5.2.1 Indirect Proof and Inequalities in One Triangle (continued) Example 3 Applying the Triangle Inequality Theorem
The Triangle Inequality Theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. In other words, the length of any side of a triangle must be less than the sum of the other two sides.
When given the length of three segments, the Triangle Inequality Theorem can be used to determine whether the three segments can form a triangle. If the three lengths do not satisfy the Triangle Inequality Theorem, then those three segments cannot form a triangle.
Use the Triangle Inequality Theorem to determine whether each given trio of side lengths can form a triangle. The Triangle Inequality Theorem must hold for all three sides. In other words, it is not sufficient to show that the length of one side is less than the sum of the other two sides. This fact must be demonstrated for all three sides, so it takes three inequalities to show that three given side lengths form a triangle. However, having one side length that is greater than the sum of the other two sides is sufficient information for failing the Triangle Inequality Theorem. The third trio of side lengths is given as expressions and a value of the variable is given, a = 4. So, first substitute 4 into each expression to find the side length and then compare each side length to the sum of the other two sides to determine whether a triangle is formed.
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