Question
A triangle has one side of length 29 units and another of length 40 units. Determine the range in which the length of the third side must lie.
Answers
GPT 3.5
Let's call the length of the third side of the triangle $x$. 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.
Therefore, we have two inequalities:
$x + 29 > 40$ (since one side has a length of 29 units and the other has a length of 40 units)
$x + 40 > 29$ (since one side has a length of 29 units and the other has a length of 40 units)
Simplifying these inequalities, we get:
$x > 11$
$x > -11$
But since the length of a side cannot be negative, the second inequality is already satisfied.
Therefore, the range in which the length of the third side must lie is $x > 11$.
Therefore, we have two inequalities:
$x + 29 > 40$ (since one side has a length of 29 units and the other has a length of 40 units)
$x + 40 > 29$ (since one side has a length of 29 units and the other has a length of 40 units)
Simplifying these inequalities, we get:
$x > 11$
$x > -11$
But since the length of a side cannot be negative, the second inequality is already satisfied.
Therefore, the range in which the length of the third side must lie is $x > 11$.