In a triangle, the sum of the lengths of any two sides must be greater than the length of the third side. This is known as the Triangle Inequality Theorem.
Using this theorem, we can determine which of the given sets of lengths could be sides of a triangle:
a) 5, 8, 3
5 + 8 = 13, which is less than 3. This violates the Triangle Inequality Theorem, so these lengths cannot form a triangle.
b) 8, 2, 8
2 + 8 = 10, which is less than 8. This violates the Triangle Inequality Theorem, so these lengths cannot form a triangle.
c) 9, 9, 9
All three sides are equal, so these lengths can form an equilateral triangle.
d) 2, 8, 14
2 + 8 = 10, which is less than 14. Also, 2 + 14 = 16, which is greater than 8. These lengths satisfy the Triangle Inequality Theorem, so they can form a triangle.
Therefore, the sets of lengths that could be sides of a triangle are c) 9, 9, 9 and d) 2, 8, 14.