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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
$let AB=a ; AM=CN=a−R ⇒65=23+2(a−R) ⇒42=2(a−R) ⇒21=a−R let AD=b ⇒b−R=23+AM ⇒b−R=23+a−R ⇒b=23+a ⇒65^2 =a^2 +(23+a)^2 ⇒4225=2a^2 +46a+529 ⇒2a^2 +46a−3696=0 ⇒a^2 +23a−1848=0 ⇒(a−33)(a+56)=0 ⇒ { ((a=33)),((b=56)) :} ⇒area of ABCD=1848 cm^2$
$l e t A B = a; A M = C N = a - R$ $\Rightarrow 65 = 23 + 2 (a - R)$ $\Rightarrow 42 = 2 (a - R)$ $\Rightarrow 21 = a - R$ $l e t A D = b$ $\Rightarrow b - R = 23 + A M$ $\Rightarrow b - R = 23 + a - R$ $\Rightarrow b = 23 + a$ $\Rightarrow 65^{2} = a^{2} + {(23 + a)}^{2}$ $\Rightarrow 4225 = 2 a^{2} + 46 a + 529$ $\Rightarrow 2 a^{2} + 46 a - 3696 = 0$ $\Rightarrow a^{2} + 23 a - 1848 = 0$ $\Rightarrow (a - 33) (a + 56) = 0$ $\Rightarrow {\begin{cases} a = 33 \\ b = 56 \end{cases} \Rightarrow a r e a o f A B C D = 1848 {c m}^{2}$
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