Question

Which choice shows three lengths that cannot be the lengths of the three sides of a triangle?

A. 2 cm, 8 cm, 8 cm
B. 2 cm, 3 cm, 6 cm
C. 4 cm, 5 cm, 7 cm
D. 5 cm, 6 cm, 9 cm

Answer 1

To determine which choice shows three lengths that cannot be the lengths of the sides of a triangle, we can use the triangle inequality theorem. This theorem states that for any triangle with sides of lengths \(a\), \(b\), and \(c\) (where \(a \leq b \leq c\)), the following must hold true:

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

To evaluate each option:

A. \(2 \, \text{cm}, 8 \, \text{cm}, 8 \, \text{cm}\)
- \(2 + 8 = 10 > 8\) (true)
- \(2 + 8 = 10 > 8\) (true)
- \(8 + 8 = 16 > 2\) (true)
This set can form a triangle.

B. \(2 \, \text{cm}, 3 \, \text{cm}, 6 \, \text{cm}\)
- \(2 + 3 = 5 \not> 6\) (false)
This set cannot form a triangle.

C. \(4 \, \text{cm}, 5 \, \text{cm}, 7 \, \text{cm}\)
- \(4 + 5 = 9 > 7\) (true)
- \(4 + 7 = 11 > 5\) (true)
- \(5 + 7 = 12 > 4\) (true)
This set can form a triangle.

D. \(5 \, \text{cm}, 6 \, \text{cm}, 9 \, \text{cm}\)
- \(5 + 6 = 11 > 9\) (true)
- \(5 + 9 = 14 > 6\) (true)
- \(6 + 9 = 15 > 5\) (true)
This set can form a triangle.

The only choice that cannot form a triangle is:

**B. 2 cm, 3 cm, 6 cm**