To find the length of $IL$, we can apply the Intersecting Chords Theorem (also known as the Power of a Point Theorem). This theorem states that if two chords intersect inside a circle, the products of the lengths of the segments of each chord are equal.

Given the information from the diagram:

• The chord $HI$ is divided into segments $HL = 8$ and $IL = x$ (the length we are trying to find).
• The chord $JK$ is divided into segments $JL = 5$ and $LK = 11$.

According to the theorem: $HL \cdot IL = JL \cdot LK$ Substituting the known values: $8 \cdot x = 5 \cdot 11$ $8x = 55$ Solving for $x$: $x = \frac{55}{8} = 6.875$

Therefore, the length of $IL$ is approximately 6.9 (rounded to the nearest tenth).

Would you like more details on the theorem or any other part of the solution?

Here are 5 questions to expand on this:

1. What other applications can the Intersecting Chords Theorem have in geometry?
2. How can we use this theorem to solve problems involving external secants?
3. What other circle theorems relate to chord properties?
4. How does this theorem apply to tangents and secants?
5. Can this theorem be applied to three-dimensional figures?

Tip: When solving geometric problems, always check if there are special theorems or properties that apply to the configuration, such as the Power of a Point theorem for circle intersections.