Triangle Inequality Theorem
The Triangle Inequality Theorem is a fundamental principle in geometry that asserts that, for any triangle, the sum of the lengths of any two sides must always be greater than or equal to the length of the third side. This can be expressed mathematically for a triangle with sides labeled AB, BC, and CA, stating that AB + BC ≥ CA, BC + CA ≥ AB, and AB + CA ≥ BC. This theorem emphasizes an intuitive understanding of distances in Euclidean space, where the shortest path between two points is a straight line. Consequently, if you were to traverse the two other sides of a triangle, it would inherently be a longer route than the direct straight line between those two points.
In practical applications, the theorem is often used in problems where students are asked to determine if a triangle with specific side lengths can exist. This involves checking the sums of the lengths of pairs of sides against the length of the third side. For example, if given side lengths of 4, 16, and 7 inches, one would find that the condition is violated (4 + 7 < 16), indicating that such a triangle cannot exist. Understanding the Triangle Inequality Theorem is essential for deeper studies in geometry and serves as a basis for further exploration of mathematical concepts related to shapes and their properties.
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Subject Terms
Triangle Inequality Theorem
The triangle inequality theorem states that, given any triangle, the sum of the lengths of two sides of the triangle will always be more than or equivalent to the length of the third side of the triangle. Stated another way, if a triangle ABC has sides AB, BC, and CA, then according to the triangle inequality theorem AB + BC ≥CA, BC + CA≥AB, and AB + CA ≥ BC. In order for this relationship to no longer hold true, there would have to be a triangle that had an area of zero, which would essentially be equivalent to a single point.
Overview
Another way of understanding the triangle inequality theorem is to say that any side of a triangle will always be shorter than the combined lengths of the other two sides. Put in these terms, the triangle inequality theorem makes good intuitive sense, because one of the fundamental qualities of living in three dimensional Euclidean space is that the shortest possible distance between two points is a straight line. It then follows that for the two points at either end of a line segment that makes up one side of a triangle, the shortest distance between those two points is a straight line from one to the other. If one were to add together the lengths of the other two sides of the triangle, one would essentially be trying to go from one end of the line segment to the other end without travelling along the straight line connecting them. Because the straight line connecting them is the shortest distance between the points, it stands to reason that any other route, such as the route described by following the other two sides of the triangle, must be longer.
There are several types of problems that are frequently encountered by mathematics students concerning the triangle inequality theorem. The simplest type of problem involves a question that provides the lengths of the three sides of a hypothetical triangle, and asks one to simply determine whether or not it is possible for a triangle with the given dimensions to exist. Solving this type of problem requires one to take each set of two sides of the triangle, add their lengths together, and compare the result of this calculation with the length of the third side. If the sum is greater than the length of the third side, then the triangle that was proposed is indeed possible. If the sum is less than the length of the third side, then the proposed triangle is not possible, meaning that it could not exist. For example, one might be told that a triangle has sides of length 4, 16 and 7 inches, respectively. To find out if a triangle could actually have sides of these lengths, one would compare 4 + 16 to 7, 16 + 7 to 4, and 4 + 7 to 16. Making this comparison shows that 20 ≥ 7, 23 ≥ 4, and 11 ≤ 16. Because the last set of sides add up to a sum that is less than the remaining side, this triangle could not exist.
Bibliography
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