The Triangle Inequality Theorem states that 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 sides \(AB\), \(BD\), and \(AD\).
Given:
- \(AB = 3 , \text{cm}\)
- \(BD = 5 , \text{cm}\)
Let \(AD = x , \text{cm}\). We must satisfy the following inequalities based on the Triangle Inequality Theorem:
-
\(AB + BD > AD\): \[ 3 + 5 > x \implies 8 > x \implies x < 8 \]
-
\(AB + AD > BD\): \[ 3 + x > 5 \implies x > 2 \]
-
\(AD + BD > AB\): \[ x + 5 > 3 \implies x > -2 \quad \text{(always true since lengths are positive)} \]
Combining these inequalities, we have: \[ 2 < x < 8 \]
Possible lengths for \(AD\) must be greater than 2 cm and less than 8 cm.
From the given response options, the possible lengths of \(AD\) that fall between 2 cm and 8 cm are:
- 4 cm
- 7 cm
Thus, the two possible lengths for \(AD\) are 4 cm and 7 cm.