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avatar+1114 

A semicircle and circle inscribed inside a square. Find the radius of the circle.

 

 Dec 31, 2023
 #1
avatar+129270 
+1

From the center of the  small circle  draw a perpendicular to  the bottom  edge of the  square

  .....the distance = 4-r   = A

From  the  center of the semi-circle draw a segment to the point where the first segment intersects the base

.....this distance is 2-r  = B

Connect the center of the  small circle to the center of the semi-circle....the distance is 2 + r = C

 

We have a right triangle  such that

 

C^2 = A^2 + B^2

 

(2 + r)^2  = (4- r)^2  + (2-r)^2    simplify

 

r^2 + 4r + 4 =  r^2 -8r + 16  + r^2 - 4r + 4

 

r^2 + 4r + 4  =  2r^2 - 12r + 20

 

r^2  - 16r + 16  =  0

 

r^2 -16r   =  -16

 

r^2 -16r  + 64   = -16 + 64

 

(r - 8)^2  =  48

 

r - 8  =   sqrt (48)           or      r - 8 = -sqrt (48)

 

The first value for r is too large

 

The second value produces   r =  8 -sqrt (48)  ≈  1.07

 

cool cool cool

 Dec 31, 2023

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