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A War-Time Calculation: Oh Geometry Don't Fail Me Now!

A deep, rushing stream separates two hostile forces. To use trench mortars from Q, a point on one side of the stream against a machine gun nest at P, a point on the other side of the stream, and not to waste ammunition, it is desired to know accurately the distance PQ. Show how this distance PQ may be obtained.

Solution:

Source: J. Moore & J. Mira, The Gist of Mathematics, 1942