Difference between revisions of "Pythagorean Inequality"

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The Pythagorean Inequality is a generalization of the [[Pythagorean Theorem]]. The Theorem states that in a [[right triangle]] with sides of length <math>a \leq b \leq c</math> we have <math>a^2 + b^2 = c^2</math>.  The Inequality extends this to [[obtuse triangle| obtuse]] and [[acute triangle]]s. The inequality says:
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The Pythagorean Inequality is a generalization of the [[Pythagorean Theorem]], which states that in a [[right triangle]] with sides of length <math>a \leq b \leq c</math> we have <math>a^2 + b^2 = c^2</math>.  This Inequality extends this to [[obtuse triangle| obtuse]] and [[acute triangle]]s. The inequality says:
  
 
For an acute triangle with sides of length <math>a \leq b \leq c</math>, <math>a^2+b^2>c^2</math>. For an obtuse triangle with sides <math>a \leq b \leq c</math>, <math>a^2+b^2<c^2</math>.  
 
For an acute triangle with sides of length <math>a \leq b \leq c</math>, <math>a^2+b^2>c^2</math>. For an obtuse triangle with sides <math>a \leq b \leq c</math>, <math>a^2+b^2<c^2</math>.  

Revision as of 11:53, 2 August 2013

The Pythagorean Inequality is a generalization of the Pythagorean Theorem, which states that in a right triangle with sides of length $a \leq b \leq c$ we have $a^2 + b^2 = c^2$. This Inequality extends this to obtuse and acute triangles. The inequality says:

For an acute triangle with sides of length $a \leq b \leq c$, $a^2+b^2>c^2$. For an obtuse triangle with sides $a \leq b \leq c$, $a^2+b^2<c^2$.

This inequality is a direct result of the Law of Cosines, although it is also possible to prove without using trigonometry.

See also

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