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 have:
- \( AB = 3 \) cm
- \( BD = 5 \) cm
- Let \( AD = x \) cm
We need to apply the Triangle Inequality Theorem:
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\( AB + BD > AD \) \[ 3 + 5 > x \implies 8 > x \]
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\( AB + AD > BD \) \[ 3 + x > 5 \implies x > 2 \]
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\( AD + BD > AB \) \[ x + 5 > 3 \implies x > -2 \quad \text{(This inequality is always true since lengths are positive)} \]
From the first inequality, \( x < 8 \), and from the second inequality, \( x > 2 \). So, combining these gives: \[ 2 < x < 8 \]
This means the possible lengths of \( AD \) can be any value between 2 cm and 8 cm, not including 2 cm and 8 cm themselves.
Now looking at the provided options:
- 4 cm (valid)
- 1 cm (not valid)
- 7 cm (valid)
- 2 cm (not valid)
- 10 cm (not valid)
Thus, the possible lengths of \( AD \) are 4 cm and 7 cm.
2 answers