Perimeter of Triangle

Hello, RubyRed!

Who assigned this? . . . It is an ugly problem!

Quote:

Given a point $P(a,b)$ with $0 < b < a$ ,
determine the minimum perimeter of a triangle with one vertex at $P$ ,
one on the x-axis, and one on the line $y=x$

Code:

| * | * | R * | (y,y)o | * * * | * * * P | * * o(a,b) | * * * | * * * - + - - - - - - - - o - - - - - | (x,0) | Q

Let $Q(x,0)$ be on the x-axis.

Let $R(y,y)$ be on the line $y = x$

Then: . $\text{Perimeter} \;=\;\overline{PQ} + \overline{QR} + \overline{RP}$

. . $P \;=\;\bigg[(x-a)^2 + b^2\bigg]^{\frac{1}{2}} + \bigg[(x-y)^2 + y^2\bigg]^{\frac{1}{2}} + \bigg[(y-a)^2 + (y-b)^2\bigg]^{\frac{1}{2}}$

. . $P \;=\;\bigg[x^2-2ax + a^2+b^2\bigg]^{\frac{1}{2}} + \bigg[x^2-2xy + 2y^2\bigg]^{\frac{1}{2}} + \bigg[2y^2 - 2(a+b)y + (a^2+b^2)\bigg]^{\frac{1}{2}}$

And now solve the system: . $\begin{Bmatrix}\dfrac{\partial P}{\partial x} &=& 0 \\ \\[-3mm] \dfrac{\partial P}{\partial y} &=& 0 \end{Bmatrix}$
Good luck!