Answer :

GinnyAnswer
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]

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