# Triangle Inequality: True or False?

• blinder
In summary, the Triangle Inequality states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side, and this is always true for any triangle. However, it can be false for degenerate triangles with one or more sides of zero length. The Triangle Inequality is used in geometry to determine the validity of triangles and is also used in proofs and theorems related to triangles. It cannot be applied to other shapes as it is specific to triangles. If it is violated, it means that the given side lengths cannot form a valid triangle, which could lead to errors in calculations or proofs.
blinder
Does the triangle inequality hold true for three vectors that says the norm(u+v+w)<=norm(u)+norm(v)+norm(w)...true or false

true because norm(u+v+w) = norm( (u+v)+w)
<= norm(u+v) + norm(w)
<= (norm(u) + norm(v)) + norm(w)
<= norm(u) + norm(v) + norm(w)

True. The triangle inequality holds true for three vectors. This inequality states that the sum of the lengths of any two sides of a triangle must be greater than or equal to the length of the third side. In this case, the norm of u+v+w represents the length of the third side, while the norm of u, v, and w represent the lengths of the other two sides. Therefore, the inequality holds true and can be applied to three vectors.

## 1. Is the Triangle Inequality always true?

Yes, the Triangle Inequality states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. This is always true for any triangle.

## 2. Can the Triangle Inequality be false for certain types of triangles?

Yes, the Triangle Inequality can be false for degenerate triangles, which are triangles with one or more sides that have zero length. In these cases, the sum of the lengths of two sides may be equal to the length of the third side, leading to a false Triangle Inequality.

## 3. How is the Triangle Inequality used in geometry?

The Triangle Inequality is used to determine whether a set of three given side lengths can form a valid triangle. It is also used in proofs and theorems related to triangles, such as the Pythagorean Theorem.

## 4. Can the Triangle Inequality be applied to other shapes besides triangles?

No, the Triangle Inequality is specific to triangles and cannot be applied to other shapes. It is based on the unique properties and relationships of the sides and angles in a triangle.

## 5. What happens if the Triangle Inequality is violated?

If the Triangle Inequality is violated, it means that the given side lengths cannot form a valid triangle. This could lead to errors in calculations or proofs that involve the triangle.

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