According to the Triangle Inequality Theorem, the length of any side of a triangle must be less than the sum of the lengths of the other two sides and greater than the difference of the lengths of the other two sides.
Let the length of side AD be represented as \( x \). The sides of triangle ABD are AB (3 cm), BD (5 cm), and AD (x).
Using the Triangle Inequality Theorem, we have the following inequalities:
-
\( AB + BD > AD \)
\( 3 + 5 > x \)
\( 8 > x \) or \( x < 8 \) -
\( AB + AD > BD \)
\( 3 + x > 5 \)
\( x > 2 \) -
\( AD + BD > AB \)
\( x + 5 > 3 \)
\( x > -2 \) (This condition is always satisfied since x is positive.)
Combining the two relevant inequalities, we have:
- \( x > 2 \)
- \( x < 8 \)
Thus, the possible values for \( x \) (the length of AD) must be in the range \( 2 < x < 8 \).
From the provided options:
- 1 cm (not valid)
- 10 cm (not valid)
- 4 cm (valid)
- 7 cm (valid)
- 2 cm (not valid)
So, the two possible lengths of AD are 4 cm and 7 cm.
2 answers