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Question Number 17884 by b.e.h.i.8.3.417@gmail.com last updated on 11/Jul/17

Commented by b.e.h.i.8.3.417@gmail.com last updated on 11/Jul/17

$$diagonalsoftrapezoid:ABCD,create$$$$4triangles.areaoftwothistriangles$$$$areequailto:a^2,and,b^2.$$$$findareaoftrapezoidintermsof:a,b.$$$$oneofdiagonals,cutanotherwith$$$$agivenratio.$$$$trialcase:$$$$when:S_{CDE}=a^2,S_{ABE}=b^2$$$$\Rightarrow S_{ABCD}={(a+b)}^2$$

Answered by mrW1 last updated on 12/Jul/17

$$\mathrm{let}\frac{\mathrm{CE}}{\mathrm{BE}}=\frac{\mathrm{DE}}{\mathrm{AE}}=\frac{\mathrm{x}}{1}$$$$$$$$\mathrm{case}1:$$$$\left[\mathrm{ABE}\right]={\mathrm{a}}^2\mathrm{and}\left[\mathrm{CDE}\right]={\mathrm{b}}^2$$$$\left[\mathrm{ACE}\right]=\frac{\mathrm{x}}{1}{\mathrm{a}}^2=\frac{1}{\mathrm{x}}{\mathrm{b}}^2$$$$\Rightarrow {\mathrm{x}}^2=\frac{{\mathrm{b}}^2}{{\mathrm{a}}^2}$$$$\Rightarrow \mathrm{x}=\frac{\mathrm{b}}{\mathrm{a}}$$$$\left[\mathrm{ABCD}\right]={\mathrm{a}}^2+{\mathrm{b}}^2+2\left[\mathrm{ACE}\right]={\mathrm{a}}^2+{\mathrm{b}}^2+2\frac{\mathrm{b}}{\mathrm{a}}{\mathrm{a}}^2$$$$\Rightarrow \left[\mathrm{ABCD}\right]={(\mathrm{a}+\mathrm{b})}^2$$$$$$$$\mathrm{case}2:$$$$\left[\mathrm{ABE}\right]={\mathrm{a}}^2\mathrm{and}\left[\mathrm{ACE}\right]={\mathrm{b}}^2$$$$\frac{\mathrm{x}}{1}=\frac{{\mathrm{b}}^2}{{\mathrm{a}}^2}$$$$\left[\mathrm{CDE}\right]=\frac{\mathrm{x}}{1}{\mathrm{b}}^2=\frac{{\mathrm{b}}^4}{{\mathrm{a}}^2}$$$$\left[\mathrm{ABCD}\right]={\mathrm{a}}^2+{2\mathrm{b}}^2+\left[\mathrm{CDE}\right]={\mathrm{a}}^2+{2\mathrm{b}}^2+\frac{{\mathrm{b}}^4}{{\mathrm{a}}^2}=\frac{{\mathrm{a}}^4+{2\mathrm{a}}^2{\mathrm{b}}^2+{\mathrm{b}}^4}{{\mathrm{a}}^2}$$$$\Rightarrow \left[\mathrm{ABCD}\right]=\frac{{({\mathrm{a}}^2+{\mathrm{b}}^2)}^2}{{\mathrm{a}}^2}$$

Commented by b.e.h.i.8.3.417@gmail.com last updated on 12/Jul/17

$$thanksmaster.onecaseisremain:$$$$\left[CAE\right]=a^2,\left[BDE\right]=b^2.$$

Commented by mrW1 last updated on 12/Jul/17

$$\mathrm{this}\mathrm{is}\mathrm{not}\mathrm{a}\mathrm{possible}\mathrm{case},$$$$\mathrm{since}\left[CAE\right]=\left[BDE\right]\mathrm{always}$$$$$$$$\mathrm{but}\left[CAE\right]=\left[BDE\right]={\mathrm{b}}^2\mathrm{alone}\mathrm{is}\mathrm{not}\mathrm{enough}\mathrm{to}$$$$\mathrm{determine}\left[\mathrm{ABCD}\right]\mathrm{as}\mathrm{we}\mathrm{can}\mathrm{see}\mathrm{from}$$$$\mathrm{case}2.$$

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