By Tracy Kompelien
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Title: 2-D Shapes Are at the back of the Drapes!
Author: Kompelien, Tracy
Publisher: Abdo Group
Publication Date: 2006/09/01
Number of Pages: 24
Binding style: LIBRARY
Library of Congress: 2006012570
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Extra info for 2-D Shapes Are Behind the Drapes!
Geometric Mean 2 8 B e a u t i f u l Geometry the return flight, the same wind now becomes a tailwind, increasing the ground speed to 550 mph. At what speed would the aircraft have to fly in still air to complete the round trip in the same stretch of time as when the wind was blowing? Your first impulse might be to give the answer as 500 mph, the aircraft’s own airspeed and the arithmetic mean of 450 and 550. But first impulses can be wrong. Let the distance between the cities be d and the required speed, v.
Of course, from any given triple we can generate infinitely many others by multiplying it by an arbitrary integer; for example, the triple (6, 8, 10) is just the triple (3, 4, 5) magnified by a factor of 2. Such nonprimitive triples represent similar triangles and are essentially equivalent. 1 In Book X of the Elements, Euclid gives an algorithm for generating every primitive Pythagorean triple (there are infinitely many of them); we give it in the appendix. 1 simple configuration holds within it some surprises.
1 N ot e : 1. This chapter is based on an article by Maor published in The Mathematics Teacher (May 1979, pp. 363– 367). 13 One Theorem, Three Proofs T heorem 13 of Book VI of Euclid tells us how to find the geometric mean (the mean proportion, as the Greeks called it) of two line segments. In essence, it says that in a right triangle, the altitude h divides the hypotenuse into two segments m and n such that h/m = n/h. From this it follows that h2 = mn, so that h is the geometric mean of m and n.