To determine whether the given sets of side lengths can form a triangle, we can use the triangle inequality theorem. This theorem states that for any triangle with side lengths \( a \), \( b \), and \( c \), the following conditions must hold:
- \( a + b > c \)
- \( a + c > b \)
- \( b + c > a \)
Let's analyze each set of side lengths:
A. \( m = 3 , \text{cm}, n = 16 , \text{cm}, p = 23 , \text{cm} \)
- \( 3 + 16 = 19 > 23 \) (False)
- \( 3 + 23 = 26 > 16 \) (True)
- \( 16 + 23 = 39 > 3 \) (True)
Since the first condition is false, these lengths cannot form a triangle.
B. \( m = 7 , \text{cm}, n = 16 , \text{cm}, p = 19 , \text{cm} \)
- \( 7 + 16 = 23 > 19 \) (True)
- \( 7 + 19 = 26 > 16 \) (True)
- \( 16 + 19 = 35 > 7 \) (True)
All conditions are satisfied, so these lengths can form a triangle.
C. \( m = 7 , \text{cm}, n = 16 , \text{cm}, p = 27 , \text{cm} \)
- \( 7 + 16 = 23 > 27 \) (False)
- \( 7 + 27 = 34 > 16 \) (True)
- \( 16 + 27 = 43 > 7 \) (True)
Since the first condition is false, these lengths cannot form a triangle.
D. \( m = 7 , \text{cm}, n = 12 , \text{cm}, p = 23 , \text{cm} \)
- \( 7 + 12 = 19 > 23 \) (False)
- \( 7 + 23 = 30 > 12 \) (True)
- \( 12 + 23 = 35 > 7 \) (True)
Since the first condition is false, these lengths cannot form a triangle.
Conclusion:
Among the options provided, only option B \( (7 , \text{cm}, 16 , \text{cm}, 19 , \text{cm}) \) can be the side lengths of a triangle.