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Question Number 209637 by mr W last updated on 17/Jul/24

Commented by mr W last updated on 17/Jul/24

$$findminimumofAB+BC+CD=?$$

Commented by Frix last updated on 17/Jul/24

$$\mathrm{I}\mathrm{get}2\sqrt{7}$$

Commented by mr W last updated on 17/Jul/24

$${that}^{\prime }scorrectsir!$$$$yourmethod?$$

Commented by Frix last updated on 17/Jul/24

$$\mathrm{I}\mathrm{will}\mathrm{post}\mathrm{it}\mathrm{later}.$$

Commented by Frix last updated on 19/Jul/24

$$\mathrm{I}‘‘{\mathrm{fold}}^{″}\mathrm{it}$$$$A=\left(\begin{array}{c}2\\ 0\end{array}\right)$$$$B\in y=x\tan 20°$$$$C\in y=x\tan 40°$$$$D\in y=x\tan 60°\wedge \mid OD\mid =6\Rightarrow D=\left(\begin{array}{c}3\\ 3\sqrt{3}\end{array}\right)$$$$\mathrm{The}\mathrm{shortest}\mathrm{path}\mathrm{is}$$$$\mid AD\mid =2\sqrt{7}$$

Commented by mr W last updated on 19/Jul/24

$$thankssir!$$

Commented by Frix last updated on 19/Jul/24

$$\mathrm{But}\mathrm{this}\mathrm{method}{\mathrm{doesn}}^{\prime }\mathrm{t}\mathrm{work}\mathrm{for}n\mathrm{points}$$$$\mathrm{with}(n-1)\alpha ⩾180°$$

Commented by mr W last updated on 19/Jul/24

$$for(n-1)\alpha >180°thereexistsno$$$$minimumatall.$$

Answered by mr W last updated on 19/Jul/24

Commented by mr W last updated on 19/Jul/24

$${let}^{\prime }streatOA,ODasmirrors.$$$${(AB+BC+CD)}_{min}=A^{\prime }{D}^{\prime }$$$$=\sqrt{2^2+6^2-2\times 2\times 6\cos \left(3\times 20°\right)}$$$$=2\sqrt{7}$$

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