To determine the range of possible values for the third side of a triangle when given the lengths of the other two sides, we can use the triangle inequality theorem. This theorem states that for any triangle, the length of one side must be less than the sum of the lengths of the other two sides, and greater than the positive difference between the lengths of the other two sides.
Given the side lengths:
- [tex]\( a = 8 \)[/tex]
- [tex]\( b = 12 \)[/tex]
We need to find the range for the third side, which we'll call [tex]\( c \)[/tex].
The conditions set by the triangle inequality theorem are:
1. [tex]\( c \)[/tex] must be greater than the absolute difference of [tex]\( a \)[/tex] and [tex]\( b \)[/tex]:
[tex]\[
c > |a - b|
\][/tex]
2. [tex]\( c \)[/tex] must be less than the sum of [tex]\( a \)[/tex] and [tex]\( b \)[/tex]:
[tex]\[
c < a + b
\][/tex]
First, we compute the absolute difference of [tex]\( a \)[/tex] and [tex]\( b \)[/tex]:
[tex]\[
|a - b| = |8 - 12| = 4
\][/tex]
Next, we compute the sum of [tex]\( a \)[/tex] and [tex]\( b \)[/tex]:
[tex]\[
a + b = 8 + 12 = 20
\][/tex]
According to the triangle inequality, the third side [tex]\( c \)[/tex] must satisfy:
[tex]\[
4 < c < 20
\][/tex]
Thus, the range of values for the third side [tex]\( c \)[/tex] is between 4 and 20 (not inclusive).
The final answer is:
[tex]\[
(4, 20)
\][/tex]
