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 know the lengths of sides \( AB \) and \( BD \):
- \( AB = 3 \) cm
- \( BD = 5 \) cm
- Let \( AD = x \) cm (the length we need to find).
According to the Triangle Inequality Theorem, we can write the following inequalities:
-
\( AB + BD > AD \): \[ 3 + 5 > x \implies 8 > x \implies x < 8 \]
-
\( AB + AD > BD \): \[ 3 + x > 5 \implies x > 5 - 3 \implies x > 2 \]
-
\( BD + AD > AB \): \[ 5 + x > 3 \implies x > 3 - 5 \implies x > -2 \] (This inequality is always satisfied since \( x \) is positive.)
Combining the valid inequalities gives us: \[ 2 < x < 8 \]
Therefore, the possible lengths of \( AD \) (denoted as \( x \)) must be greater than 2 cm and less than 8 cm. This means the possible lengths of \( AD \) can lie in the range \( (2, 8) \).