Question Number 64545 by ajfour last updated on 19/Jul/19
Commented by ajfour last updated on 19/Jul/19
Commented by aliesam last updated on 19/Jul/19
Answered by ajfour last updated on 19/Jul/19
$${let}\:{vertex}\:{angle}\:{be}\:\mathrm{2}\theta. \\ $$$$\Rightarrow\:{n}\phi={n}\left(\frac{\pi}{\mathrm{2}}−\theta\right)=\mathrm{2}\pi \\ $$$$\Rightarrow\:\:\theta=\frac{\pi}{\mathrm{2}}−\frac{\mathrm{2}\pi}{{n}} \\ $$$${R}=\frac{\mathrm{1}}{\mathrm{sin}\:\theta}\:=\:\frac{\mathrm{1}}{\mathrm{cos}\:\left(\frac{\mathrm{2}\pi}{{n}}\right)} \\ $$$${This}\:{is}\:{clearly}\:{wrong},\:{but}\:{i} \\ $$$${dont}\:{understand},\:{how}\:{it}\:{came} \\ $$$${wrong};\:{mrW}\:{Sir},\:{please}\:{help}. \\ $$$$\left({I}\:{deleted}\:{my}\:{same}\:{previous}\right. \\ $$$${answer}\:{and}\:{your}\:{commented} \\ $$$$\left.{answer}\:{got}\:{deleted}\:{alongwith}\right). \\ $$
Commented by aliesam last updated on 19/Jul/19
Commented by aliesam last updated on 19/Jul/19
$${this}\:{relationship}\:{is}\:{true}\:{for}\:{any}\:{n} \\ $$
Commented by mr W last updated on 19/Jul/19
$${na}=\mathrm{2}\pi\:{is}\:{always}\:{true},\:{but} \\ $$$${a}=\frac{\pi}{\mathrm{2}}−\theta\:{is}\:{not}\:{true}! \\ $$$${in}\:{your}\:{diagram}\:{you}\:{assume}\:{that} \\ $$$${both}\:{green}\:{lines}\:{are}\:\bot\:{to}\:{the}\:{sides} \\ $$$${of}\:{the}\:{star},\:{because}\:{you}\:{take}\:{a}=\frac{\pi}{\mathrm{2}}−\theta, \\ $$$${but}\:{this}\:{is}\:{only}\:{the}\:{case}\:{with}\:{n}=\mathrm{5}. \\ $$$${why}\:{don}'{t}\:{you}\:{check}\:{it}\:{when}\:{you} \\ $$$${draw}\:{a}\:{figur}\:{with}\:{n}=\mathrm{7}? \\ $$
Commented by mr W last updated on 19/Jul/19
$$\Rightarrow\:\:\:{n}\left(\frac{\pi}{\mathrm{2}}−\theta\right)=\mathrm{2}\pi \\ $$$${is}\:{not}\:{generally}\:{true}\:{for}\:{a}\:{n}−{corner}\:{star}. \\ $$$${it}'{s}\:{only}\:{true}\:{for}\:{the}\:{displayed}\:{case} \\ $$$${with}\:{n}=\mathrm{5}. \\ $$$${you}'{ll}\:{see}\:{the}\:{error}\:{when}\:{you}\:{draw}\:{a} \\ $$$$\mathrm{7}−{corner}\:{star}. \\ $$
Commented by aliesam last updated on 20/Jul/19
$${are}\:{sides}\:{of}\:{the}\:{star}\:{tangent}\:{to}\:{the}\:{circle}? \\ $$$${if}\:{yes}\:,\:{then}\:{thr}\:{two}\:{sides}\:{in}\:{the}\:{star}\:{are} \\ $$$${perpenduclar}\:{to}\:{the}\:{green}\:{lines} \\ $$
Answered by mr W last updated on 19/Jul/19
Commented by mr W last updated on 19/Jul/19
$$\overset{\frown} {{CD}}\:{is}\:\mathrm{1}/{n}\:{of}\:{the}\:{circumcircle} \\ $$$$\Rightarrow\mathrm{4}\varphi=\frac{\mathrm{2}\pi}{{n}} \\ $$$$\Rightarrow\varphi=\frac{\pi}{\mathrm{2}{n}} \\ $$$$\mathrm{sin}\:\varphi=\frac{\mathrm{1}}{{R}} \\ $$$$\Rightarrow{R}=\frac{\mathrm{1}}{\mathrm{sin}\:\varphi}=\frac{\mathrm{1}}{\mathrm{sin}\:\frac{\pi}{\mathrm{2}{n}}} \\ $$
Commented by ajfour last updated on 19/Jul/19
$${Thanks}\:{Sir},\:{but}\:{i}\:{am}\:{still}\:{in} \\ $$$${doubt}\:{in}\:{the}\:{way}\:{i}\:{proceeded},\:{i} \\ $$$${shall}\:{explain}\:{how}.. \\ $$
Commented by ajfour last updated on 19/Jul/19
$${consider}\:{n}=\mathrm{4}\:\:\:\:{sir}? \\ $$
Commented by mr W last updated on 19/Jul/19
$${maybe}\:{we}\:{don}'{t}\:{mean}\:{the}\:{same}\:{with} \\ $$$${the}\:'{star}'\:{figur}.\:{you}\:{can}\:{not}\:{construct} \\ $$$${a}\:\mathrm{4}−{corner}\:{star}. \\ $$$${i}\:{assume}\:{you}\:{mean}\:{in}\:{the}\:{question}\:{a} \\ $$$${five}\:{corner}\:{star},\:{not}\:{a}\:{pentagon}. \\ $$$${this}\:{is}\:{what}\:{i}\:{mean}\:{with}\:{n}=\mathrm{7}. \\ $$
Commented by mr W last updated on 19/Jul/19
