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Two semicircles are inscribed in a 50 by 100 rectangle, as shown below.  Find R.

 

 May 5, 2020
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The  circles will be tangent  at the  center of the rectangle  = (50, 25)

 

Dropping a perpendicular  from  this point  to the  bottom side of  the rectangle  will form one leg of a right triangle  = 25

Connecting the center  of the bottom semi-circle to the  center of the  square  is the  hypotenuse of this right triangle   = R

And the other leg of thisright triangle =  50  - R

 

So we have

 

(50-R)^2  + 25^2  = R^2      simplify

 

R^2  - 100R  + 2500 + 625  = R^2

 

100R  = 3125

 

R  = 3125/100   =  125  /  4  = 31.25 

 

cool cool cool

 May 5, 2020

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