Extending Two Classic Proofs of the Pythagorean Theorem to the Law of Cosines

2017 ◽  
Vol 90 (3) ◽  
pp. 182-186
Author(s):  
Kevin K. Ferland
2018 ◽  
Vol 102 (553) ◽  
pp. 77-88
Author(s):  
Matúš Harminc ◽  
Lucia Janičková

The following observations are motivated by the facts that the area of a planar figure displayed on a screen can be expressed by a certain number of pixels; and if the figure is drawn by a plotter, then its area can be characterised by the total length of a line which fills it in.The generalisations of the Pythagorean theorem are of three kinds. Firstly, the squares on the sides of the right triangle are substituted by other geometrically similar planar figures (Euclid's Elements Book VI, Proposition 31 [1]). Secondly, the assumption of the right angle is omitted (the law of cosines), or both of these generalizations occur simultaneously (Pappus’ area theorem [2], see also H. W. Eves [3]). Thirdly, mathematical spaces other than the plane are considered (for example, de Gua-Faulhaber theorem about trirectangular tetrahedra [3], further generalised by Tinseau [4], Euclideann-spaces, Banach spaces [5], see also [6]).


Author(s):  
Glen Van Brummelen

This chapter discusses the modern approach to solving oblique triangles. Two important theorems about planar oblique triangles are the spherical and planar Law of Sines and the Law of Cosines, which is an extension of the Pythagorean Theorem applied to oblique triangles. Book I of Euclid's Elements deals primarily with the Pythagorean Theorem (Proposition 47) and its converse (Proposition 48), while Book II contains theorems that may be translated directly into various algebraic statements. The chapter considers two of the last three theorems of Book II: Proposition 12, which deals with obtuse-angled triangles, and Proposition 13, which is concerned with acute-angled triangles. It also extends the Law of Cosines to the sphere and uses it to solve astronomical and geographical problems, such as finding the distance from Vancouver to Edmonton. Finally, it describes Delambre's analogies and Napier's analogies.


Author(s):  
Vecihi S. Zambak ◽  
Budi Mulyono

In history, geometry was founded more as a practical endeavor than a theoretical one. Early developments of the branch portray philosophers' attempts to make sense of their surroundings, including the measurement of distances on earth and in space. Such a link between earth and space sciences and geometry motivated us to develop and implement a multidisciplinary lesson focusing on the conceptual understanding of the law of cosines in the context of astronomy. In our content specific STEAM lesson, the authors aimed to facilitate an understanding of the law of cosines in ninth grade students, and then apply the law in a star map task to find approximate distances between stars. The second part of the lesson also included the use of an instructional technology to support students' work with the star map task. In the conclusion, the authors discuss possible ways to improve the quality of their STEAM education efforts for the given context.


1990 ◽  
Vol 63 (5) ◽  
pp. 342-342 ◽  
Author(s):  
Sidney H. Kung
Keyword(s):  
The Law ◽  

1951 ◽  
Vol 58 (10) ◽  
pp. 698 ◽  
Author(s):  
S. L. Thompson
Keyword(s):  
The Law ◽  

1955 ◽  
Vol 48 (5) ◽  
pp. 308-309
Author(s):  
Benjamin Greenberg
Keyword(s):  
The Law ◽  

The law of cosines has always frustrated those teachers who wish to use it computationally. The transformation developed in this paper makes it more amenable to the use of logarithmic computation.


1990 ◽  
Vol 63 (5) ◽  
pp. 342 ◽  
Author(s):  
Sidney H. Kung
Keyword(s):  
The Law ◽  

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