1. **State the problem:** Find the area of the polygon with vertices P, Q, R, S, M, N given the side lengths and right angles. 2. **Analyze the shape:** The polygon can be decomposed into rectangles and right triangles using the given lengths: - QR = 210 m (horizontal) - MS = 32 m (vertical) - MN = 150 m (horizontal) - NR = 30 m (vertical) - SR = 80 m (horizontal) - Right angles at P, S, and N 3. **Decompose the polygon:** - Rectangle 1: M-N-R-S (since MN and NR are perpendicular, and SR is horizontal) - Rectangle 2: P-Q-R-S (since QR and SR are horizontal and vertical segments) - Triangle 1: P-S-M (right angle at P) 4. **Calculate areas:** - Area of rectangle M-N-R-S = length MN × height NR = $150 \times 30 = 4500$ m² - Area of rectangle P-Q-R-S = length QR × height SR = $210 \times 80 = 16800$ m² - Area of triangle P-S-M = $\frac{1}{2} \times PS \times MS$ 5. **Find PS:** Since PS is vertical and MS is vertical segment 32 m, and right angle at P, PS = MS = 32 m 6. **Calculate triangle area:** $$\text{Area} = \frac{1}{2} \times PS \times MS = \frac{1}{2} \times 32 \times 32 = 512 \text{ m}^2$$ 7. **Sum all areas:** $$\text{Total area} = 4500 + 16800 + 512 = 21812 \text{ m}^2$$ **Final answer:** The area of the polygon is $21812$ square meters.