Question Number 227938 by mr W last updated on 03/Mar/26
Commented by mr W last updated on 03/Mar/26
$${find}\:{the}\:{area}\:{of}\:{the}\:{regular}\:{hexagon}. \\ $$
Answered by TonyCWX last updated on 04/Mar/26
$$\mathrm{Extend}\:\mathrm{Line}\:{EF}\:\mathrm{and}\:{AB}\:\mathrm{to}\:\mathrm{form}\:\mathrm{an}\:\mathrm{equilateral}\:\mathrm{triangle}. \\ $$$$\mathrm{Let}\:{K}\:=\:\mathrm{Intersection}\:\mathrm{point}\:\mathrm{between}\:{IJ}\:\mathrm{and}\:{AF} \\ $$$$ \\ $$$${S}_{{AKI}} \:=\:\:\:\Rightarrow\:{S}_{{KIF}} \:=\:\mathrm{10}−\: \\ $$$$ \\ $$$${AI}\:\mathrm{is}\:\mathrm{the}\:\mathrm{median}\:\mathrm{of}\:\Delta_{{BIJ}} \: \\ $$$$\Rightarrow\:{S}_{{AIB}} \:=\:{S}_{{AIJ}} \\ $$$$\Rightarrow\:{S}_{{AIJ}} \:=\:\mathrm{8} \\ $$$$\Rightarrow{S}_{{AKJ}} \:=\:\mathrm{8}−\: \\ $$$$ \\ $$$${IF}\:\mathrm{is}\:\mathrm{the}\:\mathrm{median}\:\mathrm{of}\:\Delta_{{IJE}} \\ $$$$\Rightarrow\:{S}_{{IEF}} \:=\:{S}_{{IFJ}} \\ $$$$\Rightarrow\:{S}_{{IFJ}} \:=\:\mathrm{11} \\ $$$$\Rightarrow\:{S}_{{FKJ}} \:=\:\mathrm{11}−\left(\mathrm{10}−\:\right)\:=\:\:−\mathrm{1} \\ $$$$ \\ $$$${S}_{{AJF}} \:=\:\mathrm{8}−\:+\:−\mathrm{1}\:=\:\mathrm{9} \\ $$$$\Rightarrow\:\mathrm{Area}\:\mathrm{of}\:\mathrm{Regular}\:\mathrm{Hexagon}\:=\:\mathrm{6}\left(\mathrm{9}\right)\:=\:\mathrm{54}\:\checkmark \\ $$
Commented by TonyCWX last updated on 04/Mar/26
Commented by TonyCWX last updated on 04/Mar/26
$$\mathrm{Generalisation}: \\ $$$${S}_{{AIB}} \:=\:{A} \\ $$$${S}_{{AIF}} \:=\:{B} \\ $$$${S}_{{FIE}} \:=\:{C} \\ $$$$\Rightarrow\begin{array}{|c|}{{S}_{\mathrm{Hexagon}} \:=\:\mathrm{6}\left({A}−{B}+{C}\right)}\\\hline\end{array} \\ $$
Commented by mr W last updated on 04/Mar/26
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Answered by mr W last updated on 04/Mar/26
Commented by mr W last updated on 04/Mar/26
$${A}_{{hexagon}} =\mathrm{6}×{A}_{\:{BFC}} =\mathrm{6}×\left(\mathrm{11}+\mathrm{8}−\mathrm{10}\right)=\mathrm{54} \\ $$
Commented by TonyCWX last updated on 04/Mar/26
$${Nice}!! \\ $$