To determine the possible values of [tex]\( n \)[/tex] for the triangle with side lengths [tex]\( 20 \)[/tex] cm, [tex]\( 5 \)[/tex] cm, and [tex]\( n \)[/tex] cm, we need to apply the triangle inequality theorem.
The triangle inequality 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. This involves three conditions:
1. The sum of the lengths of the first two sides must be greater than the third side.
2. The sum of the lengths of the second and third sides must be greater than the first side.
3. The sum of the lengths of the first and third sides must be greater than the second side.
We are given:
- One side, [tex]\( a = 20 \)[/tex] cm
- Another side, [tex]\( b = 5 \)[/tex] cm
- The third side, [tex]\( c = n \)[/tex] cm
Let's apply the triangle inequality theorem:
1. [tex]\( 20 + 5 > n \)[/tex]
[tex]\[
25 > n \quad \text{or} \quad n < 25
\][/tex]
2. [tex]\( 20 + n > 5 \)[/tex]
[tex]\[
20 + n > 5 \quad \Rightarrow \quad n > -15
\][/tex]
Note: Since side lengths must be positive, [tex]\( n > -15 \)[/tex] is always satisfied and is not a constraint in our range.
3. [tex]\( 5 + n > 20 \)[/tex]
[tex]\[
5 + n > 20 \quad \Rightarrow \quad n > 15
\][/tex]
By combining these inequalities, we find the range of [tex]\( n \)[/tex]:
[tex]\[
15 < n < 25
\][/tex]
Thus, the possible values of [tex]\( n \)[/tex] must satisfy the inequality [tex]\( 15 < n < 25 \)[/tex].
Therefore, the correct description of the possible values of [tex]\( n \)[/tex] is [tex]\( 15 < n < 25 \)[/tex].
