Difference between revisions of "Triangle Inequality"

Revision as of 13:38, 17 August 2006

The Triangle Inequality says that in a nondegenerate triangle $\displaystyle ABC$ :

$\displaystyle AB + BC > AC$

$\displaystyle BC + AC > AB$

$\displaystyle AC + AB > BC$

That is, the sum of the lengths of any two sides is larger than the length of the third side. In degenerate triangles, the strict inequality must be replaced by "greater than or equal to."

The Triangle Inequality can also be extended to other polygons. The lengths $a_1, a_2, \ldots, a_n$ can only be the sides of a nondegenerate $n$ -gon if $a_i < a_1 + \ldots + a_{i -1} + a_{i + 1} + \ldots + a_n = \left(\sum_{i=1}^n a_i\right) - a_i$ for $i = 1, 2 \ldots, n$ .

Example Problems

Introductory Problems

2006 AMC 10B Problem 10

2006 AIME II Problem 2

@@ Line 9: / Line 9: @@
 That is, the sum of the lengths of any two sides is larger than the length of the third side.
 In [[degenerate]] triangles, the [[strict inequality]] must be replaced by "greater than or equal to."
+The Triangle Inequality can also be extended to other [[polygon]]s.  The lengths <math>a_1, a_2, \ldots, a_n</math> can only be the sides of a nondegenerate <math>n</math>-gon if <math>a_i < a_1 + \ldots + a_{i -1} + a_{i + 1} + \ldots + a_n = \left(\sum_{i=1}^n a_i\right) - a_i</math> for <math>i = 1, 2 \ldots, n</math>.

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Difference between revisions of "Triangle Inequality"

Revision as of 13:38, 17 August 2006

Example Problems

Introductory Problems

See Also