To determine the range of possible values for the third side of a triangle when we know the lengths of the other two sides, we can use 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.
Given:
- Side 1: 10
- Side 2: 28
We need to find the minimum and maximum possible lengths for the third side of the triangle, which we will call Side 3.
### Step-by-Step Solution:
1. Establish the minimum value for Side 3:
To ensure the triangle inequality theorem holds, the difference between the lengths of Side 1 and Side 2 (i.e., [tex]\(|\text{Side 1} - \text{Side 2}|\)[/tex]) must be less than the length of Side 3. Mathematically, this is expressed as:
[tex]\[
\text{Side 3} > |\text{Side 1} - \text{Side 2}|
\][/tex]
Substituting the given values:
[tex]\[
\text{Side 3} > |10 - 28|
\][/tex]
Simplifying the expression inside the absolute value:
[tex]\[
\text{Side 3} > | -18 |
\][/tex]
[tex]\[
\text{Side 3} > 18
\][/tex]
Since Side 3 must be strictly greater than 18, the smallest integer greater than 18 is 19. Therefore, the minimum possible value for Side 3 is 19.
2. Establish the maximum value for Side 3:
Similarly, to ensure the triangle inequality theorem holds, the sum of the lengths of Side 1 and Side 2 must be greater than the length of Side 3. Mathematically, this is expressed as:
[tex]\[
\text{Side 3} < \text{Side 1} + \text{Side 2}
\][/tex]
Substituting the given values:
[tex]\[
\text{Side 3} < 10 + 28
\][/tex]
Simplifying this expression results in:
[tex]\[
\text{Side 3} < 38
\][/tex]
Since Side 3 must be strictly less than 38, the largest integer less than 38 is 37. Therefore, the maximum possible value for Side 3 is 37.
### Conclusion:
The range of possible values for the third side (Side 3) of the triangle given Side 1 as 10 and Side 2 as 28 is from 19 to 37 inclusive. So, the number that belongs in the green box is:
[tex]\[
\boxed{19 \text{ to } 37}
\][/tex]
