To find the range of possible measures for the third side of a triangle when the other two sides are given, we 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 third side.
Given two sides of a triangle with lengths 12 and 10, let's denote the third side as [tex]\( x \)[/tex].
We need to satisfy the following conditions:
1. The sum of the lengths of any two sides must be greater than the length of the third side.
2. Similarly, the length of the third side must also be greater than the absolute difference of the two sides.
Applying these conditions:
1. For the third side [tex]\( x \)[/tex]:
[tex]\[
x + 10 > 12
\][/tex]
[tex]\[
x + 12 > 10
\][/tex]
[tex]\[
10 + 12 > x
\][/tex]
2. Simplifying these inequalities, we get:
[tex]\[
x > 2 \quad \text{(from } x + 10 > 12 \text{)}
\][/tex]
[tex]\[
x > -2 \quad \text{(from } x + 12 > 10 \text{, which is always true for positive } x \text{)}
\][/tex]
[tex]\[
x < 22 \quad \text{(from } 10 + 12 > x \text{)}
\][/tex]
Considering these results together, the third side [tex]\( x \)[/tex] must be greater than 2 and less than 22. Therefore, the range of possible measures for the third side is:
[tex]\[
2 < x < 22
\][/tex]
So, the correct answer is:
[tex]\[
2 < x < 22
\][/tex]
