A slant range with an explicit vertical component

A fictional geometry exercise places an aircraft four nautical miles horizontally from a DME facility and three nautical miles vertically above it. These are actual geometric components supplied for the model, not a pressure-altitude reading that must be treated as exact height. Ignore terrain, signal errors and equipment delays for this calculation. A learner adds the two distances and reports seven nautical miles; another ignores the vertical component and reports four. The task asks for the direct spatial distance and an explanation of the difference from ground distance. It supplies no service volume, approach chart, legal minimum or flight instruction. The result cannot establish that a real DME is available, identified or authorized for a particular procedure. A complete answer should show the relationship and retain the stated units.