This problem involves the geometry of a circle, specifically chord and segment relationships. The information provided is:

• $m = 5$ (one segment of the divided chord),
• $a = 6$ (another segment of the divided chord),
• $r = 7$ (radius of the circle),
• $b = ?$ (length of the unknown segment).

This appears to be a situation where the power of a point theorem applies. Let's break it down:

The power of a point theorem states:

$\text{For two intersecting chords in a circle: } (m)(b) = (a)(a).$

Using the given values:

• $m = 5$,
• $a = 6$,
• $b = ?$.

Solving for $b$:

$(5)(b) = (6)(6).$

$5b = 36.$

$b = \frac{36}{5} = 7.2.$

Final Answer:

The value of $b$ is $7.2$.

Would you like a detailed explanation of the theorem or additional examples? Here are related questions to explore:

1. How does the power of a point theorem generalize to secant and tangent lines?
2. What are the formulas for segment lengths involving external and internal points of a circle?
3. How can the radius of the circle be used in similar problems?
4. What are other ways to verify this solution using coordinate geometry?
5. How does this problem extend to three-dimensional geometry?

Tip: Always check for intersections and relationships between the lines and circle; many problems in circle geometry rely on similar triangle properties or the power of a point theorem.