SleepyGurl said:
Can anyone help me with this hard problem?
A right triangle has a perimeter of 12. What are the lenghts of its 3 sides?
I do know that p=side 1+side2+side3
so 12=Side1+Side2+Side3
so the length of each side would be 4, correct?
The most general formulas for deriving all integer sided right-angled Pythagorean triangles, have been known since the days of Diophantus and the early Greeks. For a right triangle with sides X, Y, and Z, Z being the hypotenuse, the lengths of the three sides of the triangle can be derived as follows: X = k(m^2 - n^2), Y = k(2mn), and Z = k(m^2 + n^2) where k = 1 for primitive triangles (X, Y, and Z having no common factor), m and n are arbitrarily selected integers, one odd, one even, usually called generating numbers, with m greater than n. The symbol ^ means "raised to the power of" such that m^2 means m squared, etc.
Assuming k = 1 and we are seeking integer answers::
m^2 - n^2 + 2mn + m^2 + n^^2 = 12
This reduces to m^2 + mn - 6 = 0
Using the quadratic formula
m = [-n+/-sqrt(n^2 + 24)]/2 = (-n+/-5)/2
For n = 1, m = 2
X = 2^2 - 1^2 = 3
Y = 2(2)1 = 4
Z = 2^2 + 1^2 = 5
3 + 4 + 5 = 12
As tkhunny points out if non-integer answers are acceptable, you have an infinite number of answers.