15 Questions
Which of the following statements is true regarding the triangle inequality?
The triangle inequality states that the sum of the lengths of any two sides must be greater than the length of the third side.
Which of the following is a correct representation of the triangle inequality using vectors and vector lengths?
$|\mathbf{x} + \mathbf{y}| \leq |\mathbf{x}| + |\mathbf{y}|$
Which of the following is a correct statement about the triangle inequality?
The triangle inequality does not apply to degenerate triangles.
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.
If x, y, and z are the lengths of the sides of a triangle, which of the following inequalities represents the triangle inequality?
$z \leq x + y$
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.
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.
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$|| \mathbf{x} + \mathbf{y} || \leq || \mathbf{x} || + || \mathbf{y} ||$, where the length $z$ of the third side has been replaced by the vector sum $\mathbf{x} + \mathbf{y}$.
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.
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 $z \leq x + y$.
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.
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.
How is the triangle inequality written using vectors and vector lengths?
The triangle inequality using vectors and vector lengths is written as $||\mathbf{x} + \mathbf{y}|| \leq ||\mathbf{x}|| + ||\mathbf{y}||$.
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.
What does the triangle inequality permit in terms of triangles?
The triangle inequality permits the inclusion of degenerate triangles.
Test your knowledge of the triangle inequality theorem and its applications with this quiz! Explore the relationship between the lengths of the sides in a triangle and discover when equality is possible.
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