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Semicircles are constructed on AB, AC, and BC. A circle is tangent to all three semicircles. Find the radius of the circle.

 

 

 Jun 22, 2024
 #1
avatar+129725 
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Let O be the cemter of the red circle and r = its radius

Let N  = the center od the right semi-circle

 

BN^2  + BO^2   = ON^2

 

1^2 + (2 -r)^2  = (1 + r)^2

 

1 + 4 - 4r +r^2 = r^2 + 2r + 1

 

5 - 4r =  2r + 1

 

4 = 6r

 

r = 4/6  = 2/3

 

 

cool cool cool

 Jun 22, 2024

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