To determine the range of possible values for the third side of the triangle, we need to use the Triangle Inequality Theorem. This 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.
Let's denote the sides of the triangle as [tex]\(a = 2\)[/tex], [tex]\(b = 7\)[/tex], and [tex]\(x\)[/tex] as the unknown length of the third side. The Triangle Inequality Theorem gives us three inequalities:
1. [tex]\(a + b > x\)[/tex]
2. [tex]\(a + x > b\)[/tex]
3. [tex]\(b + x > a\)[/tex]
Substituting the known values into these inequalities, we get:
1. [tex]\(2 + 7 > x\)[/tex], which simplifies to [tex]\(9 > x\)[/tex] or [tex]\(x < 9\)[/tex].
2. [tex]\(2 + x > 7\)[/tex], which simplifies to [tex]\(x > 5\)[/tex].
3. [tex]\(7 + x > 2\)[/tex], which simplifies to [tex]\(7 + x > 2\)[/tex], which is always true for any positive value of [tex]\(x\)[/tex] and does not provide a new constraint.
Combining these inequalities, we have:
[tex]\[5 < x < 9\][/tex]
So the range of possible values for [tex]\(x\)[/tex], the third side of the triangle, is given by the inequality:
[tex]\[5 < x < 9\][/tex]
Thus, the correct answer is:
C. [tex]\(5 < x < 9\)[/tex]
