1. **Problem Statement:** We have a quadrilateral ABCD with sides AB = 246 m, BC = 312 m, AD = 257 m, and angles $\angle DÂB = 96^\circ$ and $\angle BĈD = 78^\circ$. We need to find: - (a) The length of the fence BD. - (b) The area of the paddock ABCD. 2. **Step (a): Find length BD** - We split the quadrilateral into two triangles: $\triangle ABD$ and $\triangle BCD$. - Use the Law of Cosines in $\triangle ABD$ to find BD. The Law of Cosines formula: $$c^2 = a^2 + b^2 - 2ab \cos(\theta)$$ where $c$ is the side opposite angle $\theta$. In $\triangle ABD$, sides $AB = 246$, $AD = 257$, and angle $\angle DÂB = 96^\circ$ between them. Calculate $BD$: $$BD^2 = AB^2 + AD^2 - 2 \times AB \times AD \times \cos(96^\circ)$$ $$BD^2 = 246^2 + 257^2 - 2 \times 246 \times 257 \times \cos(96^\circ)$$ Calculate each term: $$246^2 = 60516$$ $$257^2 = 66049$$ $$\cos(96^\circ) \approx -0.104528$$ Substitute: $$BD^2 = 60516 + 66049 - 2 \times 246 \times 257 \times (-0.104528)$$ $$BD^2 = 126565 + 2 \times 246 \times 257 \times 0.104528$$ Calculate the product: $$2 \times 246 \times 257 = 126444$$ $$126444 \times 0.104528 \approx 13210.5$$ So: $$BD^2 = 126565 + 13210.5 = 139775.5$$ Find $BD$: $$BD = \sqrt{139775.5} \approx 373.8\text{ m}$$ 3. **Step (b): Find the area of paddock ABCD** - The area of ABCD is the sum of areas of $\triangle ABD$ and $\triangle BCD$. **Area of $\triangle ABD$:** Use formula: $$Area = \frac{1}{2} ab \sin(\theta)$$ where $a=AB=246$, $b=AD=257$, and $\theta=96^\circ$. Calculate: $$Area_{ABD} = \frac{1}{2} \times 246 \times 257 \times \sin(96^\circ)$$ $$\sin(96^\circ) \approx 0.9945$$ So: $$Area_{ABD} = 0.5 \times 246 \times 257 \times 0.9945 \approx 31450.5\text{ m}^2$$ **Area of $\triangle BCD$:** - We know sides $BC=312$, $CD=BD=373.8$ (from part a), and angle $\angle BĈD=78^\circ$ between them. Calculate area: $$Area_{BCD} = \frac{1}{2} \times BC \times BD \times \sin(78^\circ)$$ $$\sin(78^\circ) \approx 0.9781$$ So: $$Area_{BCD} = 0.5 \times 312 \times 373.8 \times 0.9781 \approx 57000.3\text{ m}^2$$ 4. **Total area of paddock ABCD:** $$Area_{ABCD} = Area_{ABD} + Area_{BCD} = 31450.5 + 57000.3 = 88450.8\text{ m}^2$$ **Final answers:** - (a) Length of fence $BD \approx 374\text{ m}$ - (b) Area of paddock $\approx 88451\text{ m}^2$