Bot GPT 4-o mini To find the length of \( BZ \), we can use the property of similar triangles.
Since \( \triangle AYB \sim \triangle XYZ \), we can set up a proportion based on the corresponding sides. The given information is as follows:
- \( AB = 5 \)
- \( AY = 12 \)
- \( AX = AY = 12 \) (from the problem statement)
- \( BY = BZ \)
Let's denote the lengths:
-
The side lengths of \( \triangle AYB \):
- \( AY = 12 \)
- \( AB = 5 \)
-
The side lengths of \( \triangle XYZ \):
- Corresponding side \( AX = 12 \)
- Let \( BZ = x \) (which is equal to \( BY \))
Now, we can set up a proportion based on the lengths of the similar triangles:
\[ \frac{AY}{AX} = \frac{AB}{XZ} = \frac{BY}{BZ} \]
Substituting the known values into this proportion:
\[ \frac{12}{12} = \frac{5}{XZ} = \frac{BY}{x} \]
Since \( \frac{12}{12} = 1 \), we also have:
\[ \frac{5}{XZ} = 1 \quad \text{and therefore} \quad XZ = 5 \]
Now, since \( BY = BZ = x \) and substituting back into our proportion:
\[ \frac{BY}{BZ} = 1 \implies BY = BZ \]
Now we can use what we have from \( BY \):
\[ \frac{12}{12} = 1 \quad \text{therefore, } BY \text{ and } BZ \text{ are equal.} \]
From the properties of similar triangles, we know that:
\[ \frac{AB}{AY} = \frac{BY}{BZ} = \frac{5}{12} \]
Utilizing the property \( BY = BZ \), we can say:
If \( AB = 5 \) and \( BY = x \), we can relate this through the ratios from the triangles. Setting this equal gives us:
\[ x = BY = \frac{5}{1} \cdot \frac{1}{12} = \frac{5}{12} \]
And we already defined \( BY = BZ \):
Thus,
\[ BZ = x = \frac{5}{12} \]
In conclusion, the length of \( BZ \) is:
\[ \boxed{\frac{5}{12}} \]
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