To determine the possible lengths of the third side of a triangle when given two sides, we can use the triangle inequality theorem. The 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 lengths of the three sides of the triangle as [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex], where [tex]\( a = 6 \)[/tex] and [tex]\( b = 15 \)[/tex]. The third side is [tex]\( c \)[/tex]. The triangle inequality theorem provides us with three inequalities that must all be satisfied:
1. [tex]\( a + b > c \)[/tex]
2. [tex]\( a + c > b \)[/tex]
3. [tex]\( b + c > a \)[/tex]
By substituting the known lengths [tex]\( a = 6 \)[/tex] and [tex]\( b = 15 \)[/tex] into the inequalities, we can find the range of possible values for [tex]\( c \)[/tex].
1. [tex]\( 6 + 15 > c \)[/tex]
[tex]\[ 21 > c \][/tex]
Therefore, [tex]\( c < 21 \)[/tex].
2. [tex]\( 6 + c > 15 \)[/tex]
[tex]\[ c > 9 \][/tex]
Therefore, [tex]\( c > 9 \)[/tex].
3. [tex]\( 15 + c > 6 \)[/tex]
Since [tex]\( c \)[/tex] must be a positive length, this inequality is always true and does not add additional restrictions:
[tex]\[ c > -9 \][/tex]
Combining the results from the valid inequalities, we find that the third side [tex]\( c \)[/tex] must satisfy both [tex]\( c > 9 \)[/tex] and [tex]\( c < 21 \)[/tex].
Thus, the possible lengths for the third side [tex]\( c \)[/tex] of the triangle are greater than 9 and less than 21. Therefore, the possible lengths of the third side are in the interval [tex]\((9, 21)\)[/tex].
