Question

Can the sides of a triangle have lengths 5, 11, and 16?

Answer 1

To determine if three 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 be satisfied:

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

Let's denote the lengths of the sides as \( a = 5 \), \( b = 11 \), and \( c = 16 \). We will check each of the inequalities:

1. \( 5 + 11 > 16 \)
- \( 16 > 16 \) (This is **false**)

2. \( 5 + 16 > 11 \)
- \( 21 > 11 \) (This is **true**)

3. \( 11 + 16 > 5 \)
- \( 27 > 5 \) (This is **true**)

Since the first condition \( 5 + 11 > 16 \) is not satisfied, the lengths 5, 11, and 16 cannot form a triangle. Therefore, the answer is **no**.