1. **State the problem:** We have a trapezoid with the top base length 10, the middle segment length 12, and the bottom base length 2x. The trapezoid has two pairs of congruent legs, meaning the legs are equal in length. 2. **Identify the trapezoid properties:** In an isosceles trapezoid (legs congruent), the legs are equal, and the bases are parallel. The middle segment (mid-segment) length is the average of the two bases. 3. **Formula for the mid-segment of a trapezoid:** $$\text{mid-segment} = \frac{\text{base}_1 + \text{base}_2}{2}$$ 4. **Apply the formula:** $$12 = \frac{10 + 2x}{2}$$ 5. **Solve for $x$:** Multiply both sides by 2: $$2 \times 12 = 10 + 2x$$ $$24 = 10 + 2x$$ Subtract 10 from both sides: $$24 - 10 = 2x$$ $$14 = 2x$$ Divide both sides by 2: $$\frac{14}{\cancel{2}} = \frac{2x}{\cancel{2}}$$ $$7 = x$$ 6. **Final answer:** $$\boxed{7}$$ The value of $x$ is 7.