1. **State the problem:** We are given a trapezoid with the top base labeled as $3x + 2$, the middle segment inside the trapezoid labeled $17.5$, and the bottom base labeled $21$. We need to find the value of $x$. 2. **Understand the trapezoid properties:** The trapezoid has two parallel bases (top and bottom) and two equal non-parallel sides. The middle segment labeled $17.5$ is the length of the segment connecting the midpoints of the non-parallel sides, which is the median of the trapezoid. 3. **Formula for the median of a trapezoid:** The median $m$ is the average of the lengths of the two bases: $$m = \frac{\text{base}_1 + \text{base}_2}{2}$$ 4. **Apply the formula:** Here, the median $m = 17.5$, the top base is $3x + 2$, and the bottom base is $21$. Substitute these values: $$17.5 = \frac{3x + 2 + 21}{2}$$ 5. **Multiply both sides by 2 to eliminate the denominator:** $$2 \times 17.5 = 3x + 2 + 21$$ $$35 = 3x + 23$$ 6. **Isolate $x$ by subtracting 23 from both sides:** $$35 - 23 = 3x + \cancel{23} - \cancel{23}$$ $$12 = 3x$$ 7. **Divide both sides by 3 to solve for $x$:** $$\frac{12}{3} = \frac{3x}{3}$$ $$4 = x$$ **Final answer:** $$\boxed{4}$$