The points V, W, X and Y all lie on the same line segment, in that order, such that the ratio of V, W, colon, W, X, colon, X, Y*VW*:*WX*:*XY* is equal to 5, colon, 3, colon, 3, .5:3:3. If V, Y, equals, 33, comma*VY*=33, find W, X, .*WX*

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!

The points V, W, X, and Y all lie on the same line segment, in that order, such that the ratio of VW:WX:XY is equal to 5:3:3. If VY = 33, find WX.

This problem involves finding the length of WX on a line segment, where the ratio VW:WX:XY is 5:3:3, and the total distance VY is 33. Learn how to solve this using proportionality and ratios, suitable for Grades 6-8 students.

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