The Distance to an Unapproachable Point

By Arthur N. DiVito, Ph.D. (copyright © 2000 by A. N. DiVito)



This simple, but important, experiment is a fine one to do outdoors in a team competitive manner. Before the time of radar, such distances as the height of a mountain or an enemy position during a battle-that is, distances to points which cannot be practically or safely approached-were determined by the methods of geometry or trigonometry. 


In a large classroom, or outside if possible, two objects some distance apart from each other are presented to the students. These could be as simple as a point of a desk and a mark on the blackboard. But it helps to have the objects lie at about the same height, with at least one position on a level surface so that protractor readings can be comfortably taken. Teams of students are then assigned the problem of determining how far it is from one object to the other. Essentially, a team at one object must remain alongside or behind it (as in the battle, where it is safe to move about your own side, but not towards the other side). 

Suppose the objects are at points A and B. The team at A should fix another point, on “their side” of the field so to speak, say C. The distance from A to C may be measured by the team directly (with the tape measure). Then, using the protractors, the team should try to obtain excellent measures of the angles BAC and ACB, so that all three angles of triangle ABC are known (since they sum to 180°). At this point, the distance between points and A and B can be determined by similar triangle geometry, or by right triangle trigonometry (if angle BAC is intentionally placed as 90°), or by the Law of Sines. 


Empirical verification is simply a matter of measuring off the distance with the tape measure.


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