Similar Triangles Robert Leverett Feb 07, 2007 08:10 PST
 ENTS,      The geometrical concept of similar triangles provides a powerful tool for tree measurers. However, as with the percent slope method, it is often misapplied to tree height. But, used correctly, the method gives us one more tool to work with. Let's look at the concept of similar triangles.              Similar triangles are triangles that have the same overall shape. One is just a blown-up or reduced version of another. Because similar triangles have the same shape, corresponding angles of similar triangles are equal and corresponding sides are proportional in length. It is the condition of proportionality of side length that allows us to compute tree height. But how does the process work?              Assume we have two triangles that are similar. Triangle ABC is the larger and abc the smaller where the lengths of the sides are designated by A, B, and C for the larger and a, b, and c for the smaller.   The condition of proportionality leads to the following relationships:                A/B = a/b, A/C = a/c, B/C = b/c.      These are all algebraic expressions and can be manipulated algebraically to get equivalent forms such as B/A = b/a, B/b = A/a, and b/B = a/A, etc. Now suppose that we form a big triangle ABC such that A is the tree's height and B is the baseline from the measurer to the trunk. Now, if we can form a smaller similar triangle abc where we can measure sides a, b, and c, we can measure the baseline to the tree B and can compute the tree's height by using the A/B = a/b proportionality relationship. Algebraically rearranging, we get A = B(a/b). This last formula is what often accompanies diagrams showing how to measure tree height using similar triangles. If side A is vertical and the two triangles are truly similar, then the process works. However, what happens when side A is not vertical (the line from the crown-point to the base is not vertical)? Then the process does not work and that will be the case when the crown-point is not directly over the base of the tree. Sound familiar?           Similar triangles can also be used very productively for determining crown-point offset. However, that determination requires a mult-step process that will be explained with diagrams at the April ENTS event at Cook Forest SP. In fact, all the material in these e-mails will be brought together in what I hope will be our first crack in producing an ENTS Guide to Dendromorphometry. BTW, Dendromorphometry was a term created by Gary Beluzo, partly in jest and partly in seriousness to describe the energy with we Ents expend pursuing our passion. Gary, is of course, a first class dendromorphometrist. Gary once made up certificates for John Knuerr and myself. However, the idea of formalizing Dendromorphometry as an official ENTS discipline has merit. ENTS could issue certificates to bonafide dendromorphometrists - both in fun and in all seriousness. Bob Robert T. Leverett Cofounder, Eastern Native Tree Society