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We are given two triangles with the condition that their perimeters are equal, and the goal is to find the lengths of each side for both triangles.

### Step 1: Express the Perimeter of Each Triangle
For Triangle ABC:
- \( AB = 3x + 2 \)
- \( BC = 3x \)
- \( AC = 3x \)

The perimeter of Triangle ABC is:
\[
P_{ABC} = AB + BC + AC = (3x + 2) + 3x + 3x = 9x + 2
\]

For Triangle PQR:
- \( PQ = 3x \)
- \( QR = 3x \)
- \( PR = 4x + 2 \)

The perimeter of Triangle PQR is:
\[
P_{PQR} = PQ + QR + PR = 3x + 3x + (4x + 2) = 10x + 2
\]

### Step 2: Set Perimeters Equal
Since the perimeters are equal, we set the expressions for the perimeters equal to each other:
\[
9x + 2 = 10x + 2
\]

### Step 3: Solve for \( x \)
Simplify the equation:
\[
9x + 2 = 10x + 2
\]
Subtract \( 9x \) from both sides:
\[
2 = x + 2
\]
Subtract 2 from both sides:
\[
x = 0
\]

### Step 4: Calculate Side Lengths
Now that we know \( x = 0 \), substitute \( x \) back into the expressions for the side lengths.

For Triangle ABC:
- \( AB = 3(0) + 2 = 2 \)
- \( BC = 3(0) = 0 \)
- \( AC = 3(0) = 0 \)

For Triangle PQR:
- \( PQ = 3(0) = 0 \)
- \( QR = 3(0) = 0 \)
- \( PR = 4(0) + 2 = 2 \)

### Final Answer:
For Triangle ABC:
- \( AB = 2 \)
- \( BC = 0 \)
- \( AC = 0 \)

For Triangle PQR:
- \( PQ = 0 \)
- \( QR = 0 \)
- \( PR = 2 \)

Let me know if you'd like more details or have any questions!

Here are some related questions you might consider:
1. What does it mean geometrically when two sides of a triangle are 0?
2. Can a triangle have sides of length zero?
3. What happens to the angles of a triangle if two sides have zero length?
4. How would this problem change if we had different expressions for the side lengths?
5. Could this situation occur in real-world geometry?

**Tip:** Always check if the side lengths of a triangle satisfy the triangle inequality, which states that the sum of the lengths of any two sides must be greater than the third side.

The perimeters of the triangles shown are equal. Find the side lengths of each triangle.

Learn how to solve a problem involving two triangles with equal perimeters, using algebraic expressions to find the side lengths. This problem is perfect for students in Grades 7-9 who are learning about basic algebra and geometry.

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