Geometry

Question Analysis

The task is to apply the Triangle Inequality Theorem to determine the range of possible lengths for the third side of a triangle when the lengths of two other sides are provided.

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 theorem is based on the geometric principle that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side in order to form a closed geometric figure.

Step - by - Step Solution

1. Let  and .

First, calculate :