To determine the possible values of [tex]\( n \)[/tex] for a triangle with sides measuring 20 cm, 5 cm, and [tex]\( n \)[/tex] cm, we need to 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 remaining side.
Let's denote the sides of the triangle as:
- [tex]\( a = 20 \)[/tex] cm
- [tex]\( b = 5 \)[/tex] cm
- [tex]\( c = n \)[/tex] cm
Applying the triangle inequality theorem, we get three conditions:
1. [tex]\( a + b > c \)[/tex]
2. [tex]\( a + c > b \)[/tex]
3. [tex]\( b + c > a \)[/tex]
Using these conditions with our known values:
1. [tex]\( 20 + 5 > n \)[/tex]
[tex]\[ 25 > n \][/tex]
[tex]\[ n < 25 \][/tex]
2. [tex]\( 20 + n > 5 \)[/tex]
[tex]\[ 20 + n > 5 \][/tex]
[tex]\[ n > -15 \][/tex]
Since [tex]\( n \)[/tex] must be a positive length, this inequality [tex]\( n > -15 \)[/tex] is always true for positive [tex]\( n \)[/tex].
3. [tex]\( 5 + n > 20 \)[/tex]
[tex]\[ n > 15 \][/tex]
Combining these inequalities, we get:
[tex]\[ 15 < n < 25 \][/tex]
Hence, the possible values for [tex]\( n \)[/tex] are within the range [tex]\( 15 < n < 25 \)[/tex].
The correct answer is:
[tex]\[ 15 < n < 25 \][/tex]
So, the correct choice is:
[tex]\[ \boxed{15 < n < 25} \][/tex]
