What is the length of segment XY?
Understand the Problem
The question is asking for the length of segment XY in a geometric figure involving triangles and angles. We will need to apply the properties of triangles to determine the length of XY.
Answer
The length of segment \( XY \) is approximately 4.66 units.
Answer for screen readers
The length of segment ( XY ) is approximately 4.66 units.
Steps to Solve

Identify the triangles
We have two triangles: triangle ABC and triangle XYZ. We know the lengths of sides in triangle ABC and parts of triangle XYZ. 
Use the Law of Sines on triangle ABC
To find side ( AC ), we can apply the Law of Sines:
$$ \frac{AB}{\sin(C)} = \frac{AC}{\sin(B)} $$
Where ( AB = 15 ), ( AC = 12 ), ( \angle B = 60^\circ ), and ( \angle C = 66^\circ ). 
Calculate side AC
Substituting the known values into the formula:
$$ \frac{15}{\sin(60^\circ)} = \frac{12}{\sin(66^\circ)} $$
Cross multiplying to solve for ( AC ):
$$ AC = \frac{12 \cdot \sin(60^\circ)}{\sin(66^\circ)} $$ 
Compute lengths using Law of Sines on triangle XYZ
Using the Law of Sines again for triangle XYZ to find segment ( XY ):
$$ \frac{XZ}{\sin(Y)} = \frac{XY}{\sin(Z)} $$
Where ( XZ = 4 ), ( \angle Y = 66^\circ ), and ( \angle Z = 60^\circ ). 
Calculate length XY
Substituting the known values:
$$ \frac{4}{\sin(60^\circ)} = \frac{XY}{\sin(66^\circ)} $$
Now solve for ( XY ):
$$ XY = \frac{4 \cdot \sin(66^\circ)}{\sin(60^\circ)} $$ 
Final computation
Now compute the values of ( XY ) for the final answer.
The length of segment ( XY ) is approximately 4.66 units.
More Information
This problem uses the Law of Sines, which is applicable in nonright triangles for finding missing lengths or angles. The angles' relationship and side proportions help solve for the unknown length ( XY ).
Tips
 Forgetting to convert angles to radians if using a calculator set to radian mode.
 Confusing the corresponding sides and angles when applying the Law of Sines.
 Miscalculating triangle side lengths due to incorrect sine values.