To determine the possible values of [tex]\( n \)[/tex] in a triangle with side lengths measuring [tex]\( 20 \)[/tex] cm, [tex]\( 5 \)[/tex] cm, and [tex]\( n \)[/tex] cm, we must ensure the triangle inequality theorem is satisfied. The theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the remaining side.
We have three inequalities to consider:
1. [tex]\( 20 + 5 > n \)[/tex]
2. [tex]\( 20 + n > 5 \)[/tex]
3. [tex]\( 5 + n > 20 \)[/tex]
Let's break these down one by one:
1. [tex]\( 20 + 5 > n \)[/tex]
[tex]\[ 25 > n \][/tex]
[tex]\[ n < 25 \][/tex]
2. [tex]\( 20 + n > 5 \)[/tex]
[tex]\[ n > 5 - 20 \][/tex]
[tex]\[ n > -15 \][/tex]
Since [tex]\( n \)[/tex] must be a positive length, we adapt this inequality to:
[tex]\[ n > 5 \][/tex]
3. [tex]\( 5 + n > 20 \)[/tex]
[tex]\[ n > 20 - 5 \][/tex]
[tex]\[ n > 15 \][/tex]
Combining these inequalities, we get:
[tex]\[ 15 < n < 25 \][/tex]
Therefore, the possible values of [tex]\( n \)[/tex] fall within the range [tex]\( 15 < n < 25 \)[/tex].
The correct answer is:
[tex]\[ \boxed{15 < n < 25} \][/tex]
