To determine whether three side lengths can form a triangle, we can apply the Triangle Inequality Theorem, which states that for any triangle with sides \(a\), \(b\), and \(c\), the following inequalities must hold:
- \(a + b > c\)
- \(a + c > b\)
- \(b + c > a\)
Let's evaluate each set of side lengths:
-
20, 10, 30
- \(20 + 10 = 30\) (not greater)
- \(20 + 30 = 50\) (greater)
- \(10 + 30 = 40\) (greater)
- Does not satisfy the first condition. Cannot form a triangle.
-
10, 20, 15
- \(10 + 20 = 30\) (greater)
- \(10 + 15 = 25\) (greater)
- \(20 + 15 = 35\) (greater)
- All conditions satisfied. Forms a triangle.
-
8, 8, 20
- \(8 + 8 = 16\) (not greater)
- \(8 + 20 = 28\) (greater)
- \(8 + 20 = 28\) (greater)
- Does not satisfy the first condition. Cannot form a triangle.
-
8, 4, 12
- \(8 + 4 = 12\) (not greater)
- \(8 + 12 = 20\) (greater)
- \(4 + 12 = 16\) (greater)
- Does not satisfy the first condition. Cannot form a triangle.
Conclusion:
The only set of side lengths that can form a triangle is 10, 20, 15.