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Question Number 52780 by ajfour last updated on 12/Jan/19

Commented by ajfour last updated on 13/Jan/19

$$Ifthetwosmallcircleshavethe$$$$sameradii,findlengthABin$$$$termsofsemicircleradiusR,$$$$and\mathrm{\angle }betweentheirtangent$$$$\left(redline\right)anddiameterofthe$$$$semicircle.$$

Answered by mr W last updated on 13/Jan/19

Commented by mr W last updated on 13/Jan/19

$$let\lambda =\frac{r}{R}$$$$circleB:$$$$2r=R-R\sin \theta$$$$\Rightarrow \lambda =\frac{r}{R}=\frac{1-\sin \theta }{2}$$$$\Rightarrow \sin \theta =1-2\lambda$$$$$$$$circleA:$$$$OA=R-r$$$$OD=\sqrt{{(R-r)}^2-r^2}=\sqrt{R(R-2r)}$$$$CD=R+\sqrt{R(R-2r)}=\frac{r}{\tan \frac{\theta }{2}}$$$$\Rightarrow \tan \frac{\theta }{2}=\frac{r}{R+\sqrt{R(R-2r)}}=\frac{\lambda }{1+\sqrt{1-2\lambda }}$$$$\Rightarrow 2\tan \frac{\theta }{2}=\frac{1-\sin \theta }{1+\sqrt{\sin \theta }}$$$$\Rightarrow 2\tan \frac{\theta }{2}=1-\sqrt{\sin \theta }$$$$\Rightarrow \theta \approx 21.9741°$$$$\Rightarrow \frac{r}{R}\approx 0.3129$$

Commented by ajfour last updated on 13/Jan/19

$$NiceWaySir!Thankyou.$$

Commented by Otchere Abdullai last updated on 13/Jan/19

$$wethankGodforsuchawonderful$$$$man!$$

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