To determine which set of side lengths can form a triangle, we'll apply the Triangle Inequality Theorem. The theorem states that for any three sides \( a \), \( b \), and \( c \) of a triangle, the following must be true:
- \( a + b > c \)
- \( a + c > b \)
- \( b + c > a \)
Let's evaluate each set of side lengths:
-
8, 4, 12
- \( 8 + 4 = 12 \) (not greater, so fails)
- \( 8 + 12 > 4 \) (true)
- \( 4 + 12 > 8 \) (true)
- Does not form a triangle.
-
10, 20, 15
- \( 10 + 20 = 30 > 15 \) (true)
- \( 10 + 15 = 25 > 20 \) (true)
- \( 20 + 15 = 35 > 10 \) (true)
- Forms a triangle.
-
20, 10, 30
- \( 20 + 10 = 30 \) (not greater, so fails)
- \( 20 + 30 > 10 \) (true)
- \( 10 + 30 > 20 \) (true)
- Does not form a triangle.
-
8, 8, 20
- \( 8 + 8 = 16 < 20 \) (not greater, so fails)
- \( 8 + 20 > 8 \) (true)
- \( 8 + 20 > 8 \) (true)
- Does not form a triangle.
Conclusion: The only set of side lengths that can form a triangle is 10, 20, 15.