The problem involves two concentric circles with center $O$, and the following information is given:

• $OE = 25 \, \text{cm}$ (the radius of the inner circle).
• $AE = 14 \, \text{cm}$.
• $AB = 16 \, \text{cm}$, so $BE = AB - AE = 16 - 14 = 2 \, \text{cm}$.
• $M$ is the midpoint of $AD$.

We need to find the length of $OM$.

Approach:

1. Since $M$ is the midpoint of $AD$, and $AB = 16 \, \text{cm}$, it follows that $AM = MD = \frac{AB}{2} = 8 \, \text{cm}$.

2. Using the right triangle $OEM$, we can apply the Pythagorean theorem to find $OM$.

• In triangle $OEM$, $OE = 25 \, \text{cm}$ and $EM = AE - AM = 14 - 8 = 6 \, \text{cm}$.

Now, applying the Pythagorean theorem:

$OM = \sqrt{OE^2 - EM^2}$

Let me calculate $OM$ for you.The length of $OM$ is approximately $24.27 \, \text{cm}$.

Would you like more details on this, or do you have any questions?

Here are five related questions to deepen understanding:

1. How would the solution change if $AE$ and $AB$ were different values?
2. Can the same method be applied if the circles were not concentric?
3. What geometric properties are essential in solving problems involving concentric circles?
4. How can the Pythagorean theorem be applied to find other distances in the diagram?
5. If $O$ were not the center, how would the approach change?

Tip: Always verify that you're using the correct sides of a right triangle when applying the Pythagorean theorem, ensuring that you correctly identify the hypotenuse.