Answer :
To determine the range of possible values for the third side of an acute triangle, given the lengths of the other two sides are [tex]\(10 \, \text{cm}\)[/tex] and [tex]\(16 \, \text{cm}\)[/tex], we need to consider the following principles:
1. 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.
2. Condition for an Acute Triangle: All angles in an acute triangle are less than [tex]\(90^\circ\)[/tex].
First, let's apply the Triangle Inequality Theorem:
1. [tex]\(a + b > c\)[/tex]
2. [tex]\(a + c > b\)[/tex]
3. [tex]\(b + c > a\)[/tex]
For sides [tex]\(a = 10\)[/tex] cm and [tex]\(b = 16\)[/tex] cm, considering [tex]\(c\)[/tex] to be the unknown side length:
1. [tex]\(10 + 16 > c \Rightarrow c < 26\)[/tex]
2. [tex]\(10 + c > 16 \Rightarrow c > 6\)[/tex]
3. [tex]\(16 + c > 10 \Rightarrow c > -6\)[/tex] (which is always true since side lengths can't be negative)
From the above, we derive that the third side [tex]\(c\)[/tex] must satisfy:
[tex]\[ 6 < c < 26 \][/tex]
Now, we need to ensure this range fits the criterion for forming an acute triangle. In an acute triangle, all angles must be less than [tex]\(90^\circ\)[/tex]. Given the sides in question [tex]\(10 \, \text{cm}\)[/tex], [tex]\(16 \, \text{cm}\)[/tex], and [tex]\(c \, \text{cm}\)[/tex], ensuring all three angles are less than [tex]\(90^\circ\)[/tex] would automatically be taken care of by the valid range derived through the triangle inequality for the typical acute triangle sides given.
Therefore, the correct range of possible values for the length of the third side [tex]\(c\)[/tex] of the acute triangle is:
[tex]\[ 6 < x < 26 \][/tex]
Among the given options, the one that best describes this range is:
[tex]\[ \boxed{6
1. 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.
2. Condition for an Acute Triangle: All angles in an acute triangle are less than [tex]\(90^\circ\)[/tex].
First, let's apply the Triangle Inequality Theorem:
1. [tex]\(a + b > c\)[/tex]
2. [tex]\(a + c > b\)[/tex]
3. [tex]\(b + c > a\)[/tex]
For sides [tex]\(a = 10\)[/tex] cm and [tex]\(b = 16\)[/tex] cm, considering [tex]\(c\)[/tex] to be the unknown side length:
1. [tex]\(10 + 16 > c \Rightarrow c < 26\)[/tex]
2. [tex]\(10 + c > 16 \Rightarrow c > 6\)[/tex]
3. [tex]\(16 + c > 10 \Rightarrow c > -6\)[/tex] (which is always true since side lengths can't be negative)
From the above, we derive that the third side [tex]\(c\)[/tex] must satisfy:
[tex]\[ 6 < c < 26 \][/tex]
Now, we need to ensure this range fits the criterion for forming an acute triangle. In an acute triangle, all angles must be less than [tex]\(90^\circ\)[/tex]. Given the sides in question [tex]\(10 \, \text{cm}\)[/tex], [tex]\(16 \, \text{cm}\)[/tex], and [tex]\(c \, \text{cm}\)[/tex], ensuring all three angles are less than [tex]\(90^\circ\)[/tex] would automatically be taken care of by the valid range derived through the triangle inequality for the typical acute triangle sides given.
Therefore, the correct range of possible values for the length of the third side [tex]\(c\)[/tex] of the acute triangle is:
[tex]\[ 6 < x < 26 \][/tex]
Among the given options, the one that best describes this range is:
[tex]\[ \boxed{6
