To determine the range of possible values for the third side of a triangle, we will use 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 third side.
Given sides [tex]\( a \)[/tex] and [tex]\( b \)[/tex], the inequalities that must be satisfied to form a triangle are:
1. [tex]\( a + b > c \)[/tex]
2. [tex]\( a + c > b \)[/tex]
3. [tex]\( b + c > a \)[/tex]
Here, [tex]\( a = 9 \)[/tex] and [tex]\( b = 16 \)[/tex]. The remaining side will be represented by [tex]\( c \)[/tex].
Let's evaluate each of these inequalities:
1. [tex]\( a + b > c \)[/tex]
[tex]\[
9 + 16 > c
\][/tex]
[tex]\[
25 > c \quad \text{or} \quad c < 25
\][/tex]
2. [tex]\( a + c > b \)[/tex]
[tex]\[
9 + c > 16
\][/tex]
[tex]\[
c > 16 - 9
\][/tex]
[tex]\[
c > 7
\][/tex]
3. [tex]\( b + c > a \)[/tex]
[tex]\[
16 + c > 9
\][/tex]
[tex]\[
c > 9 - 16 \quad (\text{since } 16 > 9)
\][/tex]
[tex]\[
c > -7
\][/tex]
Since the length of a side cannot be negative, this inequality does not provide additional useful information.
Combining the inequalities derived in steps 1 and 2, we have:
[tex]\[
7 < c < 25
\][/tex]
Therefore, the range of possible values for the third side [tex]\( c \)[/tex] of the triangle is:
[tex]\[
7 < c < 25
\][/tex]
