1. **Problem Statement:** Amal is at point A, directly north of Bimal at point B. A statue S is in the field with a bearing 144° from A. Angle ABS is given as 54°, distance AS = 80.9 m, and distance Bimal to flag pole F is 30 m west. We need to analyze triangle ABS and find distance AB, then find angle AFB. 2. **Why triangle ABS is useful:** Triangle ABS includes points A, B, and S with known angles (ABS = 54° and bearing giving other angles) and a known side AS = 80.9 m. This allows using the Law of Sines or Cosines to find unknown distances such as AB. 3. **Find distance AB:** - Given bearing of S from A is 144°, which means angle BAS = 144° - 90° = 54° because B is directly south of A (A north of B). - Triangle ABS has angles: - Angle at A = 54° (bearing adjustment) - Angle at B = 54° (given ABS) - Angle at S = 180° - 54° - 54° = 72° Apply the Law of Sines: $$\frac{AB}{\sin 72^\circ} = \frac{AS}{\sin 54^\circ}$$ Calculate AB: $$AB = \frac{AS \times \sin 72^\circ}{\sin 54^\circ} = \frac{80.9 \times \sin 72^\circ}{\sin 54^\circ}$$ Approximate values: $$\sin 72^\circ \approx 0.9511, \quad \sin 54^\circ \approx 0.8090$$ So, $$AB = \frac{80.9 \times 0.9511}{0.8090} \approx 95.2 \text{ metres}$$ 4. **Distance between Amal and Bimal = 95.2 metres.** 5. **Find angle AFB:** - Point F is 30 metres west of B (so BF = 30 m). - Coordinates based on B as origin: - B at (0,0) - A at (0, 95.2) north - F at (-30, 0) Vectors: $$\overrightarrow{FA} = (0 - (-30), 95.2 - 0) = (30, 95.2)$$ $$\overrightarrow{FB} = (0 - (-30), 0 - 0) = (30, 0)$$ Find angle AFB using dot product: $$\cos \theta = \frac{\overrightarrow{FA} \cdot \overrightarrow{FB}}{|FA||FB|}$$ Calculate dot product: $$\overrightarrow{FA} \cdot \overrightarrow{FB} = 30 \times 30 + 95.2 \times 0 = 900$$ Magnitudes: $$|FA| = \sqrt{30^2 + 95.2^2} = \sqrt{900 + 9063.04} = \sqrt{9963.04} \approx 99.8$$ $$|FB| = 30$$ So, $$\cos \theta = \frac{900}{99.8 \times 30} = \frac{900}{2994} \approx 0.3005$$ Therefore, $$\theta = \cos^{-1}(0.3005) \approx 72.5^\circ$$ **Final answers:** - Distance AB between Amal and Bimal is approximately 95.2 metres. - Angle AFB is approximately 72.5°.