Question

9 4 0 6 1 7 5 10. Iba constructs nABC in her notebook. ··· AB is 12 centimeters long. ··· BC is 10 centimeters long. Which measurement could be the length of ··· AC ? Select all that apply. 햠 2 cm 햡 8 cm 햢 16 cm 햣 22 cm 햤 24 cm

Answer 1

To determine which lengths could be the length of segment AC in triangle ABC, we can apply the triangle inequality theorem. This theorem states that for any triangle with sides \( a \), \( b \), and \( c \):

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

In our case, let:
- \( AB = 12 \) cm
- \( BC = 10 \) cm
- \( AC = x \) cm (where \( x \) is the unknown length we want to find)

We need to apply the triangle inequality for our triangle ABC:

1. From \( AB + BC > AC \):
\[
12 + 10 > x \quad \Rightarrow \quad 22 > x \quad \Rightarrow \quad x < 22
\]

2. From \( AB + AC > BC \):
\[
12 + x > 10 \quad \Rightarrow \quad x > -2 \quad (\text{This condition is always satisfied since } x \text{ must be positive})
\]

3. From \( BC + AC > AB \):
\[
10 + x > 12 \quad \Rightarrow \quad x > 2
\]

Now combining the valid inequalities, we find:
\[
2 < x < 22
\]

The possible measurements for \( AC \) are:
- \( 2 \) cm (not applicable since it must be greater than 2)
- \( 8 \) cm (valid because \( 2 < 8 < 22 \))
- \( 16 \) cm (valid because \( 2 < 16 < 22 \))
- \( 22 \) cm (not applicable since it must be less than 22)
- \( 24 \) cm (not applicable since it must be less than 22)

Thus, the valid measurements for \( AC \) are:

- **8 cm**
- **16 cm**