1. We are given a right trapezoid with one base of unknown length, a vertical height of 3 cm, a diagonal side of 12 cm, and a small inscribed right triangle with height 2 cm. 2. Let's denote the unknown base length as $x$ cm. 3. The trapezoid's height is 3 cm, which is the distance between the two parallel sides. 4. The small right triangle inside has a height of 2 cm and forms a right angle with the trapezoid's vertical side. 5. Since the trapezoid is right-angled, the height and one base form right angles with each other, making the trapezoid's diagonal side the hypotenuse of the right triangle formed with height 3 cm and the unknown base extension. 6. By the Pythagorean theorem, for the triangle formed by height 3 cm and length $(x-2)$ cm (since 2 cm is already taken by the small triangle), the diagonal side 12 cm satisfies: $$12^2 = 3^2 + (x-2)^2$$ 7. Simplify and solve: $$144 = 9 + (x-2)^2$$ $$144 - 9 = (x-2)^2$$ $$135 = (x-2)^2$$ 8. Take the square root: $$x-2 = \sqrt{135} = \sqrt{9 \times 15} = 3\sqrt{15}$$ 9. Finally, solve for $x$: $$x = 2 + 3\sqrt{15}$$ 10. Approximate to decimal if needed: $$x \approx 2 + 3 \times 3.873 = 2 + 11.619 = 13.619$$ **Answer:** The value in the box is $$x = 2 + 3\sqrt{15} \approx 13.62\text{ cm}$$.