In triangle ABD, AB is 3 cm long and BD is 5 cm long. According to the Triangle Inequality Theorem, which two lengths are possible lengths of AD

Question

? (1 point)
Responses

4 cm

2 cm

7 cm

1 cm

10 cm

Answer 1

According to the Triangle Inequality Theorem, for any triangle with sides \(a\), \(b\), and \(c\), the following three conditions must hold:

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

In triangle ABD, we have:

• \(AB = 3 , \text{cm}\)
• \(BD = 5 , \text{cm}\)
• Let \(AD = x , \text{cm}\)

We need to apply the Triangle Inequality Theorem to find the possible lengths of \(AD\) (denoted as \(x\)).

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

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

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

Combining these inequalities, we have: \[ 2 < x < 8 \]

The possible integer lengths for \(x\) that satisfy this inequality are 3, 4, 5, 6, and 7.

From the options given:

• 2 cm (not valid)
• 4 cm (valid)
• 7 cm (valid)
• 1 cm (not valid)
• 10 cm (not valid)

Thus, the two possible lengths for \(AD\) are 4 cm and 7 cm.