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Question Number 93166 by john santu last updated on 11/May/20

Answered by john santu last updated on 11/May/20

$$\mathrm{let}\mathrm{put}\mathrm{the}\mathrm{right}\mathrm{angle}\mathrm{at}\mathrm{B}(0,0),\mathrm{A}(0,3)$$$$\mathrm{C}(4,0).\mathrm{for}\mathrm{a}\mathrm{general}\mathrm{right}\mathrm{triangle}$$$$\mathrm{A}(0,\mathrm{a}),\mathrm{B}(0,0),\mathrm{C}(\mathrm{c},0)$$$$\mathrm{we}\mathrm{have}\mathrm{X}(x,\mathrm{y})\Rightarrow \mathrm{T}(x,\mathrm{y})=$$$${\mathrm{AX}}^2+{\mathrm{BX}}^2+{\mathrm{CX}}^2$$$$\mathrm{T}=3x^2+3y^2-2(cx+ay)+a^2+c^2$$$$wecompletesquare$$$$\mathrm{T}=3({(x-\frac{c}{3})}^2-\frac{c^2}{9})+3({(y-\frac{a}{3})}^2-\frac{a^2}{9})$$$$+a^2+c^2$$$$thatclearlyhasaminimumat$$$$(x,\mathrm{y})=(\frac{\mathrm{c}}{3},\frac{a}{3}).\mathrm{For}a=3,\mathrm{c}=4$$$$\mathrm{we}\mathrm{get}{\mathrm{T}}_{\min }=\frac{2}{3}(3^2+4^2)=\frac{50}{3}$$$$\mathrm{at}\mathrm{X}(\frac{4}{3},1)$$

Commented by i jagooll last updated on 11/May/20

$$\mathrm{cool}..==$$

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