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Question Number 208021 by efronzo1 last updated on 02/Jun/24

Answered by A5T last updated on 02/Jun/24

$$\frac{CD}{CD+r}=\frac{r}{r+AB}\Rightarrow \frac{r+AB}{r+CD}=\frac{r}{CD}$$$$\frac{r+AB}{8+r}=\frac{r+CD}{3+r}\Rightarrow \frac{r+AB}{r+CD}=\frac{8+r}{3+r}=\frac{r}{CD}$$$$\Rightarrow CD=\frac{r(3+r)}{8+r}$$$$\frac{AB}{AB+r}=\frac{r}{CD+r}=AB(CD+r)=r\left(AB\right)+r^2$$$$AB=\frac{r^2}{CD}=\frac{r(8+r)}{3+r}$$$${(r+AB)}^2+{(r+CD)}^2={(11+2r)}^2$$$$\Rightarrow {(r+AB)}^2={(11+2r)}^2-r^2{(1+\frac{3+r}{8+r})}^2$$$$AB=\sqrt{{(11+2r)}^2-r^2{(1+\frac{3+r}{8+r})}^2}-r=\frac{r(8+r)}{3+r}$$$$\Rightarrow r=12\Rightarrow Diameter=24$$

Answered by A5T last updated on 02/Jun/24

$$AB=\sqrt{8(8+2r)}=\frac{r(8+r)}{3+r}\Rightarrow r=12$$

Answered by A5T last updated on 02/Jun/24

$$\frac{r}{r+AB}=\frac{3+r}{11+2r}\Rightarrow 11r+2r^2=r(3+r)+3AB+rAB$$$$AB=\frac{r(8+r)}{3+r}$$$${(8+r)}^2=r^2+\frac{r^2{(8+r)}^2}{{(3+r)}^2}\Rightarrow r=12$$

Answered by efronzo1 last updated on 02/Jun/24

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