The Pythagorean theorem and some of its generalizations
The Pythagorean Theorem relates the sides of a right triangle. The Law of Cosines is a generalization of the Pythagorean Theorem because it also relates the sides of a triangle, not necessarily a right triangle. If a, b, c are the sides of a right triangle where c is the hypotenuse, we can put three...
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| Main Authors: | , |
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| Format: | text |
| Language: | English |
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Animo Repository
2002
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| Online Access: | https://animorepository.dlsu.edu.ph/etd_bachelors/17229 |
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| Institution: | De La Salle University |
| Language: | English |
| Summary: | The Pythagorean Theorem relates the sides of a right triangle. The Law of Cosines is a generalization of the Pythagorean Theorem because it also relates the sides of a triangle, not necessarily a right triangle.
If a, b, c are the sides of a right triangle where c is the hypotenuse, we can put three squares with sides 1, b, c on the sides of the right triangle such that the side of length a coincides with the side of the square with side a and similarly with the squares with sides b and c. The Pythagorean Theorem states that the sum of the areas of the two smaller squares is equal to the area of the larger square. The Pythagorean Theorem can be generalized by replacing the squares with semicircles, similar triangles and more generally with proportional regions. Finally, this generalization and Law of Cosines were unified into one single formula. |
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