T98- A BIT OF GEOMETRY by Fredric M. Menger, Emory University, Atlanta, GA, USA

A BIT OF GEOMETRY by Fredric M. Menger, Emory University, Atlanta, GA, USA

You are standing on the ocean shoreline at point-P while looking at a ship S far out in the water. You want to know x, your distance from the ship. So you walk along the beach until you come to point-A and measure the angle-α. Then you walk back along the shoreline to point-B and obtain a reading of angle-β. Assume you have also measured the A/B distance referred to as d. How do you determine x, the distance from P to the ship S (assuming you have a calculator with trigonometric functions with you)?

Clearly, d = d1 + d2

You know that (by definition) tan α = x/d1 and tan β = x/d2 (where “tan” refers to the tangent of the angle). Rearranging, d1 = x/tan α and d2 = x/tan β.

Substituting: d = x/tan α + x/tan β or d = x(1/tan α + 1/tan β )

and: x = d/(1/tan α + 1/tan β)

Since you have measured d, and your calculator gives you the tangents of the angles, you can determine x, the distance across the water from point P to the ship S.