To determine the possible length of the third side of a triangle when two sides are given, you can use the triangle inequality theorem. This theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the remaining side.
Let \( a = 4 \) meters and \( b = 6 \) meters be the lengths of the two sides. Let \( c \) be the length of the third side. According to the triangle inequality theorem, we have the following inequalities:
- \( a + b > c \)
- \( a + c > b \)
- \( b + c > a \)
Substituting the values of \( a \) and \( b \):
-
\( 4 + 6 > c \)
\[ 10 > c \] \[ c < 10 \] -
\( 4 + c > 6 \)
\[ c > 2 \] -
\( 6 + c > 4 \)
(This inequality is always true as \( c \) will be greater than 2).
Combining the results from the first two inequalities, we have:
\[ 2 < c < 10 \]
Therefore, the possible length of the third side \( c \) must be greater than 2 meters and less than 10 meters.