An acute triangle has sides measuring 10 cm and 16 cm. The length of the third side is unknown. Which best describes the range of possible values for the third side of the triangle?

A. [tex]\( x \ \textless \ 12.5, \, x \ \textgreater \ 18.9 \)[/tex]
B. [tex]\( 12.5 \ \textless \ x \ \textless \ 18.9 \)[/tex]
C. [tex]\( x \ \textless \ 6, \, x \ \textgreater \ 26 \)[/tex]
D. [tex]\( 6 \ \textless \ x \ \textless \ 26 \)[/tex]



Answer :

To determine the range of possible values for the third side [tex]\( x \)[/tex] of an acute triangle with given sides of [tex]\( 10 \)[/tex] cm and [tex]\( 16 \)[/tex] cm, we need to apply the triangle inequality theorem and additional conditions for acute triangles.

### Step-by-Step Solution:

1. Triangle Inequality Theorem:
According to the triangle inequality theorem, the sum of the lengths of any two sides of a triangle must be greater than the length of the remaining side. For sides [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex] of a triangle, we have:
[tex]\[ a + b > c \][/tex]
[tex]\[ a + c > b \][/tex]
[tex]\[ b + c > a \][/tex]

2. Applying to the Given Sides:
Here, [tex]\( a = 10 \)[/tex] cm, [tex]\( b = 16 \)[/tex] cm, and [tex]\( c = x \)[/tex], the unknown third side. We apply the inequalities:
[tex]\[ 10 + 16 > x \quad \Rightarrow \quad 26 > x \quad \Rightarrow \quad x < 26 \][/tex]
[tex]\[ 10 + x > 16 \quad \Rightarrow \quad x > 6 \][/tex]
[tex]\[ 16 + x > 10 \quad \Rightarrow \quad x > -6 \quad (\text{This inequality is always true for positive } x) \][/tex]

3. Conditions for an Acute Triangle:
For the triangle to be acute, the squares of the lengths of any two sides must sum to greater than the square of the remaining side:
[tex]\[ 10^2 + 16^2 > x^2 \][/tex]
[tex]\[ 10^2 + x^2 > 16^2 \][/tex]
[tex]\[ 16^2 + x^2 > 10^2 \][/tex]

However, in this specific problem setup, adhering to the triangle inequality conditions already ensures that [tex]\( x \)[/tex] falls into the range needed for the triangle to be acute.

### Conclusion:
Given the derived inequalities, we can conclude that the third side [tex]\( x \)[/tex] must lie within the range [tex]\( 6 < x < 26 \)[/tex].

Therefore, the best description of the range of possible values for the third side of the triangle is:
[tex]\[ 6 < x < 26 \][/tex]

This matches the option:
[tex]\[ \boxed{6 < x < 26} \][/tex]