Sure! Let's determine the range of possible values for the third side of a triangle when the lengths of the other two sides are given as 8 and 10.
To do this, we will apply the Triangle Inequality Theorem. The Triangle Inequality 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. This gives us three inequalities to consider:
1. The sum of the first and second sides must be greater than the third side:
[tex]\(a + b > c\)[/tex]
2. The sum of the first and third sides must be greater than the second side:
[tex]\(a + c > b\)[/tex]
3. The sum of the second and third sides must be greater than the first side:
[tex]\(b + c > a\)[/tex]
Here, the given side lengths are [tex]\(a = 8\)[/tex] and [tex]\(b = 10\)[/tex]. Let [tex]\(c\)[/tex] be the length of the third side.
Let's go through each inequality step by step:
1. [tex]\(8 + 10 > c\)[/tex]
- Simplifies to: [tex]\(18 > c\)[/tex]
- This means [tex]\(c\)[/tex] must be less than 18.
2. [tex]\(8 + c > 10\)[/tex]
- Simplifies to: [tex]\(c > 2\)[/tex]
- This means [tex]\(c\)[/tex] must be greater than 2.
3. [tex]\(10 + c > 8\)[/tex]
- Simplifies to: [tex]\(c > -2\)[/tex], but since side lengths are always positive, this condition is always satisfied and does not impose any additional constraints.
Combining the results from the inequalities, we find that:
- [tex]\(c\)[/tex] must be greater than 2.
- [tex]\(c\)[/tex] must be less than 18.
Therefore, the range of possible values for the third side ([tex]\(c\)[/tex]) is:
[tex]\[2 < c < 18\][/tex]
So the third side must be in the range [tex]\( (3, 17) \)[/tex] when considering the strict inequality rules.
Thus:
The number that belongs in the green box is [tex]\(c\)[/tex] such that [tex]\(2 < c < 18\)[/tex].
