To apply the Triangle Inequality Theorem to triangle \( ABD \), we consider the lengths \( AB = 3 \) cm, \( BD = 5 \) cm, and let \( AD \) be represented as \( x \). 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. This gives us the following inequalities:
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\( AB + BD > AD \) \[ 3 + 5 > x \quad \Rightarrow \quad 8 > x \quad \Rightarrow \quad x < 8 \]
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\( AB + AD > BD \) \[ 3 + x > 5 \quad \Rightarrow \quad x > 2 \]
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\( AD + BD > AB \) \[ x + 5 > 3 \quad \Rightarrow \quad x > -2 \quad (\text{This is always true since } x > 2) \]
Combining the valid inequalities from the above results:
- \( x > 2 \)
- \( x < 8 \)
Thus, \( AD \) must satisfy: \[ 2 < x < 8 \]
The possible lengths of \( AD \) based on this inequality are the values that fall within this range. Based on the answer choices given, the following lengths satisfy this condition:
- 4 cm
- 7 cm
So, the possible lengths of \( AD \) are \( 4 \) cm and \( 7 \) cm.