The VOR (VHF omnidirectional radio range) answered which line; distance-measuring equipment (DME) answers how far along it. This is the signal behind the DME range window the Instrumentation course drew on the HSI (horizontal situation indicator), and its one honest subtlety, the difference between what it measures and what a chart means by distance, is where most of its exam marks live.
Interrogator and transponder
DME operates in the UHF (ultra-high frequency) band, and for the first time the measurement itself is a two-way radio exchange between the system's two basic components: the interrogator in the aircraft and the transponder on the ground. The aircraft's interrogator transmits a timed interrogation; the ground station's transponder replies on a different frequency, and the airborne equipment measures the time between question and answer. Radio waves travel at a known speed, so the round-trip time, less the transponder's fixed reply delay, halves into the distance. The reply deliberately sits on a different frequency from the interrogation, so the receiver can never mistake its own transmitted pulse, or its echo, for the station's answer.
Slant range, not ground range
What the timer measures is the straight line from antenna to antenna, and that line leaves the ground the moment the aircraft does: the distance measured by DME is slant range, the hypotenuse of the triangle whose other sides are the aircraft's height and the distance over the ground. Directly overhead the station, the ground distance is zero and the DME reads the aircraft's height and nothing else. The exam's favourite arithmetic converts one side of that triangle into another:
Ground distance from slant range