Answer :
To determine the possible values of \( n \) for a triangle with side lengths \( 20 \, \text{cm}, 5 \, \text{cm} \), and \( n \, \text{cm} \), we will use 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 need to consider the three conditions of the triangle inequality theorem for the given side lengths:
1. The sum of the two given sides must be greater than the unknown side:
[tex]\[ 20 + 5 > n \implies 25 > n \implies n < 25 \][/tex]
2. The sum of the unknown side and one given side must be greater than the other given side:
[tex]\[ 20 + n > 5 \implies n > -15 \][/tex]
However, since side lengths cannot be negative, this condition simplifies to \( n > 0 \).
3. The sum of the unknown side and the other given side must be greater than the remaining given side:
[tex]\[ 5 + n > 20 \implies n > 15 \][/tex]
Combining these inequalities, we get:
[tex]\[ 15 < n < 25 \][/tex]
Thus, the possible values of \( n \) that satisfy all inequalities are \( 15 < n < 25 \).
Therefore, the correct answer is:
[tex]\[ \boxed{15 < n < 25} \][/tex]
We need to consider the three conditions of the triangle inequality theorem for the given side lengths:
1. The sum of the two given sides must be greater than the unknown side:
[tex]\[ 20 + 5 > n \implies 25 > n \implies n < 25 \][/tex]
2. The sum of the unknown side and one given side must be greater than the other given side:
[tex]\[ 20 + n > 5 \implies n > -15 \][/tex]
However, since side lengths cannot be negative, this condition simplifies to \( n > 0 \).
3. The sum of the unknown side and the other given side must be greater than the remaining given side:
[tex]\[ 5 + n > 20 \implies n > 15 \][/tex]
Combining these inequalities, we get:
[tex]\[ 15 < n < 25 \][/tex]
Thus, the possible values of \( n \) that satisfy all inequalities are \( 15 < n < 25 \).
Therefore, the correct answer is:
[tex]\[ \boxed{15 < n < 25} \][/tex]
