To determine which set of side lengths represents a triangle with 3 lines of reflectional symmetry, let's define what such a triangle looks like. A triangle with 3 lines of reflectional symmetry is known as an equilateral triangle. An equilateral triangle has all three sides of equal length.
Let's analyze each set of side lengths given in the question:
1. Set: [tex]\(3, 4, 5\)[/tex]
- This is a scalene triangle, where all three sides are of different lengths. Therefore, it does not have 3 lines of reflectional symmetry.
2. Set: [tex]\(3, 6, 9\)[/tex]
- This set of side lengths does not form a valid triangle since the sum of any two sides is not greater than the third side. Thus, it cannot have any lines of reflectional symmetry.
3. Set: [tex]\(5, 5, 5\)[/tex]
- This is an equilateral triangle, where all three sides are equal. An equilateral triangle indeed has 3 lines of reflectional symmetry because all sides are the same length, making the triangle symmetric along lines that bisect each angle and meet the opposite side head-on.
4. Set: [tex]\(5, 10, 5\)[/tex]
- This is an isosceles triangle, where only two sides are of equal length. An isosceles triangle has only one line of reflectional symmetry, through the unequal side bisecting the angle formed by the two equal sides.
Given this analysis, the set of side lengths that represents a triangle with 3 lines of reflectional symmetry is:
[tex]\[ \boxed{5, 5, 5} \][/tex]
