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

2 cm
2 cm

10 cm
10 cm

4 cm
4 cm

7 cm
7 cm

1 cm
1 cm

Answer 1

According to the Triangle Inequality Theorem, the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.

In triangle ABD, we have the sides given as follows:
- AB = 3 cm
- BD = 5 cm
- AD = x cm (the length we want to determine)

The inequalities that must be satisfied are:

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
\[
5 + x > 3 \quad \Rightarrow \quad x > -2 \quad \text{(this condition is always true since lengths are positive)}
\]

From the two inequalities we derived that are relevant (1 and 2), we conclude that:
- \( x > 2 \)
- \( x < 8 \)

Thus, \( 2 < x < 8 \).

The possible lengths of AD must be greater than 2 cm and less than 8 cm. From the provided options, the only lengths that satisfy this inequality are:

- 4 cm (which is between 2 and 8)
- 7 cm (which is also between 2 and 8)

Therefore, the two possible lengths for AD are **4 cm and 7 cm**.