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Question Number 159864 by cherokeesay last updated on 21/Nov/21

Answered by som(math1967) last updated on 22/Nov/21

$$\frac{MP}{AM}=sin30\Rightarrow AM=2R$$$$BL=KN=\sqrt{{MN}^2-{MK}^2}$$$$=\sqrt{{(R+r)}^2+{(R-r)}^2}=\sqrt{4rR}=2\sqrt{R}$$$$\frac{LC}{LN}=cot30\Rightarrow LC=\sqrt{3}LN=\sqrt{3}r=\sqrt{3}$$$$\frac{AB}{BC}=tan60$$$$\frac{2R+R}{2\sqrt{R}+\sqrt{3}}=tan60$$$$3R=2\sqrt{3R}+3$$$$3(R-1)=2\sqrt{3R}$$$$9{(R-1)}^2=12R$$$$3R^2+3-6R=4R$$$$3R^2-10R+3=0$$$$R=3$$$$\therefore AB=3\times 3=9$$$$BC=ABtan30=3\sqrt{3}$$$$AC=\sqrt{27+81}=6\sqrt{3}$$$$perimeter=(6\sqrt{3}+3\sqrt{3}+9)unit$$

Commented by som(math1967) last updated on 22/Nov/21

Commented by cherokeesay last updated on 22/Nov/21

$$thankyousir!$$

Answered by mr W last updated on 22/Nov/21

Commented by mr W last updated on 22/Nov/21

$$\frac{EF}{EG}=\sin 30°$$$$\frac{R-r}{R+r}=\frac{1}{2}$$$$\Rightarrow R=3r=3$$$$AD=DC=BC=\frac{R}{\tan 30°}=\sqrt{3}R=3\sqrt{3}$$$$AE=EC=\frac{R}{\sin 30°}=2R=6$$$$perimeterP=3+3\times 3\sqrt{3}+6=9(1+\sqrt{3})$$

Commented by som(math1967) last updated on 22/Nov/21

$$Goodwaysir$$

Commented by cherokeesay last updated on 22/Nov/21

$$verynice!$$$$thankyou!$$

Commented by Tawa11 last updated on 22/Nov/21

$$\mathrm{Great}$$

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