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

4 cm
4 cm

1 cm
1 cm

7 cm
7 cm

2 cm
2 cm

10 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:
- \( AB = 3 \) cm
- \( BD = 5 \) cm
- Let \( AD = x \) cm

We need to apply the Triangle Inequality Theorem:

1. \( AB + BD > AD \)
\[
3 + 5 > x \implies 8 > x
\]

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

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

From the first inequality, \( x < 8 \), and from the second inequality, \( x > 2 \). So, combining these gives:
\[
2 < x < 8
\]

This means the possible lengths of \( AD \) can be any value between 2 cm and 8 cm, not including 2 cm and 8 cm themselves.

Now looking at the provided options:
- 4 cm (valid)
- 1 cm (not valid)
- 7 cm (valid)
- 2 cm (not valid)
- 10 cm (not valid)

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