To solve this problem, we need to determine the perimeter of the trapezoid based on the given side lengths. Here’s the step-by-step process:

1. Understand the Given Information

The trapezoid is labeled with:

• The top base: $3x + 2$
• The bottom base: $2x + 1$
• The left side: $12$
• The right side: $16$

The perimeter of a trapezoid is the sum of all its sides: $\text{Perimeter} = \text{Top Base} + \text{Bottom Base} + \text{Left Side} + \text{Right Side}$

2. Plug in the Expressions

Substitute the given side lengths into the formula: $\text{Perimeter} = (3x + 2) + (2x + 1) + 12 + 16$

3. Simplify the Equation

Combine like terms: $\text{Perimeter} = 3x + 2x + 2 + 1 + 12 + 16$ $\text{Perimeter} = 5x + 31$

4. Solve for $x$

The problem does not directly provide the value of $x$. However, in a trapezoid with a midsegment, the midsegment is the average of the two bases. The midsegment here is labeled $2x + 4$, so we can use the equation: $\text{Midsegment} = \frac{\text{Top Base} + \text{Bottom Base}}{2}$ Substitute the given values: $2x + 4 = \frac{(3x + 2) + (2x + 1)}{2}$ Simplify the numerator: $2x + 4 = \frac{3x + 2x + 2 + 1}{2}$ $2x + 4 = \frac{5x + 3}{2}$

5. Eliminate the Fraction

Multiply through by 2 to eliminate the denominator: $2(2x + 4) = 5x + 3$ $4x + 8 = 5x + 3$

6. Solve for $x$

Rearrange to isolate $x$: $8 - 3 = 5x - 4x$ $x = 5$

7. Calculate the Perimeter

Substitute $x = 5$ into the perimeter formula: $\text{Perimeter} = 5x + 31$ $\text{Perimeter} = 5(5) + 31$ $\text{Perimeter} = 25 + 31 = 56$

Final Answer:

The perimeter of the trapezoid is 56. The correct option is: $\boxed{\text{C. 56}}$

Would you like further clarification or a breakdown of any step?

Here are 5 related questions to expand your understanding:

1. How do you calculate the perimeter of a general trapezoid?
2. What is the relationship between the bases and the midsegment of a trapezoid?
3. Why is it necessary to solve for $x$ in this problem?
4. How does the midsegment property simplify trapezoid problems?
5. What are some common applications of trapezoid geometry in real life?

Tip:

Always verify equations involving averages or midsegments, as they are often key to solving trapezoid problems!