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Category: Geometry

Question-206275

Question Number 206275 by cortano21 last updated on 11/Apr/24 Answered by HeferH24 last updated on 11/Apr/24 $$\:{CDEF}\:=\:{m} \\ $$$$\:{ABFE}\:=\:\mathrm{3}{m} \\ $$$$\:\left(\frac{\mathrm{4}}{\mathrm{6}}\right)^{\mathrm{2}} =\:\frac{\mathrm{4}}{\mathrm{9}}=\frac{\mathrm{4}{k}}{\mathrm{9}{k}} \\ $$$$\:\mathrm{5}{k}\:=\:\mathrm{4}{m} \\…

Question-206216

Question Number 206216 by cortano21 last updated on 09/Apr/24 Answered by A5T last updated on 09/Apr/24 $${HC}=\sqrt{\mathrm{108}+\mathrm{4}}=\sqrt{\mathrm{112}}=\mathrm{4}\sqrt{\mathrm{7}} \\ $$$${Let}\:{HC}\:{meet}\:{the}\:{circumcircle}\:{of}\:{the}\:{hexagon} \\ $$$${at}\:{L}\:{then},\:{HC}×{HL}={FH}×{HE}=\mathrm{8}\Rightarrow{HL}=\frac{\mathrm{2}\sqrt{\mathrm{7}}}{\mathrm{7}} \\ $$$${sin}\angle{EHC}=\frac{\sqrt{\mathrm{108}}}{\:\sqrt{\mathrm{112}}}=\frac{\mathrm{3}\sqrt{\mathrm{21}}}{\mathrm{14}}\Rightarrow{cos}\angle{EHC}=\frac{\sqrt{\mathrm{7}}}{\mathrm{14}} \\ $$$$\Rightarrow\sqrt{{x}^{\mathrm{2}}…

Question-206198

Question Number 206198 by lmcp1203 last updated on 09/Apr/24 Answered by A5T last updated on 09/Apr/24 $$\frac{{sin}\mathrm{4}\theta}{{AD}}=\frac{{sin}\left(\mathrm{90}−\mathrm{3}\theta\right)}{{AB}};\frac{{sin}\left(\theta\right)}{{AD}}=\frac{{sin}\left(\mathrm{90}−\mathrm{4}\theta\right)}{{CD}={AB}} \\ $$$$\Rightarrow\frac{{sin}\left(\mathrm{4}\theta\right)}{{sin}\left(\mathrm{90}−\mathrm{3}\theta\right)}=\frac{{sin}\left(\theta\right)}{{sin}\left(\mathrm{90}−\mathrm{4}\theta\right)}\Rightarrow{x}=\mathrm{20}° \\ $$ Answered by lmcp1203 last…