# 4. The Distance to an Unapproachable Point

One of the great practical problems for humanity is that of determining the distance to a point—such as to an enemy position in a battle, or across a crevice or a river—that is unapproachable. To model this, teams of students are taken outdoors where a flag (or some other suitable object) has been planted at some distance beyond a pre-arranged straight boundary that the students may not cross. The question, of course, is for each team to determine the distance between that flag and some marker (unique to each team) on the students’ side of the boundary.

This is resolvable at the most elementary level, by walking off about 30 meters on our side, and then surveying for the angles that form the actual triangular layout (the vertices being the flag, the marker, and the point at the end of the 30 meter walk-off). The students then make a scale model of the actual triangle. Since the actual and scale model triangles are similar, the students can obtain a good approximation for the requested distance via the scale constant. At an intermediate level, the students would force a right triangle and use right triangle trigonometry to determine the distance to the flag. At the most advanced level, the students could again use any triangle, but this time resolving the desired distance via the Law of Sines.

Clearly we have

Q. a very practical question, “How far is it to that distant point?”

R. The matter is resolved by way of elementary geometry or trigonometry. And,

V. The quality of the results are readily determined by making actual measurements for the desired distances using, say, a 50-meter tape measure.