The Triangle Inequality Theorem Quiz

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Questions and Answers

Which of the following statements is true regarding the triangle inequality?

The triangle inequality is a theorem about distances in Euclidean geometry.
The triangle inequality only applies to non-degenerate triangles.
The triangle inequality is written using vectors and vector lengths.
The triangle inequality states that the sum of the lengths of any two sides must be greater than the length of the third side. (correct)

Which of the following is a correct representation of the triangle inequality using vectors and vector lengths?

$\|\mathbf{x} + \mathbf{y}\| < \|\mathbf{x}\| + \|\mathbf{y}\|$
$\|\mathbf{x} + \mathbf{y}\| \geq \|\mathbf{x}\| + \|\mathbf{y}\|$
$\|\mathbf{x} + \mathbf{y}\| > \|\mathbf{x}\| + \|\mathbf{y}\|$
$\|\mathbf{x} + \mathbf{y}\| \leq \|\mathbf{x}\| + \|\mathbf{y}\|$ (correct)

Which of the following is a correct statement about the triangle inequality?

The triangle inequality states that the sum of the lengths of any two sides must be less than the length of the third side.
The triangle inequality does not apply to degenerate triangles. (correct)
The triangle inequality is a theorem about angles in a triangle.
The triangle inequality is only applicable in Euclidean geometry.

Which of the following is a correct statement about the triangle inequality in Euclidean geometry?

The triangle inequality states that the sum of the lengths of any two sides must be greater than the length of the third side. (A) Signup and view all the answers

If x, y, and z are the lengths of the sides of a triangle, which of the following inequalities represents the triangle inequality?

$z \leq x + y$ (C) Signup and view all the answers

Define the triangle inequality 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 equation form, this can be expressed as $z \leq x + y$, where $x$, $y$, and $z$ are the lengths of the sides of the triangle. Signup and view all the answers

What does the triangle inequality state about the possibility of equality?

The triangle inequality permits the inclusion of degenerate triangles, but some authors, especially those writing about elementary geometry, will exclude this possibility, thus leaving out the possibility of equality. Signup and view all the answers

How is the triangle inequality written using vectors and vector lengths?

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): $|| \mathbf{x} + \mathbf{y} || \leq || \mathbf{x} || + || \mathbf{y} ||$, where the length $z$ of the third side has been replaced by the vector sum $\mathbf{x} + \mathbf{y}$. Signup and view all the answers

What is the significance of equality in the triangle inequality?

Equality in the triangle inequality only occurs in the degenerate case of a triangle with zero area. Signup and view all the answers

What are the lengths of the sides of a triangle in relation to the triangle inequality?

If $x$, $y$, and $z$ are the lengths of the sides of the triangle, with no side being greater than $z$, then the triangle inequality states that $z \leq x + y$. Signup and view all the answers

What is the triangle inequality?

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. Signup and view all the answers

What is the condition for equality in the triangle inequality?

The condition for equality in the triangle inequality is only in the degenerate case of a triangle with zero area. Signup and view all the answers

How is the triangle inequality written using vectors and vector lengths?

The triangle inequality using vectors and vector lengths is written as $||\mathbf{x} + \mathbf{y}|| \leq ||\mathbf{x}|| + ||\mathbf{y}||$. Signup and view all the answers

What does the triangle inequality theorem state in Euclidean geometry?

In Euclidean geometry, the triangle inequality is a theorem about distances and it 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. Signup and view all the answers

What does the triangle inequality permit in terms of triangles?

The triangle inequality permits the inclusion of degenerate triangles. Signup and view all the answers

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Study Notes

Triangle Inequality Overview

The triangle inequality is a fundamental principle in mathematics concerning the lengths of sides in a triangle.
It states that the sum of the lengths of any two sides must be greater than the length of the remaining side.

Triangle Inequality in Euclidean Geometry

In Euclidean geometry, if x, y, and z represent the lengths of a triangle's sides, the following inequalities hold:
- x + y > z
- x + z > y
- y + z > x

Vector Representation

The triangle inequality can be expressed using vectors as follows:
- |a + b| ≤ |a| + |b|
Here, |a| denotes the length of vector a, emphasizing the relationship between vector sums and their lengths.

Equality Condition

Equality in the triangle inequality occurs when the triangle in question degenerates into a straight line, which means the three points are collinear.
Specifically, equality holds if the sides are in the ratio that allows the sum of two sides to exactly equal the third side.

Significance of the Triangle Inequality

The triangle inequality establishes the necessary conditions for the existence of a triangle based on its side lengths.
It delineates constraints on triangle side lengths, ensuring that a triangle can indeed be formed.

Implications

Beyond its geometric application, the triangle inequality has significant implications in various fields of mathematics, including analysis and topology.
It ensures that distances measured using metrics fulfill necessary conditions for stability and consistency.