I drew a square of side 5 together with its circumcircle. At the middle of the arc between a side and the circle, I drew a smaller square with two vertices on the side, and the other two on the circle.

What was the area of the smaller square?

Let the equation of the circle be  x^2+y^2=25/2    (eq1).

Then the coordinates of the  point where the small square touches the circle  in the 1st quadrant are:  (x, y)

2x and y-2.5 are the sides of the small square.

2x= y-2.5 (eq2) squared and subtracted from 4* (eq1) leads to  

y^2-y-8.75=0 ,    (eq3)
     (y-3,5)*(y+2,5)=0
The positive answer is  y=3.5, so 2x=1 and y-2,5=1

and the area of the smaller square is   1*1=1.

Edited on December 7, 2014, 10:27 pm

Posted by Ady TZIDON on 2014-12-06 01:38:09