Geometry

Question Analysis

The primary objective is to apply the Triangle Inequality Theorem to determine the smallest whole - number length 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 , three inequalities must hold: a + b>c, a + c>b, and b + c>a. When two side lengths are known, these inequalities can be used to find the range of possible values for the third side.

 

Step - by - Step Solution

1. Let and , and be the third side.

According to the theorem, the difference between the two given sides and their sum help define the range of . We know that |a - b|<c<a + b.

First, calculate : .

Then, calculate : .

So, 13 < c < 19.

2. Since

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