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In the given problem, we have a triangle \( \triangle VYZ \) where:

- \( ZX = 2x + 2 \)
- \( YW = 3x - 2 \)
- \( YX = 1 \)

We are tasked with finding the perimeter of \( \triangle VYZ \).

### Steps to Solve:

1. The perimeter of the triangle \( \triangle VYZ \) is the sum of the lengths of its sides: \( VY + YZ + VZ \).
2. From the figure, we can see that:
   - \( VY \) is a line segment connecting point \( V \) to \( Y \),
   - \( YZ \) is a line segment connecting \( Y \) to \( Z \),
   - \( VZ \) is a line segment connecting \( V \) to \( Z \).
   
   The figure shows that the lines \( ZX \), \( YW \), and \( YX \) are provided, and the triangle perimeter can likely be calculated by finding \( x \) and substituting into these equations.

Let's solve for \( x \) by analyzing the provided dataIn the given problem, we have a triangle \( \triangle VYZ \) where:

- \( ZX = 2x + 2 \)
- \( YW = 3x - 2 \)
- \( YX = 1 \)

We are tasked with finding the perimeter of \( \triangle VYZ \).

### Steps to Solve:

1. The perimeter of the triangle \( \triangle VYZ \) is the sum of the lengths of its sides: \( VY + YZ + VZ \).
2. From the figure, we can see that:
   - \( VY \) is a line segment connecting point \( V \) to \( Y \),
   - \( YZ \) is a line segment connecting \( Y \) to \( Z \),
   - \( VZ \) is a line segment connecting \( V \) to \( Z \).
   
   The figure shows that the lines \( ZX \), \( YW \), and \( YX \) are provided, and the triangle perimeter can likely be calculated by finding \( x \) and substituting into these equations.

Let's solve for \( x \) by analyzing the provided 

- \( ZX = 2x + 2 \),
- \( YW = 3x - 2 \),
- \( YX = 1 \).

We now need to derive a method to compute the actual values, check for any geometrical constraints, and find the values for the triangle sides.

Let me process this further to compute the solution.It seems there was an issue while trying to solve for \( x \) using \( YX = 1 \). This likely indicates that the equation doesn't lead to a meaningful solution for \( x \). Let's recheck the relationships provided and attempt a different approach to clarify how the geometry interacts in this scenario.

Let me attempt a new method.It looks like there was an issue trying to solve for \( x \) directly from the given equation \( YX = 1 \). In this problem, \( YX \) is given as a fixed value of 1, so there may not be a variable to solve for \( x \) as intended. The geometric relationships might require different considerations, possibly using other known geometric properties.

Let's try a revised approach. Since the perimeter of \( \triangle VYZ \) depends on finding the sides \( VY \), \( YZ \), and \( VZ \), we may need additional information from the diagram or specific triangle relations.

Would you like me to explore alternative methods, or perhaps clarify another part of the problem?

In the figure, ZX = 2x + 2, YW = 3x - 2, and YX = 1. Find the perimeter of triangle VYZ.

Learn how to calculate the perimeter of triangle VYZ using given line segments ZX = 2x + 2, YW = 3x - 2, and YX = 1. This problem explores algebraic equations and geometry concepts, suitable for students in Grades 8-10.

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