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)

Question

Responses

7 cm7 cm

4 cm4 cm

2 cm2 cm

10 cm10 cm

1 cm

Answer 1

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

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

In triangle \( ABD \), let the lengths be \( AB = 3 \) cm, \( BD = 5 \) cm, and \( AD = x \) cm.

We will apply the Triangle Inequality Theorem:

1. \( 3 + 5 > x \) implies \( 8 > x \) or \( x < 8 \)
2. \( 3 + x > 5 \) implies \( x > 2 \)
3. \( 5 + x > 3 \) is always true since \( x \) is positive.

From these inequalities, we find:

• From \( 3 + 5 > x \): \( x < 8 \)
• From \( 3 + x > 5 \): \( x > 2 \)

Combining these results, we get:

\[ 2 < x < 8 \]

Therefore, the possible lengths for \( AD \) are greater than 2 cm and less than 8 cm. Among the provided choices, the possible lengths of \( AD \) are:

• 4 cm (valid)
• 7 cm (valid)
• 2 cm (not valid since it must be greater than 2)
• 10 cm (not valid since it must be less than 8)
• 1 cm (not valid since it must be greater than 2)

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