To find the range of possible values for the third side of a triangle when two sides are given as 10 and 28, we can apply the triangle inequality theorem. This theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.
We can derive two key inequalities from this theorem:
1. The length of the third side must be less than the sum of the other two sides.
2. The length of the third side must be greater than the difference between the other two sides.
Given the two sides are 10 and 28:
1. For the first condition:
[tex]\[
\text{Sum of the two sides} = 10 + 28 = 38
\][/tex]
Thus, the third side must be less than 38.
2. For the second condition:
[tex]\[
\text{Difference between the two sides} = |10 - 28| = 18
\][/tex]
Thus, the third side must be greater than 18.
Combining these two conditions, we find that the range of possible values for the third side, [tex]\( x \)[/tex], is:
[tex]\[
18 < x < 38
\][/tex]
So, the number that belongs in the green box is 18.
