To determine the possible values of [tex]\( n \)[/tex] for a triangle with side lengths [tex]\( 20 \, \text{cm} \)[/tex], [tex]\( 5 \, \text{cm} \)[/tex], and [tex]\( n \, \text{cm} \)[/tex], 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 remaining side.
Let's examine each condition given by the triangle inequality theorem for the sides [tex]\( 20 \)[/tex], [tex]\( 5 \)[/tex], and [tex]\( n \)[/tex]:
1. First condition: [tex]\( 20 + 5 > n \)[/tex]
[tex]\[
25 > n
\][/tex]
2. Second condition: [tex]\( 20 + n > 5 \)[/tex]
[tex]\[
n > -15
\][/tex]
Since [tex]\( n \)[/tex] must be a positive length, this inequality is always satisfied (as [tex]\( n > 0 \)[/tex]) and doesn't provide a new constraint.
3. Third condition: [tex]\( 5 + n > 20 \)[/tex]
[tex]\[
n > 15
\][/tex]
Now, combining the inequalities [tex]\( 25 > n \)[/tex] and [tex]\( n > 15 \)[/tex]:
[tex]\[
15 < n < 25
\][/tex]
Therefore, the possible values of [tex]\( n \)[/tex] for which a triangle can be formed with sides [tex]\( 20 \, \text{cm} \)[/tex], [tex]\( 5 \, \text{cm} \)[/tex], and [tex]\( n \)[/tex] are [tex]\( 15 < n < 25 \)[/tex].
Thus, the correct answer is:
[tex]\[ 15 < n < 25 \][/tex]
