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The two pentagons, \( JKLNM \) and \( PQRST \), are **similar**, meaning their corresponding sides are proportional. We are asked to find the length \( x \) of \( TP \).

We can set up a proportion based on corresponding sides of these two pentagons. Let's first identify the corresponding sides from both figures:

- \( JK \) corresponds to \( PQ \)
- \( KL \) corresponds to \( QR \)
- \( LM \) corresponds to \( RS \)
- \( MN \) corresponds to \( ST \)
- \( NJ \) corresponds to \( TP \)

The corresponding sides and their lengths are:

- \( JK = 2 \), \( PQ = 3.2 \)
- \( KL = 3 \), \( QR = 4.8 \)
- \( LM = 3 \), \( RS = 4.8 \)
- \( MN = 2 \), \( ST = 3.2 \)

Now we use the proportionality property for corresponding sides. The ratio between any pair of corresponding sides is the same:

\[
\frac{JK}{PQ} = \frac{KL}{QR} = \frac{LM}{RS} = \frac{MN}{ST} = \frac{NJ}{TP}
\]

Let’s calculate the ratio using known corresponding sides. We'll use \( JK \) and \( PQ \):

\[
\frac{JK}{PQ} = \frac{2}{3.2} = \frac{5}{8}
\]

Now, to find \( TP = x \), we apply this same ratio to the corresponding sides \( NJ \) and \( TP \):

\[
\frac{NJ}{TP} = \frac{5}{8} \quad \Rightarrow \quad \frac{6}{x} = \frac{5}{8}
\]

Now, solve for \( x \):

\[
6 \times 8 = 5 \times x \quad \Rightarrow \quad 48 = 5x \quad \Rightarrow \quad x = \frac{48}{5} = 9.6
\]

Thus, the length of \( TP \) is \( \boxed{9.6} \).

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Would you like a detailed explanation or have any questions?

Here are five related questions:
1. How do you determine corresponding sides in similar polygons?
2. What is the significance of the proportionality of corresponding sides in similar figures?
3. Can the ratio between two polygons be used to find angles as well?
4. How would the problem change if the pentagons were not similar?
5. How do you set up proportions when working with similar triangles?

**Tip:** When dealing with similar figures, always check that the ratio between corresponding sides remains constant for all sides to confirm similarity.

The pentagons JKLNM and PQRST are similar. Find the length x of TP.

Learn how to find the length of TP in similar pentagons JKLNM and PQRST using proportions. This geometry problem covers key concepts of polygon similarity and proportionality of corresponding sides. Suitable for students in Grades 7-9.

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