Right triangle with side missing, i need to find the longest side...a=5x....b=13x...c=? Perimeter=75

13^2-5^2 = 144
c = 12x
30x = 75
x=2.5
a = 12.5
b = 32.5
c = 30.0

The two shorter sides of any triangle must add up to a number that is longer than the third, longest side. A right triangle usually uses the letter C to designate the hypotenuse (longest side).

For the moment, let's forget about the Xs. We have a triangle with a side length of a=5 and b=13. If c is implied to be the longest side, then it must be 13> c <18

a² + b² = c²
25 + 169 = 194
c = √194 = 13.928 (length of the presumed side c, hypotenuse without the x variable)

a + b + c = 5 + 13 + 13.928 = 31.928
75/31.928 = 2.349

5(2.349) + 13(2.349) + 13.928(2.349) =
11.75 + 30.53 + 32.72 = the length of each side A B & C
75 (the perimeter)

If b is the longest side, then c must be >8 c <13
b² = 169
c² = 169 - 25 = 144
c = 12 (if b were the longest side)

a + b + c = 30
75/30 = 2.5
So,
5(2.5) + 13(2.5) + 12(2.5) =
12.5 + 32.5 + 30 = the length of each side A B & C
75 (the perimeter)

So this problem has two solutions depending upon whether b is the longest side or c is the longest side.

There are a few parts to this. Lets first figure out what c is.
5x+13x=c
(5x)^2+(13x)^2=c
25x^2+169^2=c^2
194^2=c^2
The square toy of 194 is 13.92838828
5x+13x=13.9838828
18x=13.9838828
13.9838828+18x=75

if there is a side missing, it's merely an angle, not a tri-angle.