Question

For a craft project, four students each chose three pieces of wire from a box.

Don’s wires measure 3 inches, 5 inches, and 12 inches.
Margo’s wires measure 6 inches, 8 inches, and 14 inches.
Sonji’s wires measure 12 inches, 8 inches, and 17 inches.
Liam’s wires measure 16 inches, 8 inches, and 27 inches.

Which student chose pieces that can be used to construct a triangle?

Don
Margo
Sonji
Liam

Answer 1

To determine whether the wires chosen by each student can be used to construct 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 be met:

1. \(a + b > c\)
2. \(a + c > b\)
3. \(b + c > a\)

Let's check the wires of each student one by one.

**Don's Wires: 3 inches, 5 inches, 12 inches**
- \(3 + 5 = 8 > 12\) (False)
- Since the first condition fails, Don cannot form a triangle.

**Margo's Wires: 6 inches, 8 inches, 14 inches**
- \(6 + 8 = 14 > 14\) (False)
- The first condition fails again, so Margo cannot form a triangle.

**Sonji's Wires: 12 inches, 8 inches, 17 inches**
- \(12 + 8 = 20 > 17\) (True)
- \(12 + 17 = 29 > 8\) (True)
- \(8 + 17 = 25 > 12\) (True)
- All conditions are satisfied, so Sonji can form a triangle.

**Liam's Wires: 16 inches, 8 inches, 27 inches**
- \(16 + 8 = 24 > 27\) (False)
- The first condition fails, thus Liam cannot form a triangle.

Based on the evaluations, the only student whose wires can be used to construct a triangle is **Sonji**.