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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.

 

 Apr 27, 2024
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\((1+r)^2=(\overline{AB}-r)^2+1^2\\(1+r)^2=(2-r)^2+1\\ 1+2r+r^2=4-4r+r^2+1\\ 6r=4\\\color{blue}r=\frac{2}{3}\)

 

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

laugh !

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 Apr 28, 2024
edited by asinus  Apr 28, 2024

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