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Question Number 194116 by universe last updated on 28/Jun/23

Answered by mr W last updated on 02/Jul/23

Commented by mr W last updated on 02/Jul/23

$$R=radiusofsemicircle$$$$r=radiusofsmallcircle$$$$AD=2R$$$$OF=R-r$$$$OE=\sqrt{{(R-r)}^2-r^2}$$$$AB=R-r+\sqrt{{(R-r)}^2-r^2}$$$$CD=R-r-\sqrt{{(R-r)}^2-r^2}$$$$AB\times CD=r^2$$$$AB+CD+2\sqrt{AB\times CD}=2(R-r)+2r=2R=AD$$$$i.e.{(\sqrt{AB}+\sqrt{CD})}^2={\left(\sqrt{AD}\right)}^2$$$$\Rightarrow \sqrt{AD}=\sqrt{AB}+\sqrt{CD}$$

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