN \) corresponds to the side \( RP = 7 \) in pentagon \( PQRST \).

Now, set up the proportion based on these corresponding sides:
\[
\frac{MN}{ST} = \frac{LJ}{RP}
\]
Substitute the known values:
\[
\frac{4}{x} = \frac{5}{7}
\]

To solve for \( x \), cross-multiply:
\[
4 \cdot 7 = 5 \cdot x
\]
\[
28 = 5x
\]
Now, divide both sides by 5:
\[
x = \frac{28}{5} = 5.6
\]

Thus, the length of \( ST \) is \( x = 5.6 \).

Would you like further details or have any questions about this solution?

Here are 5 questions to deepen your understanding:
1. How do we determine which sides of two polygons correspond when they are similar?
2. What would change in the proportion if the polygons were not similar?
3. Can you set up another proportion using different pairs of corresponding sides?
4. What is the significance of similarity in polygons when solving problems like these?
5. How can we verify that our calculated value for \( x \) is correct?

**Tip:** In similar polygons, corresponding angles are equal, and the ratios of corresponding sides are proportional.

The pentagons JKLMN and PQRST are similar. Find the length x of ST.

Learn how to find the length of ST in the similar pentagons JKLMN and PQRST. This problem covers key geometric concepts, including the proportionality theorem and similarity of polygons. Suitable for students in Grades 7-9.

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