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Two quarter-circles are drawn inside a unit square.  A smaller square is inscribed in the two quarter-circles.  Find the area of the smaller square.

 

 Aug 18, 2023
 #2
avatar+128732 
+1

 

 

 

 

 

 

 

 

Let A = (0,0) B = (0,1) C = (1,1)  and D = (1,0)

 

Let the equation of the  circle on the  left be

x^2 + y^2  = 1

 

We can locate a point at the  midpoint of  both squares at E = (1/2, 1)

Connect  AE.....

And we can  call the intersection of AE and the left circle, F

And the distance  between F and  BC  =  FI  =  one side  of the square we  are  looking  for

And by similarity   BE / EA = !E/ EF

 

And the equation of the  line through AE is    y = 2x

 

Sub this into the equation of the  circle  for  y so  we  can  find the intersection point of the  line and the  circle

This will be point F

x^2 + (2x)^2  = 1

x^2 + 4x^2  = 1

5x^2 = 1

x^2 = 1/5

x=  1/sqrt 5

y = 2/sqrt 5

 

So  F = (1/sqrt 5, 2/sqrt 5)

 

The circle on the  right has the equation (x -1)^2 + y^2 =1

And by symmetry point G  is the  intersection of ED and this  circle  =  (1 - x , y) =   (1 - 1/sqrt5 , 2/sqrt5 )

 

The side of the  square =  FG  = FI =   (1 - 1/sqrt 5) - 1/sqrt5 =  1 -2/sqrt5

 

Then its area   =  (1 - 2/sqrt5)^2  =  1 - 4/sqrt 5 +  4/5  =   [ 9 + 4sqrt 5 ]  /  5

 

cool cool cool

 Aug 19, 2023

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