Question

Which set of measures in centimeters could be the side lengths of a triangle?
A. 2cm 7cm 12cm
B. 4cm 10cm 3cm
C. 5cm 3cm 8cm
D. 11cm 21cm 16cm

Answer 1

To determine whether a set of lengths can form a triangle, we can use the triangle inequality theorem. This theorem states that for any three sides of a triangle (let's call them \( a \), \( b \), and \( c \), where \( a \leq b \leq c \)), the following must hold:

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

Let's evaluate each option:

**A. 2cm, 7cm, 12cm**
- Check: \( 2 + 7 > 12 \) → \( 9 > 12 \) (False)
- This set cannot form a triangle.

**B. 4cm, 10cm, 3cm**
- Sort: \( 3, 4, 10 \)
- Check: \( 3 + 4 > 10 \) → \( 7 > 10 \) (False)
- This set cannot form a triangle.

**C. 5cm, 3cm, 8cm**
- Sort: \( 3, 5, 8 \)
- Check: \( 3 + 5 > 8 \) → \( 8 > 8 \) (False, needs to be strictly greater)
- This set cannot form a triangle.

**D. 11cm, 21cm, 16cm**
- Sort: \( 11, 16, 21 \)
- Check:
- \( 11 + 16 > 21 \) → \( 27 > 21 \) (True)
- \( 11 + 21 > 16 \) → \( 32 > 16 \) (True)
- \( 16 + 21 > 11 \) → \( 37 > 11 \) (True)
- This set can form a triangle.

Based on the analysis, the only set of measures that could form the side lengths of a triangle is:

**D. 11cm, 21cm, 16cm**.