Problem D

Cartesian Geocaching

Geocaching is a kind of outdoor treasure hunt where people use GPS devices to look for hidden objects, called caches. In Cartesian Geocaching, instead of using a GPS device, locations are described using Cartesian coordinates.

Hilde sets up a large field for Cartesian Geocaching, measuring the distances in kilometres so that the point \((1,0)\) lies \(1\) km east of the point \((0,0)\), for example.

Hilde starts at point \(A(0,0)\), then walks northwest in a straight line to some point \(B\), where she hides a cache. Then, from \(B\), she walks northeast in a straight line to point \(C(0,4)\) where she hides another cache. Finally she walks straight back to point \(A\).

How far does Hilde walk in total?