To solve this problem, we need to determine the possible range for the length of the third side of a triangle when the other two sides measure 10 and 28 units. We'll use the triangle inequality theorem, which states the following:
1. The sum of the lengths of any two sides of a triangle must be greater than the length of the third side.
2. The difference between the lengths of any two sides of a triangle must be less than the length of the third side.
Let's denote the three sides of the triangle as [tex]\( a = 10 \)[/tex], [tex]\( b = 28 \)[/tex], and [tex]\( c \)[/tex] (where [tex]\( c \)[/tex] is the unknown third side).
According to the triangle inequality theorem, we need to satisfy these inequalities:
1. [tex]\( a + b > c \)[/tex] \rightarrow [tex]\( 10 + 28 > c \)[/tex] \rightarrow [tex]\( 38 > c \)[/tex]
2. [tex]\( a + c > b \)[/tex] \rightarrow [tex]\( 10 + c > 28 \)[/tex] \rightarrow [tex]\( c > 18 \)[/tex]
3. [tex]\( b + c > a \)[/tex] \rightarrow [tex]\( 28 + c > 10 \)[/tex] \rightarrow [tex]\( c > -18 \)[/tex] (this condition is always true and does not impose any additional limitations).
So, taking into account the inequalities from steps 1 and 2:
- [tex]\( c > 18 \)[/tex]
- [tex]\( c < 38 \)[/tex]
Therefore, the range of values for [tex]\( c \)[/tex] is [tex]\( 18 < c < 38 \)[/tex].
This means that the range for the third side [tex]\( c \)[/tex] is between 19 and 37 inclusively.
The value that belongs in the green box is
[tex]\[
37
\][/tex]
