To determine which set of numbers cannot represent the three sides of a triangle, we need to use the triangle inequality theorem. The triangle inequality theorem states that for any three sides \(a\), \(b\), and \(c\) of a triangle, the following conditions must be satisfied:
1. \(a + b > c\)
2. \(a + c > b\)
3. \(b + c > a\)
We will test each given set of numbers against these conditions.
Set 1: \(\{8, 20, 27\}\)
1. \(8 + 20 > 27\) ⟹ \(28 > 27\), which is true.
2. \(8 + 27 > 20\) ⟹ \(35 > 20\), which is true.
3. \(20 + 27 > 8\) ⟹ \(47 > 8\), which is true.
This set satisfies the triangle inequality theorem.
Set 2: \(\{8, 19, 25\}\)
1. \(8 + 19 > 25\) ⟹ \(27 > 25\), which is true.
2. \(8 + 25 > 19\) ⟹ \(33 > 19\), which is true.
3. \(19 + 25 > 8\) ⟹ \(44 > 8\), which is true.
This set satisfies the triangle inequality theorem.
Set 3: \(\{13, 20, 32\}\)
1. \(13 + 20 > 32\) ⟹ \(33 > 32\), which is true.
2. \(13 + 32 > 20\) ⟹ \(45 > 20\), which is true.
3. \(20 + 32 > 13\) ⟹ \(52 > 13\), which is true.
This set satisfies the triangle inequality theorem.
Set 4: \(\{6, 8, 16\}\)
1. \(6 + 8 > 16\) ⟹ \(14 > 16\), which is false.
2. \(6 + 16 > 8\) ⟹ \(22 > 8\), which is true.
3. \(8 + 16 > 6\) ⟹ \(24 > 6\), which is true.
Since one of the conditions of the triangle inequality theorem (\(6 + 8 > 16\)) is false, this set does not satisfy the theorem and thus cannot represent the sides of a triangle.
Conclusion:
The set that could not represent the three sides of a triangle is \(\{6, 8, 16\}\).
Thus, the answer is the fourth set, [tex]\(\{6, 8, 16\}\)[/tex].
