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:
- \( AB = 3 , \text{cm} \)
- \( BD = 5 , \text{cm} \)
- Let \( AD \) be \( x , \text{cm} \).
We can apply the Triangle Inequality Theorem on the sides \( AB \), \( BD \), and \( AD \) (or \( x \)) as follows:
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For sides \( AB + BD > AD \): \[ 3 + 5 > x \implies 8 > x \implies x < 8 \]
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For sides \( AB + AD > BD \): \[ 3 + x > 5 \implies x > 2 \]
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For sides \( AD + BD > AB \): \[ x + 5 > 3 \implies x > -2 \text{ (this is always true)} \]
From these conditions, we get:
- \( x < 8 \)
- \( x > 2 \)
Thus, possible values for \( x \) (the length of AD) must be in the range \( 2 < x < 8 \).
Looking through the provided options:
- 1 cm: Not valid (less than 2)
- 2 cm: Not valid (not greater than 2)
- 10 cm: Not valid (greater than 8)
- 4 cm: Valid (between 2 and 8)
- 7 cm: Valid (between 2 and 8)
The two lengths that are possible for \( AD \) are 4 cm and 7 cm.
2 answers