Since triangles \(^XYZ\) and \(^AYB\) are similar (denoted as \(^XYZ \sim ^AYB\)), we can use the properties of similar triangles to establish a relationship between their corresponding sides.
Let’s assign the corresponding sides:
- \(YZ\) corresponds to \(AB\)
- \(YX\) corresponds to \(AY\)
- \(XY\) corresponds to \(YB\)
From the problem, we know:
- \(YB = 25 \text{ ft}\)
- \(YA = 16 \text{ ft}\)
- \(YZ = 88 \text{ ft}\)
Using the properties of similar triangles, we can set up the following proportion based on the corresponding sides:
\[ \frac{YX}{YA} = \frac{YZ}{YB} \]
Substituting the given values into the proportion:
\[ \frac{YX}{16} = \frac{88}{25} \]
Now, we can solve for \(YX\):
\[ YX = 16 \times \frac{88}{25} \]
Calculating that:
\[ YX = \frac{16 \times 88}{25} = \frac{1408}{25} = 56.32 \text{ ft} \]
Thus, the length of \(YX\) is \(56.32\) ft.