Geometry Questions

Question Number 173408 by mr W last updated on 11/Jul/22

Commented by mr W last updated on 11/Jul/22

$${the}\:{areas}\:{of}\:{three}\:{parts}\:{of}\:{a}\:{regular} \\$$$${hexagon}\:{are}\:{given}.\:{find}\:{the}\:{areas}\:{of} \\$$$${the}\:{other}\:{parts}. \\$$

Answered by aleks041103 last updated on 11/Jul/22

$${if}\:{we}\:{extend}\:{the}\:{sides}\:{we}\:{can}\:{form}\:\mathrm{2} \\$$$${equlateral}\:{triangles}.\:{In}\:{those},\:{the}\:{sum} \\$$$${of}\:{the}\:{distances}\:{from}\:{any}\:{point}\:{inside} \\$$$${to}\:{the}\:{sides}\:{is}\:{constant}\:{independent}\:{of} \\$$$${the}\:{chosen}\:{point}. \\$$$$\Rightarrow{S}_{{hex}} =\mathrm{2}\left(\mathrm{3}+\mathrm{4}+\mathrm{5}\right)=\mathrm{2}.\mathrm{12}=\mathrm{24} \\$$$${The}\:{sum}\:{of}\:{the}\:{distances}\:{from}\:{a}\:{point} \\$$$${to}\:{two}\:{opposite}\:{sides}\:{is}\:{constant}. \\$$$$\Rightarrow\mathrm{3}+{x}=\mathrm{4}+{y}=\mathrm{5}+{z}={S}/\mathrm{3}=\mathrm{8} \\$$$$\Rightarrow{x}=\mathrm{5},{y}=\mathrm{4},{z}=\mathrm{3} \\$$

Commented by aleks041103 last updated on 11/Jul/22

Commented by mr W last updated on 11/Jul/22

$${great}!\:{thanks}\:{sir}! \\$$