Final answer:
To find the shortest route back to Sam's starting point, the Pythagorean theorem is used to calculate the hypotenuse of the right triangle formed by his walk. The calculation is √(6² + 7²), which gives us approximately 9.22 miles as the shortest route.
Step-by-step explanation:
The student is asking how far Sam must walk to return to his starting point by the shortest route after walking south for 6 miles and then west for 7 miles. To find the shortest route back to the starting point, we need to use the Pythagorean theorem, since Sam's walk forms a right triangle with the south and west paths as the legs, and the shortest route being the hypotenuse.
Let's calculate the shortest route using the Pythagorean theorem: a² + b² = c², where 'a' and 'b' are the legs, and 'c' is the hypotenuse (the shortest route). Here, 'a' is 6 miles, 'b' is 7 miles, and 'c' is what we are trying to find.
6² + 7² = c²
36 + 49 = c²
85 = c²
c = √85
c ≈ 9.22 miles
Therefore, Sam must walk approximately 9.22 miles to return to his starting point by the shortest route.