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Question Number 172412 by cherokeesay last updated on 26/Jun/22

Answered by aleks041103 last updated on 26/Jun/22

distance from unknown center to left side is r. distance from center to top side is d. ⇒(6−d)^2 +(6−r)^2 =(6+r)^2 (6−d)^2 +r^2 =(6−r)^2 (6−d)^2 =(6+r−6+r)(6+r+6−r)= =24r (6−d)^2 =(6−r)^2 −r^2 =36−12r ⇒36=36r ⇒r=1

$${distance}\:{from}\:{unknown}\:{center}\:{to} \\ $$$${left}\:{side}\:{is}\:{r}. \\ $$$${distance}\:{from}\:{center}\:{to}\:{top}\:{side}\:{is}\:{d}. \\ $$$$\Rightarrow\left(\mathrm{6}−{d}\right)^{\mathrm{2}} +\left(\mathrm{6}−{r}\right)^{\mathrm{2}} =\left(\mathrm{6}+{r}\right)^{\mathrm{2}} \\ $$$$\left(\mathrm{6}−{d}\right)^{\mathrm{2}} +{r}^{\mathrm{2}} =\left(\mathrm{6}−{r}\right)^{\mathrm{2}} \\ $$$$\left(\mathrm{6}−{d}\right)^{\mathrm{2}} =\left(\mathrm{6}+{r}−\mathrm{6}+{r}\right)\left(\mathrm{6}+{r}+\mathrm{6}−{r}\right)= \\ $$$$=\mathrm{24}{r} \\ $$$$\left(\mathrm{6}−{d}\right)^{\mathrm{2}} =\left(\mathrm{6}−{r}\right)^{\mathrm{2}} −{r}^{\mathrm{2}} =\mathrm{36}−\mathrm{12}{r} \\ $$$$\Rightarrow\mathrm{36}=\mathrm{36}{r} \\ $$$$\Rightarrow{r}=\mathrm{1} \\ $$

Commented by aleks041103 last updated on 26/Jun/22

(6−d)^2 =(6−1)^2 −1^2 =24 ⇒6−d=±2(√6) ⇒d=6±2(√6) d<6 ⇒d=6−2(√6)

$$\left(\mathrm{6}−{d}\right)^{\mathrm{2}} =\left(\mathrm{6}−\mathrm{1}\right)^{\mathrm{2}} −\mathrm{1}^{\mathrm{2}} =\mathrm{24} \\ $$$$\Rightarrow\mathrm{6}−{d}=\pm\mathrm{2}\sqrt{\mathrm{6}} \\ $$$$\Rightarrow{d}=\mathrm{6}\pm\mathrm{2}\sqrt{\mathrm{6}} \\ $$$${d}<\mathrm{6} \\ $$$$\Rightarrow{d}=\mathrm{6}−\mathrm{2}\sqrt{\mathrm{6}} \\ $$

Commented by cherokeesay last updated on 26/Jun/22

Nice ! thank you.

$${Nice}\:!\:{thank}\:{you}. \\ $$

Commented by Tawa11 last updated on 26/Jun/22

Great sir

$$\mathrm{Great}\:\mathrm{sir} \\ $$

Answered by uochiton last updated on 27/Jun/22

(6+r)^2 =(6−r)^2 +(6−d)^2 (6+r)^2 −(6−r)^2 =(6−d)^2 [(6+r)+(6−r)]∙[(6+r)−(6−r)]=(6−d)^2 12∙2∙r=(6−d)^2 24∙r=(6−d)^2 (6−r)^2 =r^2 +(6−d)^2 (6−r)^2 =r^2 +24∙r 36−12∙r+r^2 =r^2 +24∙r 36∙r=36 r=1

$$\left(\mathrm{6}+{r}\right)^{\mathrm{2}} =\left(\mathrm{6}−{r}\right)^{\mathrm{2}} +\left(\mathrm{6}−{d}\right)^{\mathrm{2}} \\ $$$$ \\ $$$$\left(\mathrm{6}+{r}\right)^{\mathrm{2}} −\left(\mathrm{6}−{r}\right)^{\mathrm{2}} =\left(\mathrm{6}−{d}\right)^{\mathrm{2}} \\ $$$$\left[\left(\mathrm{6}+{r}\right)+\left(\mathrm{6}−{r}\right)\right]\centerdot\left[\left(\mathrm{6}+{r}\right)−\left(\mathrm{6}−{r}\right)\right]=\left(\mathrm{6}−{d}\right)^{\mathrm{2}} \\ $$$$\mathrm{12}\centerdot\mathrm{2}\centerdot{r}=\left(\mathrm{6}−{d}\right)^{\mathrm{2}} \\ $$$$\mathrm{24}\centerdot{r}=\left(\mathrm{6}−{d}\right)^{\mathrm{2}} \\ $$$$ \\ $$$$\left(\mathrm{6}−{r}\right)^{\mathrm{2}} ={r}^{\mathrm{2}} +\left(\mathrm{6}−{d}\right)^{\mathrm{2}} \\ $$$$\left(\mathrm{6}−{r}\right)^{\mathrm{2}} ={r}^{\mathrm{2}} +\mathrm{24}\centerdot{r} \\ $$$$\mathrm{36}−\mathrm{12}\centerdot{r}+{r}^{\mathrm{2}} ={r}^{\mathrm{2}} +\mathrm{24}\centerdot{r} \\ $$$$\mathrm{36}\centerdot{r}=\mathrm{36} \\ $$$${r}=\mathrm{1} \\ $$

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