Geometry

Question Analysis

The main focus lies in using the Triangle Inequality Theorem to determine the feasible range of lengths for the third side of a triangle, given the lengths of two other sides.

Key Concept Explanation

The Triangle Inequality Theorem states that for any 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 are known, we can determine the range of the third side  using the inequalities |a - b|<c<a + b. Geometrically, this ensures that the three sides can enclose an area to form a triangle.

Step - by - Step Solution

1. Let  and .

Calculate :

.

Calculate

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