1. **Stating the problem:** We have a right triangle formed by a baseline (horizontal), a slanted segment of length 20 m making a 30° angle with the baseline, and a vertical segment rising from the baseline. 2. **Goal:** Find the sum of the longest and shortest sides, and the length of the base. 3. **Known values:** Hypotenuse (slanted segment) $c = 20$ m, angle with baseline $\theta = 30^\circ$. 4. **Formulas:** In a right triangle, - Base (adjacent side) $= c \cos \theta$ - Vertical side (opposite side) $= c \sin \theta$ 5. **Calculate base:** $$\text{base} = 20 \times \cos 30^\circ = 20 \times \frac{\sqrt{3}}{2} = 10\sqrt{3} \approx 17.32 \text{ m}$$ 6. **Calculate vertical side:** $$\text{vertical} = 20 \times \sin 30^\circ = 20 \times \frac{1}{2} = 10 \text{ m}$$ 7. **Identify sides:** - Longest side is the hypotenuse: 20 m - Shortest side is the vertical side: 10 m 8. **Sum of longest and shortest sides:** $$20 + 10 = 30 \text{ m}$$ 9. **Final answers:** - Sum of longest and shortest sides: 30 m - Length of the base: $10\sqrt{3} \approx 17.32$ m