1. **Problem statement:** A man travels along a right-angled triangle path with base 4 km and hypotenuse 5 km. After completing the triangle, he continues 6 km in the same direction, then turns 90 degrees and travels 8 km. We need to find the total distance traveled and his distance from the starting point. 2. **Step 1: Find the missing side of the triangle.** Using the Pythagorean theorem for the right triangle: $$\text{hypotenuse}^2 = \text{base}^2 + \text{height}^2$$ Given hypotenuse = 5 km and base = 4 km, $$5^2 = 4^2 + h^2 \implies 25 = 16 + h^2 \implies h^2 = 9 \implies h = 3 \text{ km}$$ 3. **Step 2: Calculate the perimeter of the triangle (the path traveled so far).** $$\text{Perimeter} = 4 + 3 + 5 = 12 \text{ km}$$ 4. **Step 3: After completing the triangle, he continues 6 km in the same direction as the last side traveled (hypotenuse side).** So, he moves 6 km along the line of the hypotenuse beyond the triangle. 5. **Step 4: Then he turns 90 degrees and travels 8 km.** This forms a right angle with the direction of the last 6 km segment. 6. **Step 5: Calculate total distance traveled.** $$12 + 6 + 8 = 26 \text{ km}$$ 7. **Step 6: Calculate the distance from the starting point.** - The man ends up at a point offset from the start by the vector sum of the triangle's base and height plus the 6 km and 8 km legs. - The triangle's base and height are perpendicular, so the starting point to the triangle's right angle vertex is at $(4,0)$ and $(0,3)$ respectively. - The hypotenuse vector direction can be represented as $(4,3)$ (from origin to hypotenuse end). - Normalize this vector: $$\text{length} = 5$$ $$\text{unit vector} = \left(\frac{4}{5}, \frac{3}{5}\right)$$ - After completing the triangle, he moves 6 km further along this direction: $$6 \times \left(\frac{4}{5}, \frac{3}{5}\right) = \left(\frac{24}{5}, \frac{18}{5}\right) = (4.8, 3.6)$$ - Then he turns 90 degrees (perpendicular) and moves 8 km. The perpendicular vector to $(4,3)$ is $(-3,4)$ (rotated 90 degrees counterclockwise). - Normalize this perpendicular vector: $$\text{length} = 5$$ $$\text{unit perpendicular} = \left(-\frac{3}{5}, \frac{4}{5}\right)$$ - Multiply by 8 km: $$8 \times \left(-\frac{3}{5}, \frac{4}{5}\right) = \left(-\frac{24}{5}, \frac{32}{5}\right) = (-4.8, 6.4)$$ - Total displacement vector from origin: $$\text{triangle hypotenuse end} + \text{6 km vector} + \text{8 km vector} = (4,3) + (4.8,3.6) + (-4.8,6.4) = (4 + 4.8 - 4.8, 3 + 3.6 + 6.4) = (4, 13)$$ 8. **Step 7: Calculate the distance from the starting point using the displacement vector:** $$\text{distance} = \sqrt{4^2 + 13^2} = \sqrt{16 + 169} = \sqrt{185} \approx 13.6 \text{ km}$$ **Final answers:** - Total distance traveled = 26 km - Distance from starting point = approximately 13.6 km