# An Inefficient Route to the Cosine Law

An Inefficient Route to the Cosine Law 186 E. PAUL GOLDENBERG Figure 1. Coercing Pythagoras to accommodate triangles that have no right angles. illustrates quite well, I think, is how the use of good habits of mind guides a mathematical exploration and allows one to take advantage of serendipity. INVARIANT RATIOS It is easy to trip over the Law of Cosines in the midst of an introductory investigation of similar triangles, one that might be given to students ﬁrst encountering similarity. With dynamic software, draw an arbitrary triangle, place a movable point B on one side and, from it, construct a segment parallel to another side. This generally involves ﬁrst constructing a parallel line, letting it determine the desired segment, and then hiding the line. In Figure 2, we have labeled the triangle 1ADE, and the movable point B . As B slides along AD, BC moves with it, stretching or shrinking, but remaining parallel to DE. 1ABC thus may be seen as a triangle that varies in size, but remains invariant in its angle measurements; alternately it may be seen as a “family” of similar triangles all similar to 1ADE.The constant angles represent an intentionally built in invariant, but there are other invariants – in http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png "Technology, Knowledge and Learning" Springer Journals

# An Inefficient Route to the Cosine Law

, Volume 3 (2) – Oct 5, 2004
10 pages

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Publisher
Springer Journals
Subject
Education; Learning and Instruction; Mathematics Education; Educational Technology; Science Education; Creativity and Arts Education
ISSN
2211-1662
eISSN
1573-1766
DOI
10.1023/A:1009709803488
Publisher site
See Article on Publisher Site

### Abstract

186 E. PAUL GOLDENBERG Figure 1. Coercing Pythagoras to accommodate triangles that have no right angles. illustrates quite well, I think, is how the use of good habits of mind guides a mathematical exploration and allows one to take advantage of serendipity. INVARIANT RATIOS It is easy to trip over the Law of Cosines in the midst of an introductory investigation of similar triangles, one that might be given to students ﬁrst encountering similarity. With dynamic software, draw an arbitrary triangle, place a movable point B on one side and, from it, construct a segment parallel to another side. This generally involves ﬁrst constructing a parallel line, letting it determine the desired segment, and then hiding the line. In Figure 2, we have labeled the triangle 1ADE, and the movable point B . As B slides along AD, BC moves with it, stretching or shrinking, but remaining parallel to DE. 1ABC thus may be seen as a triangle that varies in size, but remains invariant in its angle measurements; alternately it may be seen as a “family” of similar triangles all similar to 1ADE.The constant angles represent an intentionally built in invariant, but there are other invariants – in

### Journal

"Technology, Knowledge and Learning"Springer Journals

Published: Oct 5, 2004

### References

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