The problem asks to find the length of side $CD$ of the rectangle, with a given perimeter of 106 units. The expressions for the sides are:

• $AB = 3x + 2$
• $BC = 2x + 1$

Steps:

1. Perimeter formula for a rectangle: $P = 2 \times (AB + BC)$ Given $P = 106$, the equation becomes: $106 = 2 \times \left((3x + 2) + (2x + 1)\right)$

2. Simplify inside the parentheses: $106 = 2 \times (5x + 3)$

3. Divide both sides by 2: $53 = 5x + 3$

4. Solve for $x$: $5x = 53 - 3$ $5x = 50$ $x = \frac{50}{5} = 10$

5. Now, calculate side $CD = BC$ using the expression $BC = 2x + 1$: $CD = 2(10) + 1 = 20 + 1 = 21$

Thus, the length of side $CD$ is 21 units.

Would you like any more details or explanations? Here are some related questions to expand your understanding:

1. How do you calculate the perimeter of any polygon, not just a rectangle?
2. What is the difference between perimeter and area in geometry?
3. Can you derive a general formula to find side lengths of a rectangle if the perimeter is known?
4. How do we handle expressions involving multiple variables in geometry?
5. Why do we divide the perimeter equation by 2 in the second step?

Tip: In any perimeter problem, focus on identifying and adding up all side lengths correctly before solving for unknowns.