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The problem involves a triangle where line segments are given specific relationships, and we need to find the length of \( ON \).

### Given:
- \( OP \parallel LM \) (indicating parallel lines).
- \( NP = ON + 4 \) (the length of \( NP \) is 4 units longer than \( ON \)).
- \( LN = 15 \) (the length of \( LN \)).
- \( NM = 21 \) (the length of \( NM \)).

We need to find the length of \( ON \).

The fact that \( OP \parallel LM \) and \( NP = ON + 4 \) implies a relationship between the segments in the triangle that we can analyze using similar triangles and proportionality.

I'll begin by working out the necessary steps and equations to solve this problem. Let's proceed with solving it now.To solve this problem, let's break down the given information and use the proportionality of similar triangles, since \( OP \parallel LM \) indicates that triangles \( ONP \) and \( NLM \) are similar.

### Step 1: Identify the proportionality of sides
From the given information:
- \( OP \parallel LM \), which means the triangles \( ONP \) and \( NLM \) are similar.
- The proportionality between the sides of similar triangles is given by:
\[
\frac{ON}{LN} = \frac{NP}{NM}
\]

### Step 2: Write the equation based on the proportionality
We know:
- \( LN = 15 \)
- \( NM = 21 \)
- \( NP = ON + 4 \)

Substitute these into the proportionality equation:
\[
\frac{ON}{15} = \frac{ON + 4}{21}
\]

### Step 3: Solve for \( ON \)
Now, we solve for \( ON \). Cross-multiply:
\[
21 \times ON = 15 \times (ON + 4)
\]

Expand both sides:
\[
21 \times ON = 15 \times ON + 60
\]

Now, collect like terms:
\[
21 \times ON - 15 \times ON = 60
\]
\[
6 \times ON = 60
\]

Solve for \( ON \):
\[
ON = \frac{60}{6} = 10
\]

### Step 4: Final Answer
The length of \( ON \) is 10 units.

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Would you like a more detailed explanation or additional questions on this problem? Here are 5 related questions for further exploration:

1. How would the answer change if \( NP \) was 3 more than \( ON \) instead of 4?
2. How does the concept of similar triangles apply in real-world problems?
3. Can this method be applied to other geometric figures besides triangles?
4. How can you prove that two triangles are similar?
5. What is the importance of parallel lines in triangle similarity?

**Tip:** When dealing with proportionality in similar triangles, always set up the ratios carefully and cross-multiply to simplify the equation step by step!

In the diagram below, OP is parallel to LM. If NP is 4 more than ON, LN = 15, and NM = 21, find the length of ON. Figures are not necessarily drawn to scale. State your answer in simplest radical form, if necessary.

Solve a geometry problem involving similar triangles and proportionality. Given OP || LM, NP is 4 more than ON, LN = 15, and NM = 21, calculate the length of ON in simplest radical form. This step-by-step solution is perfect for students in Grades 9-10.

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