Question 77435
Perimeter(P)=two times length (L) plus two times width (W) or

P=2L+2W
2L+2W=100 divide both sides by 2 
L+W=50-----------------------------------eq1

Area of rectangle (A)=length(L)*width(W) or

A=L*W-------------------------------------eq2

When we draw the diagonal in a rectangle we, in effect, divide the rectangle into two identical right triangles. Here, we can apply the Pythagorean Theorem:

L^2+W^2=x^2--------------------------------eq3

Clearly, we have enough information to solve these non-linear simultaneous equations----BUT THAT'S NOT RECOMMENDED!  Instead, lets square both sides of eq1 and we get:

(L+W)^2=50^2
L^2+2L*W+W^2=2500  lets rewrite this:

L^2+W^2+2L*W=2500 but
L^2+W^2=x^2 and
L*W=A

substitute these into the equation and we get:

x^2+2A=2500  subtract x^2 from both sides

x^2-x^2+2A=2500-x^2  collect like terms

2A=2500-x^2  divide both sides by 2

A=(2500-x^2)/2
A=1250-(x^2)/2--------------------------------ans

Frequently, when a problem appears much too difficult, there will be an easy way to solve it


Hope this helps---ptaylor