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Question Number 18799 by tawa tawa last updated on 29/Jul/17

Answered by Tinkutara last updated on 30/Jul/17

Diagonal of 1^(st)  square = ((12(√2))/2)  Diagonal of 2^(nd)  square = ((12(√2))/4)  Diagonal of 3^(rd)  square = ((12(√2))/8)  Area of 1^(st)  square = (((12(√2))/2))^2 ÷ 2 = 36  Area of 2^(nd)  square = (((12(√2))/4))^2 ÷ 2 = 9  Area of 3^(rd)  square = (((12(√2))/8))^2 ÷ 2 = (9/4)  This is a GP with a = 36 and r = (1/4) .  Sum of 10 terms = ((36)/(1 − (1/4))) = 48  Hence sum of areas of all these squares  is 48 sq. units.  Area remaining = 144 − 48 = 96 sq. units

$$\mathrm{Diagonal}\:\mathrm{of}\:\mathrm{1}^{\mathrm{st}} \:\mathrm{square}\:=\:\frac{\mathrm{12}\sqrt{\mathrm{2}}}{\mathrm{2}} \\ $$$$\mathrm{Diagonal}\:\mathrm{of}\:\mathrm{2}^{\mathrm{nd}} \:\mathrm{square}\:=\:\frac{\mathrm{12}\sqrt{\mathrm{2}}}{\mathrm{4}} \\ $$$$\mathrm{Diagonal}\:\mathrm{of}\:\mathrm{3}^{\mathrm{rd}} \:\mathrm{square}\:=\:\frac{\mathrm{12}\sqrt{\mathrm{2}}}{\mathrm{8}} \\ $$$$\mathrm{Area}\:\mathrm{of}\:\mathrm{1}^{\mathrm{st}} \:\mathrm{square}\:=\:\left(\frac{\mathrm{12}\sqrt{\mathrm{2}}}{\mathrm{2}}\right)^{\mathrm{2}} \boldsymbol{\div}\:\mathrm{2}\:=\:\mathrm{36} \\ $$$$\mathrm{Area}\:\mathrm{of}\:\mathrm{2}^{\mathrm{nd}} \:\mathrm{square}\:=\:\left(\frac{\mathrm{12}\sqrt{\mathrm{2}}}{\mathrm{4}}\right)^{\mathrm{2}} \boldsymbol{\div}\:\mathrm{2}\:=\:\mathrm{9} \\ $$$$\mathrm{Area}\:\mathrm{of}\:\mathrm{3}^{\mathrm{rd}} \:\mathrm{square}\:=\:\left(\frac{\mathrm{12}\sqrt{\mathrm{2}}}{\mathrm{8}}\right)^{\mathrm{2}} \boldsymbol{\div}\:\mathrm{2}\:=\:\frac{\mathrm{9}}{\mathrm{4}} \\ $$$$\mathrm{This}\:\mathrm{is}\:\mathrm{a}\:\mathrm{GP}\:\mathrm{with}\:{a}\:=\:\mathrm{36}\:\mathrm{and}\:{r}\:=\:\frac{\mathrm{1}}{\mathrm{4}}\:. \\ $$$$\mathrm{Sum}\:\mathrm{of}\:\mathrm{10}\:\mathrm{terms}\:=\:\frac{\mathrm{36}}{\mathrm{1}\:−\:\frac{\mathrm{1}}{\mathrm{4}}}\:=\:\mathrm{48} \\ $$$$\mathrm{Hence}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{areas}\:\mathrm{of}\:\mathrm{all}\:\mathrm{these}\:\mathrm{squares} \\ $$$$\mathrm{is}\:\mathrm{48}\:\mathrm{sq}.\:\mathrm{units}. \\ $$$$\mathrm{Area}\:\mathrm{remaining}\:=\:\mathrm{144}\:−\:\mathrm{48}\:=\:\mathrm{96}\:\mathrm{sq}.\:\mathrm{units} \\ $$

Commented by tawa tawa last updated on 30/Jul/17

Wow, God bless you sir.

$$\mathrm{Wow},\:\mathrm{God}\:\mathrm{bless}\:\mathrm{you}\:\mathrm{sir}. \\ $$

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