This phrase "The sides of a triangle are in the ratio 5:12:13" means that there exists 
a segment of some length d which is the common measure of the triangle sides such that

  the first side of the triangle has the length 5d,

  the second side of the triangle has the length 12d, and 

  the third side of the triangle has the length 13d.

Then the perimeter of the triangle is 5d + 12d + 13d = 30d = 150 m

Hence d=  = 5 m.

Now you can find the measures of the triangle sides. They are

5d = 5*5 = 25 m, 12d = 12*5 = 60 m and 13d = 13*5 = 65 m.

Now notice that 

 =  = , 

and it implies that

 = .

It means that the given triangle is a right-angled triangle with the legs of 5d = 25 m and 12d = 60 m.

Finally, you can calculate the area of the triangle. It is half the product of its legs, .25*60 = 750 .

Answer. Area of the triangle is 750 .

A triangle with sides 5, 12, and 13 represents a special right triangle. That's because 

To determine the length of each side, we let the multiplicative factor for the sides, be x

Therefore, the sides are: 5x, 12x, and 13x

With the perimeter being 150, we get: 5x + 12x + 13x = 150

30x = 150 

x = , or 5



We now have the sides as:

5(5), or  25 m

12(5), or 60 m

13(5), or 65 m



Obviously, the 2 shorter sides are the legs, with either leg being the altitude

Area:  = , or  

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