Question

Which of the following are the possible side lengths of a triangle?
A.
m = 3 cm, n = 16 cm, p = 23 cm
B.
m = 7 cm, n = 16 cm, p = 19 cm
C.
m = 7 cm, n = 16 cm, p = 27 cm
D.
m = 7 cm, n = 12 cm, p = 23 cm

Answer 1

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:

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

Let's analyze each set of side lengths:

**A. \( m = 3 \, \text{cm}, n = 16 \, \text{cm}, p = 23 \, \text{cm} \)**

1. \( 3 + 16 = 19 > 23 \) (False)
2. \( 3 + 23 = 26 > 16 \) (True)
3. \( 16 + 23 = 39 > 3 \) (True)

Since the first condition is false, these lengths cannot form a triangle.

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**B. \( m = 7 \, \text{cm}, n = 16 \, \text{cm}, p = 19 \, \text{cm} \)**

1. \( 7 + 16 = 23 > 19 \) (True)
2. \( 7 + 19 = 26 > 16 \) (True)
3. \( 16 + 19 = 35 > 7 \) (True)

All conditions are satisfied, so these lengths can form a triangle.

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**C. \( m = 7 \, \text{cm}, n = 16 \, \text{cm}, p = 27 \, \text{cm} \)**

1. \( 7 + 16 = 23 > 27 \) (False)
2. \( 7 + 27 = 34 > 16 \) (True)
3. \( 16 + 27 = 43 > 7 \) (True)

Since the first condition is false, these lengths cannot form a triangle.

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**D. \( m = 7 \, \text{cm}, n = 12 \, \text{cm}, p = 23 \, \text{cm} \)**

1. \( 7 + 12 = 19 > 23 \) (False)
2. \( 7 + 23 = 30 > 12 \) (True)
3. \( 12 + 23 = 35 > 7 \) (True)

Since the first condition is false, these lengths cannot form a triangle.

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**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.