Polygon Area 724D93

1. **State the problem:** We need to find the area of the polygon with vertices at points $$(0,5), (7,9), (15,2), (19,8)$$ above the x-axis. 2. **Formula used:** The area of a polygon given vertices $$(x_1,y_1), (x_2,y_2), \ldots, (x_n,y_n)$$ can be found using the shoelace formula: $$\text{Area} = \frac{1}{2} \left| \sum_{i=1}^{n-1} (x_i y_{i+1} - y_i x_{i+1}) + (x_n y_1 - y_n x_1) \right|$$ 3. **Apply the shoelace formula:** Label the points in order: $$(x_1,y_1) = (0,5), (x_2,y_2) = (7,9), (x_3,y_3) = (15,2), (x_4,y_4) = (19,8)$$ Calculate the sum: $$S_1 = x_1 y_2 + x_2 y_3 + x_3 y_4 + x_4 y_1 = 0 \times 9 + 7 \times 2 + 15 \times 8 + 19 \times 5 = 0 + 14 + 120 + 95 = 229$$ $$S_2 = y_1 x_2 + y_2 x_3 + y_3 x_4 + y_4 x_1 = 5 \times 7 + 9 \times 15 + 2 \times 19 + 8 \times 0 = 35 + 135 + 38 + 0 = 208$$ 4. **Calculate the area:** $$\text{Area} = \frac{1}{2} |S_1 - S_2| = \frac{1}{2} |229 - 208| = \frac{1}{2} \times 21 = 10.5$$ 5. **Interpretation:** The area of the shaded polygon is $$10.5$$ square units. **Final answer:** $$\boxed{10.5 \text{ square units}}$$