1. **State the problem:** We have trapezium ABCD with AB parallel to DC, BC perpendicular to both AB and DC, and given side lengths. We need to find the length of side AD in the form $a\sqrt{b}$. 2. **Identify given lengths:** - DC = 10 cm - BC = 8 cm - AB = 14 cm - Horizontal difference between AB and DC is 4 cm 3. **Understand the shape:** Since AB and DC are parallel and BC is perpendicular to both, ABCD is a trapezium with right angles at B and C. 4. **Find the horizontal length of AD:** The trapezium can be split into a rectangle (with width DC = 10 cm and height BC = 8 cm) and a right triangle on the left side formed by AD and the difference in horizontal lengths between AB and DC. 5. **Calculate the horizontal leg of the triangle:** The difference in horizontal lengths between AB and DC is $14 - 10 = 4$ cm. 6. **Calculate the vertical leg of the triangle:** The height is the same as BC, which is 8 cm. 7. **Use the Pythagorean theorem to find AD:** $$AD = \sqrt{(4)^2 + (8)^2} = \sqrt{16 + 64} = \sqrt{80}$$ 8. **Simplify $\sqrt{80}$:** $$\sqrt{80} = \sqrt{16 \times 5} = 4\sqrt{5}$$ **Final answer:** $$|AD| = 4\sqrt{5}$$