1. The problem statement, all variables and given/known data 3. The attempt at a solution I can't figure out how thales found the measurement of the length of this triangle without trigonometry since the sine and cosine ratios were not worked out until the 14th century I think. In any case, they certainly weren't known in thales' time. using trig the answer is rather straightforward, but without trig the only method i can think of for obtaining the length of the bottom line of a right triangle is rather crude. what thales could do is redraw the triangle on the beach since if he knew the AB length and the DAC angle he could physically draw the BAP triangle on the beach, although it would be a rather large and crude replica of the real triangle, but that's the only method i can think of. any help would be greatly appreciated.
hi bobsmith76! maybe they weren't published in tables until then, but the ancient greeks certainly knew all about similar triangles see eg Heath's Euclid's Elements, free online at http://books.google.co.uk/books?id=hhZrpywS8ZIC&printsec=frontcover&dq=euclid+elements&hl=en
I was reading about how Eratosthenes measured the Earth and I saw the word proportionate triangles. so that's probably what he did, he reduced the height of the new triangle by 1/10th or whatever and using the pythagorean theorem he could figure out by how much he should reduce the other triangle. maybe that's what you meant by similar triangles.
according to wikipedia, he actually measured the angle itself (presumably using some sort of enormous protractor ) … He also knew, from measurement, that in his hometown of Alexandria, the angle of elevation of the sun was 1/50th of a circle (7°12') south of the zenith on the solstice noon. did you look at Euclid ?