Let's assume the length of the rectangle to be "l" and the breadth to be "b".

Given, the area of the rectangle = 72 sq.meters So, we have:

l*b = 72 (equation 1)

Also, given the perimeter of the rectangle = 34 meters. We know that the perimeter of the rectangle is the sum of all sides, so we can write:

2(l+b) = 34 (equation 2)

Now, we can simplify equation 2 to get:

l+b = 17 (dividing both sides by 2)

We can use equation 1 to solve for one variable in terms of the other. Solving for l, we get:

l = 72/b (dividing both sides by b)

Substituting this value of l in equation 2, we get:

(72/b) + b = 17

Multiplying both sides by b, we get:

72 + b^2 = 17b

Bringing all terms to one side, we get:

b^2 - 17b + 72 = 0

We can solve this quadratic equation to find the value of b:

b^2 - 9b - 8b + 72 = 0

b(b-9) - 8(b-9) = 0

(b-9)(b-8) = 0

So, b = 9 or b = 8.

If b = 9, then from equation 1 we get:

l = 72/b = 72/9 = 8

If b = 8, then from equation 1 we get:

l = 72/b = 72/8 = 9

Therefore, the length and breadth of the rectangle are 8 meters and 9 meters, respectively.