To determine the length of \( YX \) in the triangles \( \triangle XYZ \) and \( \triangle AYB \), we can utilize the similarity of triangles if \( \triangle AYB \sim \triangle XYZ \).
From the given dimensions:
- \( YB = 25 , \text{ft} \)
- \( YA = 16 , \text{ft} \)
- \( YZ = 88 , \text{ft} \)
Assuming \( \triangle AYB \sim \triangle XYZ \), the ratio of the lengths of corresponding sides would be the same. Therefore, the following proportion can be set up:
\[ \frac{YA}{YB} = \frac{YZ}{YX} \]
Substituting the known values:
\[ \frac{16}{25} = \frac{88}{YX} \]
Now, we can solve for \( YX \). Cross-multiplying gives:
\[ 16 \cdot YX = 25 \cdot 88 \]
Calculating \( 25 \cdot 88 \):
\[ 25 \cdot 88 = 2200 \]
Now, we have:
\[ 16 \cdot YX = 2200 \]
Dividing both sides by 16:
\[ YX = \frac{2200}{16} \]
Calculating that gives:
\[ YX = 137.5 , \text{ft} \]
Thus, the length of \( YX \) is \( \boxed{137.5} , \text{ft} \).
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