According to the Triangle Inequality Theorem, for any triangle with sides of lengths \( a \), \( b \), and \( c \), the following inequalities must hold:
- \( a + b > c \)
- \( a + c > b \)
- \( b + c > a \)
In triangle \( ABD \), we have:
- \( AB = 3 , \text{cm} \)
- \( BD = 5 , \text{cm} \)
- Let \( AD = x , \text{cm} \)
We need to apply the triangle inequalities to find the possible lengths of \( AD \).
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\( AB + BD > AD \): \[ 3 + 5 > x \] \[ 8 > x \] or \( x < 8 \)
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\( AB + AD > BD \): \[ 3 + x > 5 \] \[ x > 2 \]
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\( BD + AD > AB \): \[ 5 + x > 3 \] This inequality is always true if \( x > -2 \) (which is always satisfied for positive lengths).
Now combining the results from the inequalities:
- From \( x < 8 \) and \( x > 2 \), we find: \[ 2 < x < 8 \]
Possible lengths that satisfy \( 2 < x < 8 \):
- The potential options you provided that fall within this range are:
- 4 cm (valid)
- 7 cm (valid)
Thus, the possible lengths of \( AD \) are:
4 cm and 7 cm.
2 answers