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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.

 

 May 4, 2024
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Find the radius of the circle.

 

\((1+r)^2=(\dfrac{\overline{AC}}{2}-r)^2+1^2\\ (1+r)^2=(\dfrac{4}{2}-r)^2+1^2\\ 1+2r+r^2=4-4r+r^2+1\\ 6r=4\\ \color{blue}r=\dfrac{2}{3}\)

The radius of the circle is \(\color{blue}\frac{2}{3}\ .\)

 

laugh  !

 May 4, 2024

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