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In the given diagram, we need to find the length of segment \( WY \).

### Step 1: Analyze the Triangle
The diagram shows triangle \( WZY \) with two intersecting lines at point \( X \), forming two parts on the triangle.

The given expressions are:
- \( WX = 2x + 11 \)
- \( XY = 4x - 5 \)
- \( XZ = XZ \), which means that \( WX \) and \( XY \) are parts of the same segment, suggesting the possibility of equal lengths or a proportion.

### Step 2: Set up an equation based on the equality or proportionality
Since both sides of \( WX \) and \( XY \) are parts of a bisected segment, we can equate them:
\[
WX = XY
\]
Thus, we can write:
\[
2x + 11 = 4x - 5
\]

### Step 3: Solve for \( x \)
Rearrange the equation to solve for \( x \):
\[
2x + 11 = 4x - 5
\]
Subtract \( 2x \) from both sides:
\[
11 = 2x - 5
\]
Add 5 to both sides:
\[
16 = 2x
\]
Now divide by 2:
\[
x = 8
\]

### Step 4: Find the length of \( WY \)
Now that we have \( x = 8 \), we can substitute it back into either expression for \( WX \) or \( XY \) to find the length of \( WY \). Let’s use \( WX \):
\[
WX = 2(8) + 11 = 16 + 11 = 27
\]

Thus, the length of segment \( WY = 27 \).

### Final Answer:
\[
WY = 27
\]

Would you like more details on any part of the solution?

### Follow-up Questions:
1. How can you check if the solution is correct by using the value of \( x \)?
2. What are the properties of a triangle when it is bisected?
3. Can the solution method be applied to other types of triangles or shapes?
4. What would happen if the triangle were an equilateral triangle?
5. How would the solution change if the expressions for \( WX \) and \( XY \) were different?

### Tip:
Always verify your solution by substituting the value of \( x \) back into the original expressions to make sure both sides are equal.

Find the measure of segment WY given WX = 2x + 11 and XY = 4x - 5 with a bisected triangle.

Learn how to solve for the length of segment WY given two algebraic expressions for WX and XY with a bisected triangle. This detailed solution covers core algebra concepts and basic geometry, suitable for students in Grades 8-10.

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