# If you travel 4 miles in one direction, turn left, travel 6 miles, turn right and travel 4 miles, then how far will you be from the starting point?

May 25, 2015

I'm not entirely clear what you are asking for, but let me address the problem you describe:

Starting from the origin $\left(0 , 0\right)$ travel a distance of $4$ units.

Let us choose to travel in the positive direction along the $x$ axis. That will take us to the point $\left(4 , 0\right)$.

Turning to the left will orient us in a positive direction parallel to the $y$ axis.

Moving forward $6$ units will add $6$ to the $y$ coordinate, taking us to the point $\left(4 , 6\right)$.

Turning to the right will orient us in a positive direction parallel to the $x$ axis.

Moving forward $4$ units will add $4$ to the $x$ coordinate,
taking us to the point $\left(8 , 6\right)$

If we drop a perpendicular onto the $x$ axis from this final point we get the point $\left(8 , 0\right)$.

The points $\left(0 , 0\right)$, $\left(8 , 0\right)$ and $\left(8 , 6\right)$ are the vertices of a right angled triangle. The distance from the origin $\left(0 , 0\right)$ to the point $\left(8 , 6\right)$ is the length of the hypotenuse of this triangle, so is equal to the positive square root of the sum of the squares of the lengths of the other two sides.

${8}^{2} + {6}^{2} = 64 + 36 = 100 = {10}^{2}$

So the distance between the start and finish points is $10$ miles.