# What is the triangle inequality theorem

## Triangle Inequality Theorem Learn about the Triangle Inequality Theorem: any side of a triangle must be shorter than the other two sides added together.

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In mathematics , the triangle inequality states that for any triangle , the sum of the lengths of any two sides must be greater than or equal to the length of the remaining side. In Euclidean geometry and some other geometries, the triangle inequality is a theorem about distances, and it is written using vectors and vector lengths norms :. In Euclidean geometry, for right triangles the triangle inequality is a consequence of the Pythagorean theorem , and for general triangles, a consequence of the law of cosines , although it may be proven without these theorems. The figure at the right shows three examples beginning with clear inequality top and approaching equality bottom. Thus, in Euclidean geometry, the shortest distance between two points is a straight line. The triangle inequality is a defining property of norms and measures of distance.

The Triangle Inequality Theorem-explained with pictures, examples, an interactive applet and several practice problems, explained step by step.
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The sum of the lengths of any two sides of a triangle is greater than the length of the third side. In the figure, the following inequalities hold. The sum of 7 and 9 is 16 and 16 is greater than The sum of 9 and 13 is 21 and 21 is greater than 7. The sum of 7 and 13 is 20 and 20 is greater than 9.

If you're seeing this message, it means we're having trouble loading external resources on our website. To log in and use all the features of Khan Academy, please enable JavaScript in your browser. Math High school geometry Geometry foundations Polygons. Triangle exterior angle example. Worked example: Triangle angles intersecting lines. Worked example: Triangle angles diagram. Triangle angle challenge problem 2.

The shortest distance between two points is a straight line. That is the heart of the triangle inequality theorem , which helps you determine quickly if a set of three numbers could be used to construct a triangle. Triangles are the simplest polygons, composed of only three sides, or line segments. Those three line segments cannot be just any random lengths, though. Only particular numbers can work, like a 3 - 4 - 5 triangle, with sides 3 units, 4 units and 5 units long:. We can immediately see that the 8-meter-long line segment is past the reach of the other two line segments.

## Triangle inequality

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## 1 thoughts on “What is the triangle inequality theorem”

1. Leto A. says:

Triangle Inequality Theorem. The sum of the lengths of any two sides of a triangle is greater than the length of the third side. In the figure, the following.