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Question Number 73137 by behi83417@gmail.com last updated on 06/Nov/19

Commented by behi83417@gmail.com last updated on 06/Nov/19

$$\mathrm{A}\stackrel{▴}{\mathrm{B}C},\mathrm{is}\mathrm{equilateral}\mathrm{with}:\mathrm{AB}=\mathrm{a}$$$$\mathrm{BDFG},\mathrm{is}\mathrm{rectangle}.$$$$\mathrm{find}:{\mathrm{S}}_{\mathrm{B}\stackrel{▴}{\mathrm{H}F}},\mathrm{in}\mathrm{terms}\mathrm{of}:\mathrm{a}.$$

Commented by mr W last updated on 06/Nov/19

$${\mathrm{\Delta }}_{BHF}isnotconstant.$$$$questioncouldbe:$$$$findmax.{\mathrm{\Delta }}_{BHF}intermsofa.$$

Commented by behi83417@gmail.com last updated on 07/Nov/19

$$\mathrm{hello}\mathrm{dear}\mathrm{mrW}.$$$$\mathrm{for}\mathrm{what}?\mathrm{please}\mathrm{explain}\mathrm{bit}\mathrm{more}\mathrm{if}\mathrm{have}$$$$\mathrm{time}.\mathrm{thank}\mathrm{you}.$$

Commented by mr W last updated on 07/Nov/19

Commented by mr W last updated on 07/Nov/19

$$h=\frac{\sqrt{3}}{2}x$$$$A=\frac{1}{2}(a-x)\frac{\sqrt{3}}{2}x=\frac{\sqrt{3}}{4}(a-x)x\ne constant$$$$\frac{dA}{dx}=0\Rightarrow x=\frac{a}{2}for{A}_{max}$$$$A_{max}=\frac{\sqrt{3}}{4}\times \frac{a}{2}\times \frac{a}{2}=\frac{\sqrt{3}{a}^2}{16}$$

Commented by behi83417@gmail.com last updated on 07/Nov/19

$$\mathrm{thank}\mathrm{you}\mathrm{sir}.$$$$\mathrm{what}\mathrm{happend}\mathrm{if}\mathrm{CD},\mathrm{be}\mathrm{angular}\mathrm{bisector}$$$$\mathrm{of}\measuredangle \mathrm{C}?$$

Commented by mr W last updated on 07/Nov/19

$$inthatcase,FG=DB=\frac{a}{\sqrt{3}},$$$$HB=CF=\frac{FG}{\cos 30°}=\frac{2a}{3}=a-x$$$$\Rightarrow x=\frac{a}{3}$$$$A=\frac{\sqrt{3}}{4}\times \frac{a}{3}\times \frac{2a}{3}=\frac{\sqrt{3}{a}^2}{18}$$

Commented by behi83417@gmail.com last updated on 07/Nov/19

$$\mathrm{dear}\mathrm{master}!\mathrm{thank}\mathrm{you}\mathrm{very}\mathrm{much}\mathrm{for}$$$$\mathrm{tring}\mathrm{this}.$$

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