Geometry

Question Analysis

The main focus is to use the Triangle Inequality Theorem to find the range of possible lengths for the third side of a triangle when the lengths of two other sides are given.

Key Concept Explanation

The Triangle Inequality Theorem states that for a triangle with side lengths , , and , the following three inequalities must hold: a + b>c, a + c>b, and b + c>a. When two side lengths  and  are known, we can find the range of the third side  by using the inequalities |a - b|<c<a + b. This is based on the geometric fact that the shortest path between two points is a straight line, and the sum of the lengths of any two sides of a triangle must be greater than the length of the third side to form a closed figure.

Step - by - Step Solution

1. Let  and .

First, calculate the difference :

.

Then, calculate the sum :

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