Let the length of each of the two equal sides be \( x \). According to the problem, the third side measures \( 2x - 6 \).
The perimeter of the triangle is given by the sum of the three sides, which is:
\[ x + x + (2x - 6) = 14 \]
Simplifying this expression, we get:
\[ 2x + 2x - 6 = 14 \]
Combining like terms leads to:
\[ 4x - 6 = 14 \]
Next, we isolate \( 4x \) by adding 6 to both sides:
\[ 4x = 20 \]
Now, dividing by 4 gives us:
\[ x = 5 \]
Now that we have the length of the two equal sides, we can find the length of the third side:
\[ \text{Third side} = 2x - 6 = 2(5) - 6 = 10 - 6 = 4 \]
Thus, the lengths of the three sides of the triangle are:
- Two equal sides: \( 5 \) m each
- Third side: \( 4 \) m
To summarize, the lengths of the sides are:
- \( 5 \) m
- \( 5 \) m
- \( 4 \) m
We can double-check that the perimeter is correct:
\[ 5 + 5 + 4 = 14 \text{ m} \]
Therefore, the solution is confirmed: the lengths of the three sides are \( 5 \) m, \( 5 \) m, and \( 4 \) m.