Question

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 ? (1 point)
Responses

10 cm
10 cm

1 cm
1 cm

2 cm
2 cm

7 cm
7 cm

4 cm

Pls get it right

Answer 1

To determine the possible lengths of \( AD \) in triangle \( ABD \) using the Triangle Inequality Theorem, we need to apply the theorem, which states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.

We have the following sides:
- \( AB = 3 \) cm
- \( BD = 5 \) cm
- \( AD = x \) (the unknown side we want to find)

According to the Triangle Inequality Theorem, we need to satisfy the following conditions:

1. \( AB + BD > AD \)
\[ 3 + 5 > x \]
\[ 8 > x \quad \text{(or } x < 8\text{)} \]

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

3. \( BD + AD > AB \)
\[ 5 + x > 3 \]
\[ x > -2 \quad \text{(always true since } x \text{ is positive)} \]

Now we combine the inequalities derived from the first two valid conditions:
- From condition 1: \( x < 8 \)
- From condition 2: \( x > 2 \)

Thus, we find that:
\[ 2 < x < 8 \]

This means that \( AD \) can take any value between 2 cm and 8 cm.

Looking at the provided options:
- 10 cm (not valid)
- 1 cm (not valid)
- 2 cm (not valid as it must be greater than 2)
- 7 cm (valid as it is between 2 and 8)
- 4 cm (valid as it is between 2 and 8)

Therefore, the possible lengths of \( AD \) are **4 cm** and **7 cm**.