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Question-202276




Question Number 202276 by professorleiciano last updated on 23/Dec/23
Answered by professorleiciano last updated on 23/Dec/23
Area(retangulo)=4×6=24m^2   Area(triangulo I)=3×6=18/2=9m^2   Area(triangulo II)=3×4=12/2=6m^2   Area(triangulo III)=3×4=12/2=6m^2   Area(total)=24m^2 +9m^2 +6m^2 +6m^2   =45m^2   Alternativa (a)
$${Area}\left({retangulo}\right)=\mathrm{4}×\mathrm{6}=\mathrm{24}{m}^{\mathrm{2}} \\ $$$${Area}\left({triangulo}\:{I}\right)=\mathrm{3}×\mathrm{6}=\mathrm{18}/\mathrm{2}=\mathrm{9}{m}^{\mathrm{2}} \\ $$$${Area}\left({triangulo}\:{II}\right)=\mathrm{3}×\mathrm{4}=\mathrm{12}/\mathrm{2}=\mathrm{6}{m}^{\mathrm{2}} \\ $$$${Area}\left({triangulo}\:{III}\right)=\mathrm{3}×\mathrm{4}=\mathrm{12}/\mathrm{2}=\mathrm{6}{m}^{\mathrm{2}} \\ $$$${Area}\left({total}\right)=\mathrm{24}{m}^{\mathrm{2}} +\mathrm{9}{m}^{\mathrm{2}} +\mathrm{6}{m}^{\mathrm{2}} +\mathrm{6}{m}^{\mathrm{2}} \\ $$$$=\mathrm{45}{m}^{\mathrm{2}} \\ $$$${Alternativa}\:\left({a}\right) \\ $$
Commented by professorleiciano last updated on 23/Dec/23
Commented by mr W last updated on 24/Dec/23
a better and more general method  see below.
$${a}\:{better}\:{and}\:{more}\:{general}\:{method} \\ $$$${see}\:{below}. \\ $$
Answered by mr W last updated on 24/Dec/23
Commented by mr W last updated on 24/Dec/23
A_k =(((x_(k+1) −x_k )(y_k +y_(k+1) ))/2)   determinant ((k,x_k ,y_k ,(x_(k+1) −x_k ),(y_k +y_(k+1) ),A_k ),(1,1,1,0,8,0),(2,1,7,4,(14),(28)),(3,5,7,2,(18),(18)),(4,7,(11),3,(22),(33)),(5,(10),(11),(−5),(12),(−30)),(6,5,1,(−4),2,(−4)),((7(=1)),1,1,╱,╱,(45)))  Area of polygon = ΣA_k =45 ✓  ⇒answer (a)
$${A}_{{k}} =\frac{\left({x}_{{k}+\mathrm{1}} −{x}_{{k}} \right)\left({y}_{{k}} +{y}_{{k}+\mathrm{1}} \right)}{\mathrm{2}} \\ $$$$\begin{array}{|c|c|c|c|c|c|c|c|}{{k}}&\hline{{x}_{{k}} }&\hline{{y}_{{k}} }&\hline{{x}_{{k}+\mathrm{1}} −{x}_{{k}} }&\hline{{y}_{{k}} +{y}_{{k}+\mathrm{1}} }&\hline{{A}_{{k}} }\\{\mathrm{1}}&\hline{\mathrm{1}}&\hline{\mathrm{1}}&\hline{\mathrm{0}}&\hline{\mathrm{8}}&\hline{\mathrm{0}}\\{\mathrm{2}}&\hline{\mathrm{1}}&\hline{\mathrm{7}}&\hline{\mathrm{4}}&\hline{\mathrm{14}}&\hline{\mathrm{28}}\\{\mathrm{3}}&\hline{\mathrm{5}}&\hline{\mathrm{7}}&\hline{\mathrm{2}}&\hline{\mathrm{18}}&\hline{\mathrm{18}}\\{\mathrm{4}}&\hline{\mathrm{7}}&\hline{\mathrm{11}}&\hline{\mathrm{3}}&\hline{\mathrm{22}}&\hline{\mathrm{33}}\\{\mathrm{5}}&\hline{\mathrm{10}}&\hline{\mathrm{11}}&\hline{−\mathrm{5}}&\hline{\mathrm{12}}&\hline{−\mathrm{30}}\\{\mathrm{6}}&\hline{\mathrm{5}}&\hline{\mathrm{1}}&\hline{−\mathrm{4}}&\hline{\mathrm{2}}&\hline{−\mathrm{4}}\\{\mathrm{7}\left(=\mathrm{1}\right)}&\hline{\mathrm{1}}&\hline{\mathrm{1}}&\hline{\diagup}&\hline{\diagup}&\hline{\mathrm{45}}\\\hline\end{array} \\ $$$${Area}\:{of}\:{polygon}\:=\:\Sigma{A}_{{k}} =\mathrm{45}\:\checkmark \\ $$$$\Rightarrow{answer}\:\left({a}\right) \\ $$

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