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. Given triangle ABD with sides AB = 3 cm, BD = 5 cm, and AD = x cm, we can set up the following inequalities:
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\( AB + BD > AD \) \[ 3 + 5 > x \implies 8 > x \implies x < 8 \]
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\( AB + AD > BD \) \[ 3 + x > 5 \implies x > 2 \]
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\( BD + AD > AB \) \[ 5 + x > 3 \implies x > -2 \quad \text{(this condition is always satisfied since } x > 2\text{)} \]
From these inequalities, we have two conditions for \( x \):
- \( x < 8 \)
- \( x > 2 \)
This means that \( x \) must satisfy: \[ 2 < x < 8 \]
From the provided responses, the possible lengths of \( AD \) that satisfy this range are:
- 4 cm
- 7 cm
Thus, the two possible lengths of AD are 4 cm and 7 cm.
2 answers