1. **Stating the problem:** We have a triangle HPT with sides TP = 90 m, TH = 245 m, and angle $\angle HTP = 13^\circ$. We want to verify the given answers: - Distance from P to H = 159 m - Angle $\angle PHT = 7^\circ$ - Area of triangle HPT = 2480 m² 2. **Using the Law of Cosines to find PH:** The Law of Cosines states: $$PH^2 = TP^2 + TH^2 - 2 \times TP \times TH \times \cos(\angle HTP)$$ Substitute values: $$PH^2 = 90^2 + 245^2 - 2 \times 90 \times 245 \times \cos(13^\circ)$$ Calculate: $$PH^2 = 8100 + 60025 - 44100 \times \cos(13^\circ)$$ Calculate $\cos(13^\circ) \approx 0.9744$: $$PH^2 = 68125 - 44100 \times 0.9744 = 68125 - 42956.64 = 25168.36$$ Then: $$PH = \sqrt{25168.36} \approx 158.6$$ Rounded to nearest integer: $$PH \approx 159$$ 3. **Using the Law of Sines to find angle $\angle PHT$:** Law of Sines: $$\frac{\sin(\angle PHT)}{TP} = \frac{\sin(\angle HTP)}{PH}$$ Rearranged: $$\sin(\angle PHT) = \frac{TP}{PH} \times \sin(\angle HTP)$$ Substitute values: $$\sin(\angle PHT) = \frac{90}{158.6} \times \sin(13^\circ)$$ Calculate $\sin(13^\circ) \approx 0.2249$: $$\sin(\angle PHT) = 0.5675 \times 0.2249 = 0.1277$$ Then: $$\angle PHT = \arcsin(0.1277) \approx 7.3^\circ$$ Rounded to nearest integer: $$\angle PHT \approx 7^\circ$$ 4. **Calculating the area of triangle HPT:** Area formula using two sides and included angle: $$\text{Area} = \frac{1}{2} \times TP \times TH \times \sin(\angle HTP)$$ Substitute values: $$\text{Area} = \frac{1}{2} \times 90 \times 245 \times \sin(13^\circ)$$ Calculate: $$\text{Area} = 11025 \times 0.2249 = 2480.2$$ Rounded to nearest integer: $$\text{Area} \approx 2480$$ **Final conclusion:** All given answers are correct: - Distance PH = 159 m - Angle $\angle PHT = 7^\circ$ - Area = 2480 m²