Pythagorean Theorem and non-Right TrianglesDate: 03/09/2002 at 13:16:54 From: Katy Palmen Subject: Pythagorean Theorem Why doesn't the Pythagorean Theorem work for triangles other than right triangles? I can provide counter examples to show that it doesn't work for specific triangles, such as equilateral or obtuse, but I can't figure out a general explanation or proof. Thank you for your assistance! Date: 03/09/2002 at 22:34:40 From: Doctor Jeremiah Subject: Re: Pythagorean Theorem Hi Katy, The Pythagorean formula is a special case of a more general equation. The full equation is the Cosine Law: C^2 = A^2 + B^2 - 2AB cos(c) You will notice that this equation only degenerates into the Pythagorean formula when cos(c) is equal to zero. And that can only happen when angle c is 90 degrees. That extra term is the secret to why it only works for 90-degree angles. It all goes back to the geometrical representation of the Pythagorean theorem, which you can find in the Dr. Math FAQ: http://mathforum.org/dr.math/faq/faq.pythagorean.html If the angle is not 90 degrees, you end up with something where the two squares don't add up to the third: For example, 3 squared plus 4 squared equals 5 squared, and the sides of the triangle are 3, 4, and 5 when the angle is 90 degrees, but if you make the angle 77.36 degrees then the side lengths become 4, 4, and 5. And 4 squared plus 4 squared does not equal 5 squared. You can cut out some squares of paper of different sizes: 3x3, 4x4, 5x5, 6x6, 7x7, 8x8, and see which ones work for right-angled triangles. However these numbers are right and the cosine law can figure out the long side: C^2 = A^2 + B^2 - 2AB cos(c) C^2 = 4^2 + 4^2 - 2(4)(4) cos(77.36) C^2 = 16 + 16 - 32 cos(77.36) C^2 = 32 - 32 cos(77.36) C^2 = 32 - 7 C^2 = 25 C = 5 So we can use the cosine law to figure out what the angle is: C^2 = A^2 + B^2 - 2AB cos(c) 5^2 = 3^2 + 4^2 - 2(3)(4) cos(c) 25 = 9 + 16 - 24 cos(c) 25 - 9 - 16 = -24 cos(c) 0 = -24 cos(c) 0 = cos(c) arccos(0) = c 90 = c Here is an entry from the archives that gives more examples: Basics of Trigonometry http://mathforum.org/dr.math/problems/coley.12.12.01.html - Doctor Jeremiah, The Math Forum http://mathforum.org/dr.math/ |
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