Solution:

Given, the sides of a right triangle are in the ratio 3:5:7.
Let the sides be 3x, 5x and 7x.

Step 1: Finding Perimeter
The perimeter of the triangle is given as 300m.

Therefore, 3x + 5x + 7x = 300
or, 15x = 300
or, x = 20

Hence, the sides of the right triangle are 60m, 100m, and 140m.

Step 2: Finding Area
The area of the right triangle is given as 1500√3.

Using the formula for the area of a right triangle, we have:
Area = (1/2) x Base x Height

Let the height corresponding to the longest side be h.

Then, the area of the triangle is given by:
(1/2) x 140 x h = 1500√3
or, 70h = 1500√3
or, h = (1500√3)/70

Hence, the height corresponding to the longest side is (1500√3)/70 m.

Step 3: Verification
We can verify the solution as follows:

Using the Pythagorean theorem, we can check that the sides satisfy the conditions of a right triangle.

Longest side = 140m, so we have:
140^2 = 60^2 + 100^2
or, 19600 = 3600 + 10000
or, 19600 = 13600 + 6000

Therefore, the sides satisfy the conditions of a right triangle.

Using the formula for the area of a right triangle, we can also verify that the area is 1500√3.

Area = (1/2) x Base x Height
= (1/2) x 140 x [(1500√3)/70]
= 1500√3

Therefore, the solution is verified.