To determine the possible values of [tex]\( n \)[/tex] that form a valid triangle with sides measuring 20 cm and 5 cm, we will use the Triangle Inequality Theorem. This theorem states that for any triangle, the sum of the lengths of any two sides must be greater than the length of the third side. Specifically, for sides [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex] such that [tex]\( a \leq b \leq c \)[/tex], the following inequalities must hold:
1. [tex]\( a + b > c \)[/tex]
2. [tex]\( a + c > b \)[/tex]
3. [tex]\( b + c > a \)[/tex]
Let's label our sides as:
- [tex]\( a = 5 \)[/tex] cm
- [tex]\( b = n \)[/tex] cm
- [tex]\( c = 20 \)[/tex] cm
Now, we will apply the Triangle Inequality Theorem:
1. [tex]\( 5 + n > 20 \)[/tex]
Solving this inequality:
[tex]\[
n > 15
\][/tex]
2. [tex]\( 5 + 20 > n \)[/tex]
Solving this inequality:
[tex]\[
25 > n \quad \text{or} \quad n < 25
\][/tex]
3. [tex]\( n + 20 > 5 \)[/tex]
This inequality simplifies to:
[tex]\[
n > -15
\][/tex]
Since [tex]\( n \)[/tex] is a positive length, this inequality does not provide any new restrictions because it is always true for any positive [tex]\( n \)[/tex].
Therefore, the two key constraints for [tex]\( n \)[/tex] are:
[tex]\[
15 < n < 25
\][/tex]
Among the choices provided:
- [tex]\( 5 < n < 15 \)[/tex]
- [tex]\( 5 < n < 20 \)[/tex]
- [tex]\( 15 < n < 20 \)[/tex]
- [tex]\( 15 < n < 25 \)[/tex]
The correct description of the range for [tex]\( n \)[/tex] is:
[tex]\[
15 < n < 25
\][/tex]
Thus, the answer is:
[tex]\[
\boxed{15 < n < 25}
\][/tex]
