https://file.aiquickdraw.com/mathbot/file/1725982419751j7ktzkmg.jpg

We can solve this problem using the **Intersecting Chords Theorem**. According to this theorem, if two chords intersect inside a circle, the products of the lengths of the segments of each chord are equal. 

The two chords intersect at point \( N \), and the theorem gives us the equation:

\[
JK \times KN = LN \times NM
\]

From the diagram:

- \( JK = 11 \)
- \( KN = 5 \)
- \( LM = 10 \)
- We are looking for \( LN \)

Using the formula:

\[
11 \times 5 = LN \times 10
\]

Now, let's solve for \( LN \):

\[
55 = LN \times 10
\]

\[
LN = \frac{55}{10} = 5.5
\]

So, the length of \( LN \) is \( 5.5 \).

Let me know if you'd like more details or have any questions!

Here are 5 related questions to extend your understanding:
1. How does the Intersecting Chords Theorem apply to other intersecting segments within a circle?
2. If \( JK \) were doubled, how would that affect the length of \( LN \)?
3. What if the chords intersect outside the circle? How would the formula change?
4. How can you verify the lengths using trigonometry and the angles involved?
5. What other theorems are useful for solving problems involving circle geometry?

**Tip:** Always ensure to label intersecting points clearly when applying geometric theorems.

Given the circle below with chords JK and LM. Find the length of LN. Round to the nearest tenth if necessary.

Learn how to solve for the length of LN using the Intersecting Chords Theorem in a circle. In this geometry problem, you will use key values for the chords JK and LM to find the missing segment length. Suitable for Grades 9-12.

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