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Question Number 211188 by Ari last updated on 30/Aug/24

Answered by mr W last updated on 30/Aug/24

AB=x  BC=y  x^2 +y^2 =12^2   x+y=16  (x+y)^2 =16^2   x^2 +y^2 +2xy=16^2   12^2 +2xy=16^2   ⇒red area=((xy)/2)=((16^2 −12^2 )/4)=28

$${AB}={x} \\ $$$${BC}={y} \\ $$$${x}^{\mathrm{2}} +{y}^{\mathrm{2}} =\mathrm{12}^{\mathrm{2}} \\ $$$${x}+{y}=\mathrm{16} \\ $$$$\left({x}+{y}\right)^{\mathrm{2}} =\mathrm{16}^{\mathrm{2}} \\ $$$${x}^{\mathrm{2}} +{y}^{\mathrm{2}} +\mathrm{2}{xy}=\mathrm{16}^{\mathrm{2}} \\ $$$$\mathrm{12}^{\mathrm{2}} +\mathrm{2}{xy}=\mathrm{16}^{\mathrm{2}} \\ $$$$\Rightarrow{red}\:{area}=\frac{{xy}}{\mathrm{2}}=\frac{\mathrm{16}^{\mathrm{2}} −\mathrm{12}^{\mathrm{2}} }{\mathrm{4}}=\mathrm{28} \\ $$

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