Question

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?

Answer 1

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

Starting from the origin `(0,0)` 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 `(4,0)`.

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 `(4,6)`.

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 `(8,6)`

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

The points `(0,0)`, `(8,0)` and `(8,6)` are the vertices of a right angled triangle. The distance from the origin `(0, 0)` to the point `(8,6)` 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.