The shortest distance the joggers must travel to return to their starting point is 7.81 miles.
To find the shortest distance the joggers must travel to return to their starting point, we can use the Pythagorean theorem, as the southward and eastward distances form a right triangle. The theorem states that the square of the length of the hypotenuse (the shortest distance, in this case) is equal to the sum of the squares of the other two sides:
a^2 + b^2 = c^2
Here, a is the southward distance (6 miles), and b is the eastward distance (5 miles). We need to find c, the hypotenuse.
(6 miles)^2 + (5 miles)^2 = c^2
36 + 25 = c^2
61 = c^2
Now, take the square root of both sides to find c:
c = √61
c ≈ 7.81 miles