To determine if a set of side lengths can form a triangle, we can use the triangle inequality theorem. This theorem states that for any triangle with sides of lengths \( a \), \( b \), and \( c \), the following conditions must hold true:
- \( a + b > c \)
- \( a + c > b \)
- \( b + c > a \)
Let's evaluate the sets of side lengths provided:
-
5, 3, 2
- \( 5 + 3 = 8 > 2 \) (True)
- \( 5 + 2 = 7 > 3 \) (True)
- \( 3 + 2 = 5 \not> 5 \) (False)
This set cannot form a triangle.
-
3, 6, 9
- \( 3 + 6 = 9 \not> 9 \) (False)
- This set cannot form a triangle.
-
3, 7, 8
- \( 3 + 7 = 10 > 8 \) (True)
- \( 3 + 8 = 11 > 7 \) (True)
- \( 7 + 8 = 15 > 3 \) (True)
This set can form a triangle.
-
4, 1, 6
- \( 4 + 1 = 5 \not> 6 \) (False)
- This set cannot form a triangle.
Based on the evaluations, the only set of side lengths that will form a triangle is 3, 7, 8.