Question

In triangle ABD, AB is 3 cm long and BD is 5 cm long. According to the Triangle Inequality Theorum, which two lengths are possible lengths of AD?
Answers
A. 4 cm
B. 2 cm
C. 7 cm
D. 10 cm
E. 1 cm

Answer 1

To determine the possible lengths of \( AD \) in triangle \( ABD \), we can apply the Triangle Inequality Theorem, which states that for any triangle with sides \( a \), \( b \), and \( c \):

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

In this case, we have:

- \( AB = 3 \, \text{cm} \)
- \( BD = 5 \, \text{cm} \)
- \( AD = x \, \text{cm} \) (the length we want to find)

Applying Triangle Inequality to triangle \( ABD \):

1. \( AB + BD > AD \):
\[
3 + 5 > x \quad \Rightarrow \quad 8 > x \quad \Rightarrow \quad x < 8
\]

2. \( AB + AD > BD \):
\[
3 + x > 5 \quad \Rightarrow \quad x > 2
\]

3. \( BD + AD > AB \):
\[
5 + x > 3 \quad \Rightarrow \quad x > -2 \quad \text{(this condition is always satisfied since lengths are positive)}
\]

From the inequalities we analyzed, \( AD \) must satisfy:
- \( 2 < x < 8 \)

Now, we will filter the options:
- A. \( 4 \, \text{cm} \) (valid, since \( 2 < 4 < 8 \))
- B. \( 2 \, \text{cm} \) (not valid, since it cannot be equal to 2)
- C. \( 7 \, \text{cm} \) (valid, since \( 2 < 7 < 8 \))
- D. \( 10 \, \text{cm} \) (not valid, since \( 10 > 8 \))
- E. \( 1 \, \text{cm} \) (not valid, since \( 1 < 2 \))

Therefore, the possible lengths of \( AD \) are:
- A. 4 cm
- C. 7 cm