Which set of side lengths represents a triangle with 3 lines of reflectional symmetry? \( 3,4,5 \) \( 3,6,9 \) \( 5,5,5 \) \( 5,10,5 \)

Alright, I need to determine which set of side lengths represents a triangle with 3 lines of reflectional symmetry. Let's break this down step by step. First, I recall that a triangle with 3 lines of reflectional symmetry is an equilateral triangle. In an equilateral triangle, all three sides are equal, and all three angles are 60 degrees. This symmetry means that the triangle looks the same after being reflected across any of its three medians. Now, let's examine each option: 1. **3, 4, 5**: This set of side lengths is known as a Pythagorean triple, which forms a right-angled triangle. However, it's not equilateral because all sides are not equal. Therefore, it doesn't have 3 lines of reflectional symmetry. 2. **3, 6, 9**: This set of side lengths is also a multiple of the first set (3, 4, 5). Similar to the first option, it forms a right-angled triangle but with sides in the ratio 1:2:3. Since not all sides are equal, it doesn't have 3 lines of reflectional symmetry. 3. **5, 5, 5**: This set of side lengths represents an equilateral triangle because all three sides are equal. In an equilateral triangle, there are three lines of reflectional symmetry, each corresponding to the medians. This means the triangle looks the same after being reflected across any of these lines. 4. **5, 10, 5**: This set of side lengths is not a valid triangle because the sum of any two sides must be greater than the third side. Here, 5 + 5 = 10, which is not greater than 10. Therefore, this set does not form a valid triangle, let alone one with 3 lines of reflectional symmetry. After evaluating all options, it's clear that only the set **5, 5, 5** satisfies the condition of having 3 lines of reflectional symmetry.