1. **Problem statement:** We have a right triangle with a base of 20 cm, an angle of 64° adjacent to the base, and we want to find the vertical side length $x$ to the nearest centimetre. 2. **Formula used:** In a right triangle, the tangent of an angle is the ratio of the opposite side to the adjacent side: $$\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}$$ 3. **Apply the formula:** Here, $\theta = 64^\circ$, the opposite side is $x$, and the adjacent side is 20 cm. $$\tan(64^\circ) = \frac{x}{20}$$ 4. **Solve for $x$:** Multiply both sides by 20: $$x = 20 \times \tan(64^\circ)$$ 5. **Calculate the value:** Using a calculator, $$\tan(64^\circ) \approx 2.0503$$ So, $$x = 20 \times 2.0503 = 41.006$$ 6. **Round to nearest centimetre:** $$x \approx 41 \text{ cm}$$ **Final answer:** $x = 41$ cm.