Perimeter of triangle problem

SleepyGurl

New member
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?

tkhunny

Moderator
Staff member
A "Right" triangle?

It's a perimeter, so Side1+Side2+Side3 = 12. You got that.

Assuming Side3 has the greatest length, there is the Pythagorean Theorem:

(Side1)^2 + (Side2)^2 = (Side3)^2

This is where we have to quit. There is not enough information to do any more. How sure are you that you copied the problem correctly and completely?

Was it a "Right" triangle or an "Equilateral" triangle?
Was it just a "Right" triangle, or an "Isosceles" right triangle?

As stated, there are many, many possibly answers.

TchrWill

Full Member
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
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.

tkhunny

Moderator
Staff member
Very good.

Sleepy, Did the problem state that we were looking for "Integer" or "Whole Number" solutions?

SleepyGurl

New member
It doesn't say if it is "Integer" or "Whole Numbers", I totally understand after explaining it, you guys are awesome, thanks so much for your talents and time to help me understand these problems better, THANKS!!!