To determine the range of possible values for the third side of a triangle when two sides are given, we use the properties of the triangle inequality theorem. This theorem states that for any triangle, the sum of any two sides must be greater than the third side. We'll denote the two given sides as [tex]\(a = 5\)[/tex] and [tex]\(b = 16\)[/tex]. Let's denote the third side as [tex]\(c\)[/tex].
According to the triangle inequality theorem, the following inequalities must hold true:
1. [tex]\(a + b > c\)[/tex]
2. [tex]\(a + c > b\)[/tex]
3. [tex]\(b + c > a\)[/tex]
We'll apply these inequalities to our given values to find the valid range for [tex]\(c\)[/tex]:
1. [tex]\(a + b > c\)[/tex]:
[tex]\[
5 + 16 > c \Rightarrow 21 > c
\][/tex]
2. [tex]\(a + c > b\)[/tex]:
[tex]\[
5 + c > 16 \Rightarrow c > 16 - 5 \Rightarrow c > 11
\][/tex]
3. [tex]\(b + c > a\)[/tex]:
[tex]\[
16 + c > 5 \Rightarrow c > 5 - 16 \Rightarrow c > -11
\][/tex]
However, since [tex]\(c\)[/tex] must be positive and the other two inequalities already cover this condition, we do not need to consider this particular inequality further.
From the first two relevant inequalities, we deduce that:
[tex]\[
11 < c < 21
\][/tex]
Therefore, the range of values that [tex]\(c\)[/tex] can take is between [tex]\(11\)[/tex] and [tex]\(21\)[/tex]. Any value within this range will satisfy the triangle inequality theorem, ensuring that a triangle can be formed from the sides of lengths [tex]\(5\)[/tex], [tex]\(16\)[/tex], and [tex]\(c\)[/tex].
The green box should contain the range:
[tex]\[
(11, 21)
\][/tex]
