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?

Question

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