To determine the range of possible values for the third side of a triangle, given the lengths of the two other sides, we can use the triangle inequality theorem. This theorem states the following for any triangle with sides [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex]:
1. The length of any one side must be less than the sum of the lengths of the other two sides.
2. The length of any one side must be more than the absolute difference of the lengths of the other two sides.
Let's denote the given sides as:
[tex]\[ a = 16 \][/tex]
[tex]\[ b = 27 \][/tex]
We need to find the range for the third side, [tex]\( x \)[/tex], such that it satisfies the triangle inequality theorem.
First, calculate the lower bound:
[tex]\[ x > |a - b| \][/tex]
[tex]\[ x > |16 - 27| \][/tex]
[tex]\[ x > 11 \][/tex]
Next, calculate the upper bound:
[tex]\[ x < a + b \][/tex]
[tex]\[ x < 16 + 27 \][/tex]
[tex]\[ x < 43 \][/tex]
Therefore, the range of values for the third side [tex]\( x \)[/tex] must satisfy:
[tex]\[ 11 < x < 43 \][/tex]
So, the number that belongs in the green box is:
[tex]\[ 43 \][/tex]
