### Instruction: While walking on a plane surface, a traveler first headed 19 miles north, then 16 miles west, then 3 miles south and finally 4 miles east. How many miles from the starting point was the traveler after these four legs of the journey?### Response: I want to find the distance between the starting point and the ending point of the journey, which are both on the same plane.
I can use the Pythagorean theorem to find the length of the hypotenuse of a right triangle whose legs are the horizontal and vertical displacements of the journey.
To find the horizontal displacement, I need to subtract the distance traveled east from the distance traveled west, since they are in opposite directions.
Similarly, to find the vertical displacement, I need to subtract the distance traveled south from the distance traveled north.
The horizontal displacement is 16 - 4 = 12 miles, and the vertical displacement is 19 - 3 = 16 miles.
Now I can plug these values into the Pythagorean theorem: h^2 = 12^2 + 16^2, where h is the hypotenuse.
Simplifying, I get h^2 = 144 + 256 = 400.
Taking the square root of both sides, I get h = 20.