a + b > c
a + c > b
b + c > a
If any of these inequalities is false, then the three segments cannot form a triangle. For example, if a = 5, b = 7, and c = 13, then the first inequality is false because 5 + 7 is not greater than 13. Therefore, these three segments cannot form a triangle.
How to Find the Range of Possible Lengths for the Third Side of a Triangle
To find the range of possible lengths for the third side of a triangle when two sides are given, you need to use the triangle inequality theorem in reverse. For example, if you know that two sides of a triangle have lengths 8 and 10, then you can find the range of possible lengths for the third side by solving the following inequalities:
8 + 10 > x
8 + x > 10
10 + x > 8
The first inequality gives you the upper bound for x, which is 18. The second and third inequalities give you the lower bound for x, which is 2. Therefore, the range of possible lengths for the third side is from 2 to 18, excluding these endpoints. In other words, 2 Practice Problems
Here are some practice problems for you to apply what you have learned. You can check your answers using the links provided.
Determine whether the segments with lengths 9, 12, and 15 can form a triangle. [Answer]
Determine whether the segments with lengths 3, 4, and 7 can form a triangle. [Answer]
If two sides of a triangle have lengths 5 and 9, what is the range of possible lengths for the third side? [Answer]
If two sides of a triangle have lengths 6 and 11, what is the range of possible lengths for the third side? [Answer]
References
The following sources were used to create this article:
[Unit 5 - Triangle Inequality THEOREM - Main Ideas/Questions Notes/Examples Triangle Inequality - Studocu]
[Chapter 5 - Relationships in Triangles - Get Ready for Chapter 5]
[Berkeley Heights Public Schools / Homepage]
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