bound on area of right triangle
We may bound the area of a right triangle in terms of its perimeter
.
The derivation of this bound is a good exercise in constrained
optimization using Lagrange multipliers.
Theorem 1.
If a right triangle has perimeter P, then its area is bounded as
A≤3-2√24P2 |
with equality when one has an isosceles right triangle.
Proof.
Suppose a triangle has legs of length x and y. Then its hypotenuse
has length √x2+y2, so the perimeter is given as
P=x+y+√x2+y2. |
The area, of course, is
A=12xy. |
We want to maximize A subject to the constraint that P be constant. This means that the gradient of A will be proportional to the gradient of P. That is to say, for some constant λ, we will have
∂A∂x | = | λ∂P∂x | ||
∂A∂y | = | λ∂P∂y |
Together with the constraint, these form a system of three equations for the three quantities x, y, and λ. Writing them out explicitly,
12y | = | λ(1+x√x2+y2) | ||
12x | = | λ(1+y√x2+y2) | ||
P | = | x+y+√x2+y2 |
Not that we cannot have λ=0 because that would mean that all sides of our triangle would have zero length. Hence, we may eliminate λ between the first two equations to obtain
x(1+x√x2+y2)=y(1+y√x2+y2), |
which may be manipulated to yield
(x-y)(1+x+y√x2+y2)=0. |
We have two case to consider — either the first factor or the second factor may equal zero. If the second factor equals zero,
1+x+y√x2+y2=0, |
move the “1” to the other side of the equation and cross-multiply to obtain
x+y=-√x2+y2. |
Since we want x≥0 and y≥0 but the right-hand side is non-positive, the only option would be to have a trianagle of zero area. The other possibility was to have the second factor equal zero, which would give
x-y=0. |
In this case, x equals y. Imposing this condition on the constraint, we see that
P=(2+√2)x, |
so we have the solution
x | = | P2+√2=2-√22P | ||
y | = | P2+√2=2-√22P. |
∎
Title | bound on area of right triangle |
---|---|
Canonical name | BoundOnAreaOfRightTriangle |
Date of creation | 2013-03-22 16:30:41 |
Last modified on | 2013-03-22 16:30:41 |
Owner | rspuzio (6075) |
Last modified by | rspuzio (6075) |
Numerical id | 17 |
Author | rspuzio (6075) |
Entry type | Theorem |
Classification | msc 51-00 |