The problem involves a circle with center $O$, where the radius of the circle is $15 \, \text{cm}$. The chords $AB$ and $XY$ are parallel to each other. Additionally, $OM = XN = 9 \, \text{cm}$, where $M$ and $N$ are midpoints of the chords $AB$ and $XY$, respectively.

We are asked to find:

1. The length of chord $AB$.
2. The length of segment $MN$.

Step 1: Find $AB$

Since $M$ is the midpoint of chord $AB$, and $OM$ is perpendicular to $AB$, we can apply the Pythagorean Theorem in triangle $OMA$. The radius $OQ$ is the hypotenuse, $OM$ is one leg, and half of $AB$ (let's call it $\frac{AB}{2}$) is the other leg.

From the Pythagorean Theorem: $OQ^2 = OM^2 + \left(\frac{AB}{2}\right)^2$ Substitute the known values: $15^2 = 9^2 + \left(\frac{AB}{2}\right)^2$ $225 = 81 + \left(\frac{AB}{2}\right)^2$ $\left(\frac{AB}{2}\right)^2 = 144$ $\frac{AB}{2} = 12$ $AB = 24 \, \text{cm}$

Step 2: Find $MN$

Now, consider the perpendicular distance between the two parallel chords, which can be calculated by the difference between $OM$ and $ON$, given that $OM = XN = 9 \, \text{cm}$.

Since $MN$ is the horizontal distance between points $M$ and $N$ along the parallel lines, and using the properties of parallel chords, the length $MN$ is simply the vertical separation between the midpoints, which equals $XN - OM = 9 - 9 = 0$. Therefore, $MN = 0$ does not hold geometrical relevance in this context unless it's asking for the actual overlap distance.

Would you like further clarifications or details? Here are five additional questions related to the geometry of circles:

1. How do we calculate the perpendicular distance from a point to a chord?
2. What is the general relationship between the radius of a circle and its chords?
3. How can we prove that two chords are parallel in a circle?
4. How does the Pythagorean Theorem apply to right triangles formed in circles?
5. What is the significance of midpoints in geometric constructions involving chords?

Tip: Remember that in problems involving parallel chords, symmetry plays a key role in simplifying calculations.