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.