Given that triangles \( AYB \sim XYZ \), we can use the properties of similar triangles to find the lengths of corresponding sides.
We know:
- \( AB = 5 \)
- \( AY = 12 \)
- \( AY = AX \)
- \( BY = BZ \)
Since \( AYB \sim XYZ \), the corresponding sides of the triangles are proportional. We can set up the proportion based on the corresponding sides:
\[ \frac{AB}{XZ} = \frac{AY}{XY} \]
We're not given the length of \( XZ \) or \( XY \), but knowing that \( AY = AX = 12 \) allows us to express other sides in terms of these lengths.
We need to express \( BY \) and \( BZ \).
Using the triangle similarity, we know that:
\[ \frac{BY}{AY} = \frac{BZ}{AY} \]
Given that \( BY = BZ \), we have:
\[ BY = BZ \]
Now to find the length of \( BZ \): \[ \frac{BY}{12} = \frac{5}{XZ} \]
Since \( BZ = BY \), we substitute and get:
\[ BZ = BY \]
Now, to clarify the relationship further, if \( XZ \) corresponds to \( AB \) (which equals 5), we look for \( ZY \) corresponding to \( AY \):
Setting a ratio for similarity, we find \( ZY \):
Let’s say \( BZ = k \):
- Then, the proportion using the corresponding sides can be written as: \[ \frac{k}{12} = \frac{5}{AY} \]
Where \( AY \) corresponds directly to \( AB \):
Resolving for \( BZ \): Given \( BY = BZ = k\),
Assuming all other ratios are consistent, let's delegate ourselves to solve this final step. If triangles were fully expanded, let's resolve this ratio clearly:
Using the ratios from those side lengths. We can assert:
\[ k = \frac{5 \cdot 12}{k} \leftrightarrow k^2 = 60 \]
Taking the square root :
- Find \( 5x = 12 , \Rightarrow x = \frac{12}{5} \).
Thus inserting that as \( k \):
Hence:
\[ k = \frac{60}{12} = 5, \]
So \( BZ = 5 \) units.
As an answer in integer and mixed format, the final output concludes at:
\[ \text{The length of } BZ \text{ is } 5. \]
However, clear correlation holds at \( BZ = 5 \) final.
\[ \boxed{5} \]