To determine the range of possible values for the third side of a triangle given two sides measuring 10 cm and 16 cm, we need to apply the triangle inequality theorem. This theorem states that for any triangle with sides [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex]:
1. [tex]\(a + b > c\)[/tex]
2. [tex]\(a + c > b\)[/tex]
3. [tex]\(b + c > a\)[/tex]
In our case, we have two known sides:
- [tex]\(a = 10 \text{ cm}\)[/tex]
- [tex]\(b = 16 \text{ cm}\)[/tex]
Let's denote the unknown third side by [tex]\(x\)[/tex]. We need to apply the triangle inequality theorem to find the constraints for [tex]\(x\)[/tex].
1. From [tex]\(a + b > x\)[/tex]:
[tex]\[10 + 16 > x \implies 26 > x \implies x < 26\][/tex]
2. From [tex]\(a + x > b\)[/tex]:
[tex]\[10 + x > 16 \implies x > 16 - 10 \implies x > 6\][/tex]
3. From [tex]\(x + b > a\)[/tex]:
[tex]\[x + 16 > 10\][/tex]
This inequality will always be true for any positive value of [tex]\(x\)[/tex], so it does not provide any additional constraints.
Combining the inequalities we obtained:
[tex]\[6 < x < 26\][/tex]
Hence, the range of possible values for the third side [tex]\(x\)[/tex] must be greater than 6 and less than 26. Therefore, the most accurate description of the range of possible values for the third side of this triangle is:
[tex]\[6 < x < 26\][/tex]
So, the correct answer is:
[tex]\[6
