Let the sides be:
5x , 12x , 13x
Now , perimeter of triangle = sum of all sides.
so, 390 = 5x + 12x + 13x.
390 = 30x.
x = 390/30.
x = 13.
Now , the longest of the one is 13x = 13×13 =169 (ans)

Given:
- The perimeter of a triangular field is 390m.
- The sides of the triangle are in the ratio of 5:12:13.

To find:
- The length of the altitude corresponding to the longest side.

Approach:
- Let the sides of the triangle be 5x, 12x, and 13x, where x is a common factor.
- The perimeter of the triangle is the sum of its sides, so we have 5x + 12x + 13x = 390.
- Simplifying the equation, we get 30x = 390.
- Dividing both sides by 30, we find that x = 13.

Calculations:
- Now that we have the value of x, we can calculate the lengths of the sides of the triangle.
- The lengths of the sides are 5x, 12x, and 13x, so the lengths are 5(13), 12(13), and 13(13).
- Simplifying, we get 65, 156, and 169.
- The longest side of the triangle is 169.

Calculating the altitude:
- To calculate the altitude corresponding to the longest side, we need to find the height of the triangle.
- The area of a triangle can be calculated using the formula: Area = (base * height) / 2.
- Since the base of the triangle is the longest side, the height will be the altitude corresponding to the longest side.
- We can rearrange the formula to solve for the height: height = (2 * Area) / base.
- The area of a triangle can also be calculated using Heron's formula: Area = √[s(s-a)(s-b)(s-c)], where s is the semi-perimeter and a, b, and c are the lengths of the sides.
- The semi-perimeter is half of the perimeter, so s = 390 / 2 = 195.
- Substituting the values into Heron's formula, we get Area = √[195(195-65)(195-156)(195-169)].
- Simplifying, we find Area = √[195(130)(39)(26)] = √[195(101,400)] = √19,764,000.
- Using a calculator, we find that the area is approximately 4449.21 square units.

- Now, we can substitute the values into the formula for height: height = (2 * Area) / base = (2 * 4449.21) / 169 = 26.34 meters.

Answer:
- The length of the altitude corresponding to the longest side of the triangle is approximately 26.34 meters.