Question

Show that a rectangle inscribed in a circle is a square

Answer 1

Answer:

let ABCD is a rectangle inscribed in the circle of given radius r

AB=x , BD = y

area = x*y

also, x^2 + y^2 = 4r^2

for maximum area, dA/dx = 0,

d^2A/dx^2<0

dA/ dx = d/dx[x (4r^2- x^2)^1/2]= 0

8r^2x - 4x^3/2√4r^2x^2 - x^4 = 0

x= √2r, also, d^2A/dx^2<0 for x=√2r

and for x = √2r , y= √2r

Hence, for maximum area ABCD must be square