What mathematical relationship is used to calculate horizontal distance given slant range and altitude from DME?

The mathematical relationship used to calculate horizontal distance from DME (Distance Measuring Equipment) involves a triangle formed between the aircraft, the point on the ground directly below the aircraft, and the DME station itself. In this context, the slant range is the actual line-of-sight distance from the aircraft to the DME station, the altitude is the vertical height of the aircraft above the ground, and the horizontal distance (often referred to as plan range) is the distance along the ground to the DME station.

The correct relationship involves using the Pythagorean theorem, which relates the sides of a right triangle. According to this theorem, the square of the hypotenuse (in this case, the slant range) is equal to the sum of the squares of the other two sides (the altitude and the horizontal distance). Specifically, this is formulated as:

Slant range² = height² + plan range².

Rearranging this equation to isolate the plan range gives us:

Plan range² = slant range² - height².

This derived relationship establishes that to find the horizontal distance, we need to account for both the slant range and the altitude.

The selected answer mentions an equation where the slant range is squared,