To determine the possible range for the third side of a triangle when given two of its sides, we can utilize the triangle inequality theorem. This theorem states that for any triangle, the sum of the lengths of any two sides must be greater than the length of the remaining side.
Given the sides of the triangle as 198 and 222, let’s denote them as [tex]\(a\)[/tex] and [tex]\(b\)[/tex]. The third side will be denoted as [tex]\(c\)[/tex]. According to the triangle inequality theorem, we have three conditions to satisfy:
1. [tex]\(a + b > c\)[/tex]
2. [tex]\(a + c > b\)[/tex]
3. [tex]\(b + c > a\)[/tex]
Substituting [tex]\(a = 198\)[/tex] and [tex]\(b = 222\)[/tex], the conditions become:
1. [tex]\(198 + 222 > c\)[/tex]
2. [tex]\(198 + c > 222\)[/tex]
3. [tex]\(222 + c > 198\)[/tex]
Let's analyze these inequalities one by one:
1. [tex]\(198 + 222 > c\)[/tex]
[tex]\[
420 > c \quad \text{or} \quad c < 420
\][/tex]
2. [tex]\(198 + c > 222\)[/tex]
[tex]\[
c > 222 - 198 \quad \text{or} \quad c > 24
\][/tex]
3. [tex]\(222 + c > 198\)[/tex]
[tex]\[
c > 198 - 222 \quad \text{or} \quad c > -24
\][/tex]
Since [tex]\(c\)[/tex] must be positive in the context of triangle sides, [tex]\(c > -24\)[/tex] is always true and doesn't provide any new information. Therefore, we focus on the significant inequalities:
[tex]\[
24 < c < 420
\][/tex]
Thus, combining these results, the third side of the triangle must be greater than 24 and less than 420. Hence, the range of possible values for the third side is:
[tex]\[
(24, 420)
\][/tex]
This range can be expressed in interval notation as:
[tex]\[
24 < c < 420
\][/tex]
