To determine the possible values of [tex]\( n \)[/tex] for the side lengths of a triangle, we need to apply the Triangle Inequality Theorem. This theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. We have side lengths of [tex]\( 20 \, \text{cm} \)[/tex], [tex]\( 5 \, \text{cm} \)[/tex], and [tex]\( n \, \text{cm} \)[/tex].
So, we will consider the following three cases:
1. [tex]\( \text{Side1} + \text{Side2} > \text{Side3} \)[/tex]
2. [tex]\( \text{Side1} + \text{Side3} > \text{Side2} \)[/tex]
3. [tex]\( \text{Side2} + \text{Side3} > \text{Side1} \)[/tex]
Let's apply these inequalities to our sides:
### 1. [tex]\( 20 \, \text{cm} + 5 \, \text{cm} > n \)[/tex]
[tex]\[ 20 + 5 > n \][/tex]
[tex]\[ 25 > n \][/tex]
[tex]\[ n < 25 \][/tex]
### 2. [tex]\( 20 \, \text{cm} + n > 5 \, \text{cm} \)[/tex]
[tex]\[ 20 + n > 5 \][/tex]
[tex]\[ n > 5 - 20 \][/tex]
[tex]\[ n > -15 \][/tex]
Since [tex]\( n \)[/tex] represents a length and must be greater than 0, the inequality [tex]\( n > -15 \)[/tex] does not provide any additional restriction. Thus, we maintain [tex]\( n > 0 \)[/tex].
### 3. [tex]\( 5 \, \text{cm} + n > 20 \, \text{cm} \)[/tex]
[tex]\[ 5 + n > 20 \][/tex]
[tex]\[ n > 20 - 5 \][/tex]
[tex]\[ n > 15 \][/tex]
### Combining the inequalities
From the above steps, we have:
1. [tex]\( n < 25 \)[/tex]
2. No additional restriction from [tex]\( n > -15 \)[/tex] since [tex]\( n > 0 \)[/tex] already included this.
3. [tex]\( n > 15 \)[/tex]
Combining these results, the possible values for [tex]\( n \)[/tex] that satisfy all the conditions are:
[tex]\[ 15 < n < 25 \][/tex]
Therefore, the correct answer is:
[tex]\[ \boxed{15 < n < 25} \][/tex]
