https://file.aiquickdraw.com/mathbot/file/17335334347084ni2bmz2.jpg

To solve the problem, we can use the property of similar triangles: corresponding sides of similar triangles are proportional. Let's analyze the given data and solve it step-by-step.

---

### Step 1: Identify corresponding sides
From the diagram:
- \( LMN \) is similar to \( OPQ \).
- Corresponding sides are:
  - \( LM \) corresponds to \( OP \),
  - \( MN \) corresponds to \( PQ \),
  - \( LN \) corresponds to \( OQ \).

### Step 2: Set up the proportion
We know:
- \( LN = 23 \),
- \( MN = 17 \),
- \( OQ = 11.2 \).

The proportion for finding \( OP \) is:
\[
\frac{LM}{OP} = \frac{LN}{OQ}.
\]

Substitute the known values:
\[
\frac{17}{OP} = \frac{23}{11.2}.
\]

---

### Step 3: Solve for \( OP \)
Cross-multiply to solve:
\[
17 \times 11.2 = 23 \times OP.
\]

Simplify:
\[
190.4 = 23 \times OP.
\]

Solve for \( OP \):
\[
OP = \frac{190.4}{23} \approx 8.3.
\]

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### Final Answer:
The length of side \( OP \) is approximately **8.3 units**.

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Would you like further explanation or details on this solution? Here are related questions for exploration:

1. How can we verify that two triangles are similar using angles or sides?
2. What happens to the ratio of corresponding sides in a scale enlargement or reduction?
3. How can the Pythagorean theorem help confirm side lengths in similar triangles?
4. What are some real-world applications of similar triangles?
5. What are the properties of proportional relationships in geometry?

**Tip:** Always check that your proportions match the correct corresponding sides when solving similar triangle problems!

The problem involves finding the length of side OP in two similar triangles, LMN and OPQ. The side lengths are given as LM = 17, LN = 23, and OQ = 11.2, and we need to determine the length of OP.

Learn how to solve the problem of finding the length of side OP in similar triangles LMN and OPQ. This step-by-step solution covers proportional relationships and the property of similar triangles. Suitable for students in Grades 7-9.

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