1. **Problem Statement:** We have a field with a baseline $XY = 400$ m and several points with given distances. We need to: a) Draw the map using a scale of 1 cm to 50 m. b) Find the area of the field in hectares. 2. **Scale Drawing:** The scale is 1 cm : 50 m, so every 50 m in reality corresponds to 1 cm on the map. 3. **Coordinates and Distances:** - Baseline $XY = 400$ m corresponds to $\frac{400}{50} = 8$ cm on the map. - Point $Y$ is at $(200,320)$ m, which corresponds to $(\frac{200}{50}, \frac{320}{50}) = (4, 6.4)$ cm. - Points $D$, $C$, and $B$ are at distances 150 m, 150 m, and 225 m from certain points, which will be used to locate them. 4. **Area Calculation:** We use the coordinates of the polygon vertices to find the area using the Shoelace formula. 5. **Coordinates of vertices in meters:** - $X = (0,0)$ - $Y = (400,0)$ (since baseline $XY=400$ m along x-axis) - $F = (205,170)$ - $E = (200,320)$ - $D$, $C$, $B$ are located based on given distances: - $D$ is 150 m from $E$ (200,320) - $C$ is 150 m from $F$ (205,170) - $B$ is 225 m from $X$ (0,0) 6. **Assuming approximate coordinates for $D$, $C$, and $B$ based on distances and directions (not fully specified), we approximate polygon vertices as:** $X(0,0)$, $B(?,?)$, $C(?,?)$, $D(?,?)$, $E(200,320)$, $F(205,170)$, $Y(400,0)$. 7. **Since exact coordinates for $B$, $C$, and $D$ are not fully given, we calculate area using the trapezoidal approximation or given data:** 8. **Area in square meters:** Given the polygon shape and distances, the area is approximately $400 \times 320 = 128000$ m$^2$ (assuming rectangular approximation). 9. **Convert to hectares:** $$\text{Area in hectares} = \frac{128000}{10000} = 12.8$$ hectares. **Final answers:** - a) Draw the map with 1 cm representing 50 m. - b) Area of the field is approximately $12.8$ hectares.