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Question Number 156789 by cortano last updated on 15/Oct/21

Commented by john_santu last updated on 15/Oct/21

let AB=a ; AM=CN=a−R  ⇒65=23+2(a−R)  ⇒42=2(a−R)  ⇒21=a−R  let AD=b  ⇒b−R=23+AM  ⇒b−R=23+a−R  ⇒b=23+a  ⇒65^2 =a^2 +(23+a)^2   ⇒4225=2a^2 +46a+529  ⇒2a^2 +46a−3696=0  ⇒a^2 +23a−1848=0  ⇒(a−33)(a+56)=0  ⇒ { ((a=33)),((b=56)) :} ⇒area of ABCD=1848 cm^2

$${let}\:{AB}={a}\:;\:{AM}={CN}={a}−{R} \\ $$$$\Rightarrow\mathrm{65}=\mathrm{23}+\mathrm{2}\left({a}−{R}\right) \\ $$$$\Rightarrow\mathrm{42}=\mathrm{2}\left({a}−{R}\right) \\ $$$$\Rightarrow\mathrm{21}={a}−{R} \\ $$$${let}\:{AD}={b} \\ $$$$\Rightarrow{b}−{R}=\mathrm{23}+{AM} \\ $$$$\Rightarrow{b}−{R}=\mathrm{23}+{a}−{R} \\ $$$$\Rightarrow{b}=\mathrm{23}+{a} \\ $$$$\Rightarrow\mathrm{65}^{\mathrm{2}} ={a}^{\mathrm{2}} +\left(\mathrm{23}+{a}\right)^{\mathrm{2}} \\ $$$$\Rightarrow\mathrm{4225}=\mathrm{2}{a}^{\mathrm{2}} +\mathrm{46}{a}+\mathrm{529} \\ $$$$\Rightarrow\mathrm{2}{a}^{\mathrm{2}} +\mathrm{46}{a}−\mathrm{3696}=\mathrm{0} \\ $$$$\Rightarrow{a}^{\mathrm{2}} +\mathrm{23}{a}−\mathrm{1848}=\mathrm{0} \\ $$$$\Rightarrow\left({a}−\mathrm{33}\right)\left({a}+\mathrm{56}\right)=\mathrm{0} \\ $$$$\Rightarrow\begin{cases}{{a}=\mathrm{33}}\\{{b}=\mathrm{56}}\end{cases}\:\Rightarrow{area}\:{of}\:{ABCD}=\mathrm{1848}\:{cm}^{\mathrm{2}} \\ $$$$ \\ $$

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