Question

a dog runs 4 miles toward north and turns right, runs 1 mile again, turns to its right, runs another 5 miles. it then, turns left and runs 2 miles; turns to its left and runs 5 miles, and finally turns to its right. what is the smallest distance between the dog's final point and the initial point, and in which direction is the dog facing?

Answer 1

• 5 miles
• east

Explanation:

The dog's route and final facing direction are shown in the attached diagram.

Turns

The sequence of facing directions for right turns is ...

{N, E, S, W, N}

The sequence of facing directions for left turns is the reverse:

{N, W, S, E, N}

In total, the dog made 3 right turns and 2 left turns. The net change in facing direction is 3-2 = 1 right turn. The dog's initial facing direction is north, so its final facing direction is east, one right turn from north.

Distance

The distances traveled are ...

N 4, E 1, S 5, E 2, N 5.

The net change in position is north (4-5+5) = 4 miles, and east (1 +2) = 3 miles. The distance between the final point and the initial point is the hypotenuse of a right triangle with legs 4 miles and 3 miles. The Pythagorean theorem tells us that distance is ...

c² = a² +b²

c = √(a² +b²) = √(4² +3²) = √25 = 5

The smallest distance between the dog's final and initial points is 5 miles.