Question 798196
Here's a map showing the explorer's path.
Each square in the grid is 1 mile by 1 mile.
The explorer goes along the red arrow path from point A to point B.
The distance (as the crow flies) from A to B is the hypotenuse (AB) of the green right triangle ABC.
{{{drawing(300,300,1,11,1,11,
grid(1),blue(arrow(8.5,4.5,8.5,7.5)),locate(8.4,8,N),
green(rectangle(2.3,3.3,2,3)),green(line(2,9,7,3)),
green(line(7,3,2,3)),green(line(2,9,2,3)),
red(circle(2,9,0.15)),red(line(2.15,9,6,9)),red(arrow(2.15,9,6,9)),
red(line(6,9,6,2)),red(arrow(6,6,6,2)),
red(line(6,2,9,2)),red(arrow(6,2,9,2)),
red(line(9,2,9,3)),red(arrow(9,2,9,3)),
red(line(9,3,7,3)),red(arrow(9,3,7.15,3)),
red(circle(7,3,0.15)),locate(1.7,3,C),
locate(1.7,9,red(A)),locate(7.1,2.9,red(B))
)}}} {{{AC=6}}} and {{{CB=5}}} so {{{AB^2=6^2+5^2=36+25=61}}}--> {{{AB=sqrt(61)}}} is about 7.8 miles.
 
NOTE:
Without drawing the map, we could calculate total distance to the East as the sum of the eastward stretches of the path:
{{{4+3+(-2)=5}}}.
Similarly, we could calculate total distance to the South as the sum of the southward stretches of the path:
{{{7+(-1)=6}}}.