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Question Number 183628 by a.lgnaoui last updated on 27/Dec/22

surface du rectangle?

$${surface}\:{du}\:{rectangle}? \\ $$

Commented by a.lgnaoui last updated on 27/Dec/22

Commented by Frix last updated on 27/Dec/22

a+9=b+2=r  ⇒  b=a+7∧r=a+9    a^2 +b^2 =r^2   a^2 +(a+7)^2 =(a+9)^2   a^2 −4a−32=0  (a−8)(a+4)=0  ⇒  a=8∧b=15∧r=17  Area = 120

$${a}+\mathrm{9}={b}+\mathrm{2}={r} \\ $$$$\Rightarrow \\ $$$${b}={a}+\mathrm{7}\wedge{r}={a}+\mathrm{9} \\ $$$$ \\ $$$${a}^{\mathrm{2}} +{b}^{\mathrm{2}} ={r}^{\mathrm{2}} \\ $$$${a}^{\mathrm{2}} +\left({a}+\mathrm{7}\right)^{\mathrm{2}} =\left({a}+\mathrm{9}\right)^{\mathrm{2}} \\ $$$${a}^{\mathrm{2}} −\mathrm{4}{a}−\mathrm{32}=\mathrm{0} \\ $$$$\left({a}−\mathrm{8}\right)\left({a}+\mathrm{4}\right)=\mathrm{0} \\ $$$$\Rightarrow \\ $$$${a}=\mathrm{8}\wedge{b}=\mathrm{15}\wedge{r}=\mathrm{17} \\ $$$$\mathrm{Area}\:=\:\mathrm{120} \\ $$

Answered by HeferH last updated on 27/Dec/22

(R − 9)^2  + (R − 2)^2 =R^2    R^2  + 81 − 18R + R^2  +4 − 4R = R^2    R^2  − 22R + 85  =  0    (R − 5)(R − 17) = 0    R = 9 + ... ⇒ R≠5    R = 17   Area = (17−9)(17−2)=8∙ 15 = 120

$$\left({R}\:−\:\mathrm{9}\right)^{\mathrm{2}} \:+\:\left({R}\:−\:\mathrm{2}\right)^{\mathrm{2}} ={R}^{\mathrm{2}} \\ $$$$\:{R}^{\mathrm{2}} \:+\:\mathrm{81}\:−\:\mathrm{18}{R}\:+\:{R}^{\mathrm{2}} \:+\mathrm{4}\:−\:\mathrm{4}{R}\:=\:{R}^{\mathrm{2}} \\ $$$$\:{R}^{\mathrm{2}} \:−\:\mathrm{22}{R}\:+\:\mathrm{85}\:\:=\:\:\mathrm{0}\: \\ $$$$\:\left({R}\:−\:\mathrm{5}\right)\left({R}\:−\:\mathrm{17}\right)\:=\:\mathrm{0}\: \\ $$$$\:{R}\:=\:\mathrm{9}\:+\:...\:\Rightarrow\:{R}\neq\mathrm{5}\: \\ $$$$\:{R}\:=\:\mathrm{17} \\ $$$$\:{Area}\:=\:\left(\mathrm{17}−\mathrm{9}\right)\left(\mathrm{17}−\mathrm{2}\right)=\mathrm{8}\centerdot\:\mathrm{15}\:=\:\mathrm{120} \\ $$

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