The Triangle Inequality Theorem states that for any triangle, the sum of the lengths of any two sides must be greater than the length of the remaining side. In triangle ABD, we have:
- AB = 3 cm
- BD = 5 cm
- Let AD = x cm (the length we want to find).
According to the Triangle Inequality Theorem, we need to satisfy the following inequalities:
-
AB + BD > AD: \( 3 + 5 > x \) \( 8 > x \) or \( x < 8 \)
-
AB + AD > BD: \( 3 + x > 5 \) \( x > 2 \)
-
AD + BD > AB: \( x + 5 > 3 \) \( x > -2 \) (This does not impose any additional restrictions, as x is positive.)
Combining the relevant inequalities, we have: \( 2 < x < 8 \)
Now, we check the provided options:
1 cm – not valid (because x must be greater than 2) 2 cm – not valid (because x must be greater than 2) 4 cm – valid (because 2 < 4 < 8) 10 cm – not valid (because x must be less than 8) 7 cm – valid (because 2 < 7 < 8)
Thus, the possible lengths of AD are 4 cm and 7 cm.