1. **State the problem:** We have two similar rectangles D and E. Rectangle D has width 8 mm and height $x$ mm. Rectangle E has height 10 mm and width $5x$ mm. We need to find the value of $x$. 2. **Use the property of similar rectangles:** Corresponding sides are proportional. This means: $$\frac{\text{width of D}}{\text{width of E}} = \frac{\text{height of D}}{\text{height of E}}$$ 3. **Write the proportion:** $$\frac{8}{5x} = \frac{x}{10}$$ 4. **Cross multiply:** $$8 \times 10 = x \times 5x$$ $$80 = 5x^2$$ 5. **Divide both sides by 5:** $$\frac{80}{\cancel{5}} = \frac{5x^2}{\cancel{5}}$$ $$16 = x^2$$ 6. **Take the square root of both sides:** $$x = \pm \sqrt{16}$$ $$x = \pm 4$$ 7. **Since $x$ represents a length, it must be positive:** $$x = 4$$ **Final answer:** $x = 4$ mm.