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Question Number 3607 by Rasheed Soomro last updated on 16/Dec/15

$$Constructalinesegmentof{\mathit{a}}^{2}units$$$$usingrulerandcompassonly,when$$$$alinesegmentof\mathit{a}unitsisgiven.$$

Answered by prakash jain last updated on 16/Dec/15

$$\mathrm{Let}\mathrm{us}\mathrm{say}\mathrm{line}\mathrm{segment}\mathrm{is}\mathrm{AB}.$$$$\mathrm{Draw}\mathrm{a}\left(\mathrm{BC}\right){\mathrm{\perp }}^r\mathrm{to}\mathrm{AB}\mathrm{at}\mathrm{B}\mathrm{of}\mathrm{length}1\mathrm{unit}\mathrm{let}$$$$\mathrm{us}\mathrm{point}\mathrm{is}\mathrm{C}.$$$$\mathrm{ABC}\mathrm{is}\mathrm{right}\mathrm{angled}\mathrm{triangle}\mathrm{with}\mathrm{\angle }\mathrm{B}=90°$$$$\tan \mathrm{C}=\frac{\mathrm{AB}}{\mathrm{BC}}=a$$$$\mathrm{At}\mathrm{point}\mathrm{B}\mathrm{draw}\mathrm{a}\mathrm{circle}\mathrm{of}\mathrm{radius}\mathrm{AB}.$$$$\mathrm{Extend}\mathrm{BC}\mathrm{let}\mathrm{us}\mathrm{it}\mathrm{cuts}\mathrm{the}\mathrm{circle}\mathrm{at}\mathrm{D}.$$$$\mathrm{BD}=a.$$$$\mathrm{Draw}\mathrm{a}\mathrm{line}l\mathrm{parallel}\mathrm{to}\mathrm{line}\mathrm{AC}\mathrm{passing}\mathrm{thru}$$$$\mathrm{B}.$$$$\mathrm{Draw}{\mathrm{\perp }}^r\mathrm{to}\mathrm{line}\mathrm{BD}\mathrm{at}\mathrm{D}.\mathrm{let}\mathrm{us}\mathrm{say}$$$$\mathrm{it}\mathrm{cuts}\mathrm{line}l\mathrm{at}\mathrm{E}.$$$$\mathrm{Since}l\left(\mathrm{BE}\right)\parallel \mathrm{to}\mathrm{line}\mathrm{AC}\mathrm{and}\mathrm{CBD}\mathrm{is}\mathrm{a}\mathrm{straight}$$$$\mathrm{line}.$$$$\mathrm{\angle }ACB=\mathrm{\angle }EBD=x$$$$\tan x=a=\frac{\mathrm{AB}}{\mathrm{BC}}=\frac{\mathrm{DE}}{\mathrm{BD}}$$$$\mathrm{BD}=a\Rightarrow \mathrm{DE}=a^2$$

Commented by Rasheed Soomro last updated on 16/Dec/15

$$\mathit{e}\mathcal{XCEL}\mathit{l}\mathit{e}\mathit{n}\mathit{t}\mathcal{S}\mathit{i}\mathit{r}!$$

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