To determine the possible values of [tex]\( n \)[/tex] for the third side of a triangle given side lengths of [tex]\( 20 \)[/tex] cm and [tex]\( 5 \)[/tex] cm, we use the triangle inequality theorem. The theorem states:
1. The sum of the lengths of any two sides of a triangle must be greater than the length of the third side.
2. This requirement must be satisfied for all three combinations of the sides.
Given the sides [tex]\( a = 20 \)[/tex] cm, [tex]\( b = 5 \)[/tex] cm, and [tex]\( c = n \)[/tex] cm, we can derive the inequalities as follows:
1. [tex]\( a + b > c \)[/tex]
2. [tex]\( a + c > b \)[/tex]
3. [tex]\( b + c > a \)[/tex]
Substituting the given side lengths into the inequalities:
1. [tex]\( 20 + 5 > n \)[/tex]
[tex]\[ 25 > n \][/tex]
[tex]\[ n < 25 \][/tex]
2. [tex]\( 20 + n > 5 \)[/tex]
[tex]\[ n > -15 \][/tex]
Since [tex]\( n \)[/tex] must be a positive length, we disregard [tex]\( n > -15 \)[/tex] as it is always true for any positive number.
3. [tex]\( 5 + n > 20 \)[/tex]
[tex]\[ n > 15 \][/tex]
Combining these inequalities gives:
[tex]\[ 15 < n < 25 \][/tex]
Therefore, the correct range for the side length [tex]\( n \)[/tex] is [tex]\( 15 < n < 25 \)[/tex].
From the given multiple-choice options, the correct answer is:
[tex]\[ 15 < n < 25 \][/tex]
