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

7 cm
7 cm

10 cm
10 cm

4 cm
4 cm

2 cm
2 cm

1 cm
1 cm

Answer 1

According to the Triangle Inequality Theorem, for any triangle with sides of lengths \( a \), \( b \), and \( c \), the following inequalities 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 inequalities to find the possible lengths of \( AD \).

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

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

3. \( BD + AD > AB \):
\[ 5 + x > 3 \]
This inequality is always true if \( x > -2 \) (which is always satisfied for positive lengths).

Now combining the results from the inequalities:
- From \( x < 8 \) and \( x > 2 \), we find:
\[ 2 < x < 8 \]

Possible lengths that satisfy \( 2 < x < 8 \):
- The potential options you provided that fall within this range are:
- 4 cm (valid)
- 7 cm (valid)

Thus, the possible lengths of \( AD \) are:

**4 cm and 7 cm.**