Question

If a triangle has sides that measure 8cm and 12cm, which of the following could be the third side?(2 points)
Responses

20 cm
20 cm

24 cm
24 cm

3 cm
3 cm

5 cm

Answer 1

To determine the possible lengths of the third side of a triangle when the other two sides measure 8 cm and 12 cm, we can use the triangle inequality theorem. The triangle inequality states that for any triangle with sides \( a\), \( b\), and \( c\):

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

Let the lengths of the sides be \( a = 8 \) cm, \( b = 12 \) cm, and \( c \) be the unknown side. We can set up the inequalities:

1. \( 8 + 12 > c \) → \( 20 > c \) → \( c < 20 \)
2. \( 8 + c > 12 \) → \( c > 4 \)
3. \( 12 + c > 8 \) → \( c > -4\) (this inequality will always be true for positive lengths)

Combining these conditions, we find:

4 < c < 20

Now we can evaluate the given options for the third side:

- **20 cm**: Not possible (c must be less than 20).
- **24 cm**: Not possible (c must be less than 20).
- **3 cm**: Not possible (c must be greater than 4).
- **5 cm**: Possible (c is greater than 4 and less than 20).

Thus, the only possible length for the third side is **5 cm**.