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 \(\( a \)\) and \(\( b \)\), the inequalities that must be satisfied to form a triangle are:
1. \(\( a + b > c \)\)
2. \(\( a + c > b \)\)
3. \(\( b + c > a \)\)
Here, \(\( a = 9 \)\) and \(\( b = 16 \)\). The remaining side will be represented by \(\( c \)\).
Let's evaluate each of these inequalities:
1. \(\( a + b > c \)\)
\(\[
9 + 16 > c
\]\)
\(\[
25 > c \quad \text{or} \quad c < 25
\]\)
2. \(\( a + c > b \)\)
\(\[
9 + c > 16
\]\)
\(\[
c > 16 - 9
\]\)
\(\[
c > 7
\]\)
3. \(\( b + c > a \)\)
\(\[
16 + c > 9
\]\)
\(\[
c > 9 - 16 \quad (\text{since } 16 > 9)
\]\)
\(\[
c > -7
\]\)
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:
\(\[
7 < c < 25
\]\)
Therefore, the range of possible values for the third side \(\( c \)\) of the triangle is:
\(\[
7 < c < 25
\]\)
