Two sides of a triangle have the same length. The third side measures $7 \mathrm{~m}$ less than twice the common length. The perimeter of the triangle is $13 \mathrm{~m}$. What are the lengths of the three sides?

Step 1 ：Let's denote the common length of the two sides of the triangle as x. Then the third side is 2x - 7.

Step 2 ：The perimeter of the triangle is the sum of the lengths of all sides, which gives us the equation 4x - 7 = 13.

Step 3 ：Solving this equation, we find that x = 5.

Step 4 ：Substituting x = 5 into the expressions for the sides, we find that the lengths of the three sides are 5 m, 5 m, and 3 m.

Step 5 ：Final Answer: The lengths of the three sides are \(\boxed{5 \mathrm{~m}}\), \(\boxed{5 \mathrm{~m}}\), and \(\boxed{3 \mathrm{~m}}\).