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 the sides given as follows:
- AB = 3 cm
- BD = 5 cm
- AD = x cm (the length we want to determine)
The inequalities that must be satisfied are:
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AB + BD > AD \[ 3 + 5 > x \quad \Rightarrow \quad 8 > x \quad \Rightarrow \quad x < 8 \]
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AB + AD > BD \[ 3 + x > 5 \quad \Rightarrow \quad x > 2 \]
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BD + AD > AB \[ 5 + x > 3 \quad \Rightarrow \quad x > -2 \quad \text{(this condition is always true since lengths are positive)} \]
From the two inequalities we derived that are relevant (1 and 2), we conclude that:
- \( x > 2 \)
- \( x < 8 \)
Thus, \( 2 < x < 8 \).
The possible lengths of AD must be greater than 2 cm and less than 8 cm. From the provided options, the only lengths that satisfy this inequality are:
- 4 cm (which is between 2 and 8)
- 7 cm (which is also between 2 and 8)
Therefore, the two possible lengths for AD are 4 cm and 7 cm.
2 answers