To solve this problem, we can apply the Law of Sines from trigonometry.

Given that we have the following details from the problem:

• $\angle A = 41^\circ$
• $\angle BDC = 54^\circ$
• Distance $AD = y = 13$
• We are asked to find $x = DC$

Steps to solve:

1. Determine $\angle DBC$: Since $\triangle BDC$ forms a triangle, the sum of angles in a triangle is always $180^\circ$. We already know two angles: $\angle BDC = 54^\circ$ and $\angle A = 41^\circ$. Therefore, to find the third angle, use the following: $\angle B = 180^\circ - (54^\circ + 41^\circ) = 85^\circ$

2. Use the Law of Sines: The Law of Sines states: $\frac{y}{\sin(\angle BDC)} = \frac{x}{\sin(\angle A)}$ Plugging in the known values: $\frac{13}{\sin(54^\circ)} = \frac{x}{\sin(41^\circ)}$

3. Solve for $x$: Now, solve for $x$: $x = \frac{13 \cdot \sin(41^\circ)}{\sin(54^\circ)}$

I'll calculate this next.The value of $x$ is approximately 11 when rounded to the nearest whole number.

Would you like further clarification on the process or more details?

Here are 5 related questions to expand on this:

1. How can the Law of Cosines be used to solve similar problems?
2. What would change if one of the angles in the triangle were obtuse?
3. How would the solution differ if we knew the length of $BC$ instead of $AD$?
4. Can you solve for the height $h$ using the given information?
5. How would the result change if $\angle BDC$ were smaller?

Tip: The Law of Sines is especially useful when you're dealing with non-right triangles and have information about angles and side lengths.