# Lagrange multipliers and triangles

1. Apr 4, 2005

### physicsss

Use Lagrange Multipliers to prove that the triangle with the maximum area that has a given perimeter p is equilateral.
[Hint: Use Heron’s formula for the area of a triangle: A = sqrt[s(s - x)(s - y)(s - z)] where s = p/2 and x, y, and z are the lengths of the sides.]

I have no idea how to do this.

2. Apr 4, 2005

### dextercioby

U have a function of 3 varibles (the area) and a constraint depending on these 3 variables (the perimeter is constant).So basically construct the constaint "area" function and then apply the theory...

Daniel.

3. Apr 4, 2005

### HallsofIvy

Staff Emeritus
Are you saying that you don't know what "Lagrange multipliers" are?

The problem is to maximize $$A= \sqrt{s(s-x)(s-y)(s-z)}$$ subject to the condition x+ y+ z= p.

One nice thing about "Lagrange multipliers" is that we can find important information
(like x= y= z) without having to find x, y, z specifically- eliminate the "multiplier" $$\lambda$$ from the equations and see what happens.

Last edited: Apr 6, 2005