Question

E) 4. If a triangle has sides that measure 8cm and 12cm, which of the following could be the third side?
(2 points)
20 cm
5 cm
24 cm
3 cm

Answer 1

To determine what the third side can be, we apply the triangle inequality theorem. This theorem states that for any triangle with sides of lengths \( a \), \( b \), and \( c \), the following must hold true:

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

In this case, let \( a = 8 \) cm, \( b = 12 \) cm, and \( c \) be the unknown third side. We need to check each option to see if it satisfies the triangle inequality.

1. **Option: 20 cm**
- \( 8 + 12 > 20 \) → \( 20 > 20 \) (not true)
- \( 8 + 20 > 12 \) → \( 28 > 12 \) (true)
- \( 12 + 20 > 8 \) → \( 32 > 8 \) (true)
- Fails the first condition.

2. **Option: 5 cm**
- \( 8 + 12 > 5 \) → \( 20 > 5 \) (true)
- \( 8 + 5 > 12 \) → \( 13 > 12 \) (true)
- \( 12 + 5 > 8 \) → \( 17 > 8 \) (true)
- All conditions are satisfied.

3. **Option: 24 cm**
- \( 8 + 12 > 24 \) → \( 20 > 24 \) (not true)
- \( 8 + 24 > 12 \) → \( 32 > 12 \) (true)
- \( 12 + 24 > 8 \) → \( 36 > 8 \) (true)
- Fails the first condition.

4. **Option: 3 cm**
- \( 8 + 12 > 3 \) → \( 20 > 3 \) (true)
- \( 8 + 3 > 12 \) → \( 11 > 12 \) (not true)
- \( 12 + 3 > 8 \) → \( 15 > 8 \) (true)
- Fails the second condition.

The only option that satisfies all triangle inequality conditions is **5 cm**. Thus, the third side could be:

**5 cm**