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Finding Distance Using the Pythagorean Theorem

About 2600 years ago lived the founder of Greek philosophy; Thales. Little is known about him but, if the ancient stories are true, he was able to predicted a year in which a solar eclipse turned the "day into night." Thales was not the first to unravel the intricacies of mathematics and astronomy. His prediction of an eclipse could not have been accomplished without the knowledge, observations, and geometric facts known before him in Egypt and Babylon. He probably began a systematic organization of these teachings just as Euclid (of big-time geometry fame) later organized knowledge in the Elements. Thales is supposed to have made two difficult measurements: · The distance of ships at sea from the shore. · The height of the Great Pyramid in Egypt. We will show how a solution to the ship to shore problem relies upon the Pythagorean theorem. Suppose you want to find the distance from a boat at position B and a tree on the shore at position A (see Figure 1.). Starting from point A, walk along the shore any distance in a direction perpendicular to the line AB. Put a marker at point S, then walk the same distance (as you did from A to S ) to C. Make a right angle turn at C and walk away from the shore until you reach point E ; the point from which your ship and your marker at S are exactly lined up. The distance from C to E will now be exactly the same as the distance from A to B ; the distance of the ship to the shore. Problem 1: Using the above scheme, find the distance C to E, if CS = 5 and ES = 13.

B

C

S (Stake)

C

A

E

Figure 1.

Problem 2: About 370 years after Thales, another Greek mathematician, Eratosthenes, used a similar approach to find the radius of the Earth. Assume the triangle given in Figure 2. is the one Eratosthenes used to find the radius of the earth. Find R if the distances are in thousands of kilometers.

R 8 Figure 2.

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