We are given that the points $V$, $W$, $X$, and $Y$ lie on the same line segment, and the ratio of the distances $VW:WX:XY$ is 5:3:3. We also know that the total distance $VY = 33$, and we are asked to find $WX$.

Step-by-Step Solution

1. Let the distances be proportional to the given ratio: Since the ratio of the segments $VW:WX:XY = 5:3:3$, we can express each segment as multiples of some common factor $k$. That is: $VW = 5k, \quad WX = 3k, \quad XY = 3k$

2. Use the total distance $VY$: The total distance $VY$ is the sum of $VW$, $WX$, and $XY$: $VY = VW + WX + XY$ Substituting the expressions for $VW$, $WX$, and $XY$: $33 = 5k + 3k + 3k$ Simplifying: $33 = 11k$ Solving for $k$: $k = \frac{33}{11} = 3$

3. Find $WX$: Now that we know $k = 3$, we can find $WX$ by substituting $k$ into the expression for $WX$: $WX = 3k = 3(3) = 9$

Thus, the length of $WX$ is $\boxed{9}$.

Would you like more details or have any questions? Here are some related questions:

1. What is the length of $VW$?
2. What is the length of $XY$?
3. How do we verify the total length of $VY$ after finding $k$?
4. If $WX$ were instead in the ratio 4:3:5, how would the solution change?
5. What is the importance of proportionality in geometric problems involving line segments?

Tip: When dealing with ratios, always express unknown lengths in terms of a common variable for easier calculation!