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Question Number 158069 by zainaltanjung last updated on 30/Oct/21

Answered by Rasheed.Sindhi last updated on 30/Oct/21

$$Letthesideofoldhexagon=a$$$${▴}_{old}=6\sqrt{\frac{3a}{2}{(\frac{3a}{2}-a)}^3}$$$${▴}_{old}=6\sqrt{\frac{3a}{2}{\left(\frac{a}{2}\right)}^3}=6\sqrt{3{\left(\frac{a}{2}\right)}^4}$$$$=6{\left(\frac{a}{2}\right)}^2\sqrt{3}=6\left(\frac{a^2}{4}\right)\sqrt{3}=\frac{3a^2\sqrt{3}}{2}$$$$Sideofnew6-gon=3a$$$${▴}_{new}=6\sqrt{\frac{9a}{2}{(\frac{9a}{2}-3a)}^3}$$$$=6\sqrt{\frac{9a}{2}{\left(\frac{3a}{2}\right)}^3}=6\sqrt{3{\left(\frac{3a}{2}\right)}^4}$$$$=6{\left(\frac{3a}{2}\right)}^2\sqrt{3}=6\left(\frac{9a^2}{4}\right)\sqrt{3}$$$$=\frac{27a^2\sqrt{3}}{2}$$$$\frac{{▴}_{new}}{{▴}_{old}}=\frac{\frac{27a^2\sqrt{3}}{2}}{\frac{3a^2\sqrt{3}}{2}}=\frac{27a^2\sqrt{3}}{3a^2\sqrt{3}}=\frac{9}{1}=9:1$$

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