To determine the range of possible values for the third side of an acute triangle given sides of 10 cm and 16 cm, let's break it down step-by-step:
1. Triangle Inequality Theorem:
- For any triangle with sides [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex], the sum of the lengths of any two sides must be greater than the length of the third side. This gives us three inequalities:
- [tex]\(a + b > c\)[/tex]
- [tex]\(a + c > b\)[/tex]
- [tex]\(b + c > a\)[/tex]
2. Applying the Triangle Inequality Theorem to Find the Range:
- Let [tex]\(a = 10\)[/tex], [tex]\(b = 16\)[/tex], and [tex]\(c\)[/tex] be the unknown side.
- From the triangle inequality theorem, we have:
[tex]\[
10 + 16 > c \implies c < 26
\][/tex]
[tex]\[
10 + c > 16 \implies c > 6
\][/tex]
[tex]\[
16 + c > 10 \implies c > -6 \quad \text{(always true for positive c)}
\][/tex]
Combining these, we get:
[tex]\[
6 < c < 26
\][/tex]
3. Condition for an Acute Triangle:
- For a triangle to be acute, the square of the length of the longest side must be less than the sum of the squares of the other two sides.
- In an acute triangle with sides [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] (where [tex]\( c \)[/tex] is the maximum side length), the following must be true:
[tex]\[
c^2 < a^2 + b^2
\][/tex]
In this scenario, we need to ensure that the third side still gives the triangle a shape where the square of the longest side is less than the sum of the squares of the other two sides.
Therefore, the correct range for the length of the third side that makes the triangle acute is:
[tex]\[
6 < x < 26
\][/tex]
So, the best answer from the given choices is:
[tex]\[
\boxed{6
