Difference between revisions of "Pythagorean inequality"
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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 | + | 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 states: |
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>. | ||
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[[Category:Geometry]] | [[Category:Geometry]] | ||
| + | [[Category:Inequalities]] | ||
[[Category:Geometric Inequalities]] | [[Category:Geometric Inequalities]] | ||
Latest revision as of 08:04, 7 June 2023
The Pythagorean inequality is a generalization of the Pythagorean theorem, which states that in a right triangle with sides of length 
we have 
. This inequality extends this to obtuse and acute triangles. The inequality states:
For an acute triangle with sides of length , 
. For an obtuse triangle with sides , 
.
This inequality is a direct result of the Law of cosines, although it is also possible to prove without using trigonometry.
