1. **State the problem:** We have a trapezium ABCD with sides AD = $x$ cm, BC = $x$ cm, AB = $2x$ cm, and DC = $2x + 4$ cm. The perimeter is 38 cm. We need to find the length of AD, which is $x$. 2. **Write the perimeter formula:** The perimeter $P$ of a trapezium is the sum of all its sides: $$P = AD + BC + AB + DC$$ 3. **Substitute the given expressions:** $$38 = x + x + 2x + (2x + 4)$$ 4. **Simplify the equation:** $$38 = x + x + 2x + 2x + 4$$ $$38 = 6x + 4$$ 5. **Isolate $x$:** $$38 - 4 = 6x$$ $$34 = 6x$$ 6. **Divide both sides by 6:** $$\frac{34}{\cancel{6}} = \cancel{6}x$$ $$x = \frac{34}{6}$$ 7. **Simplify the fraction:** $$x = \frac{17}{3} \approx 5.67$$ **Final answer:** The length of AD is $\boxed{\frac{17}{3} \text{ cm} \approx 5.67 \text{ cm}}$.