To solve this problem, we need to find the measure of side $XY$ in quadrilateral $WXYZ$, given that quadrilateral $STUV$ is similar to $WXYZ$. Here's the approach:

Step 1: Set up the ratio of corresponding sides

Since the quadrilaterals are similar, the ratio of corresponding sides is constant. We are given the following:

• Side $ST = 2$ corresponds to side $WX = 6$.
• Side $UV = 5.7$ corresponds to side $XY$ (which we are solving for).

Step 2: Use the ratio of similarity

The ratio of the sides can be set up as follows: $\frac{ST}{WX} = \frac{UV}{XY}$

Substitute the known values: $\frac{2}{6} = \frac{5.7}{XY}$

Step 3: Solve for $XY$

To find $XY$, cross-multiply: $2 \cdot XY = 6 \cdot 5.7$

This simplifies to: $2 \cdot XY = 34.2$

Now, divide by 2: $XY = \frac{34.2}{2} = 17.1$

Conclusion:

The length of side $XY$ is $17.1$.

Would you like any further clarifications?

Here are some related questions to explore:

1. How do we confirm that two quadrilaterals are similar?
2. What other properties are preserved under similarity in polygons?
3. Can you apply the same method to find other sides of the quadrilateral $WXYZ$?
4. What would happen if the ratio between the sides was reversed?
5. How does this method change if quadrilaterals were not similar but congruent?

Tip: Always ensure that corresponding sides are clearly identified when working with similar figures.