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Question Number 38079 by ajfour last updated on 21/Jun/18

Commented by ajfour last updated on 21/Jun/18

$F i n d c o l o u r e d a r e a A B C D i n$ $t e r m s o f a, b, c, d .$ $I t i s t h e a r e a c o m m o n t o t w o$ $s q u a r e s o f s i d e l e n g t h a .$
Commented by Rasheed.Sindhi last updated on 22/Jun/18

${What}^{'} s the source of your questions, sir ajfour ?$
Commented by ajfour last updated on 22/Jun/18

$I c r e a t e t h e p h y s i c s a n d g e o m e t r y$ $o n e s m y s e l f . I n t e g r a t i o n a n d$ $a l g e b r a f r o m b o o k s, s o m e t i m e s .$
Commented by Rasheed.Sindhi last updated on 22/Jun/18
You're creative-mind Sir!
	Answered by ajfour last updated on 21/Jun/18
	$let sin θ = ((d−b)/a) x_A =c−(y_A −y_B )tan θ x_A = (a−b)tan θ area△ABD = (1/2)(a−x_A )(a−b) =(((a−b))/2)[a−c+(a−b)tan θ] y_C =b+(a−c)tan θ y_D −y_C =a−b−(a−c)tan θ area△BCD =(1/2)(y_D −y_C )(a−c) =(((a−c))/2)[a−b−(a−c)tan θ] Area ABCD = area△ABD +area△BCD = (1/2){(a−b)[a−c+(a−b)tan θ] +(a−c)[a−b−(a−c)tan θ]} =(1/2){a^2 −ac−ab−ac+bc+a^2 −ac−ab+bc +(a^2 −2ab+b^2 −a^2 +2ac−c^2 )tan θ} =(a^2 −ab−ac+bc) +(1/2)[(a−b)^2 −(a−c)^2 ](((d−b))/(√(a^2 −(d−b)^2 ))) } . =(a−c)(a−b) +(c−b)(a−((b+c)/2))(((d−b))/(√(a^2 −(d−b)^2 ))) .$
	$l e t \sin θ = \frac{d - b}{a}$ $x_{A} = c - (y_{A} - y_{B}) \tan θ$ $x_{A} = (a - b) \tan θ$ $a r e a △ A B D = \frac{1}{2} (a - x_{A}) (a - b)$ $= \frac{(a - b)}{2} [a - c + (a - b) \tan θ]$ $y_{C} = b + (a - c) \tan θ$ $y_{D} - y_{C} = a - b - (a - c) \tan θ$ $a r e a △ B C D = \frac{1}{2} (y_{D} - y_{C}) (a - c)$ $= \frac{(a - c)}{2} [a - b - (a - c) \tan θ]$ $A r e a A B C D = a r e a △ A B D$ $+ a r e a △ B C D$ $= \frac{1}{2} {(a - b) [a - c + (a - b) \tan θ]$ $+ (a - c) [a - b - (a - c) \tan θ]}$ $= \frac{1}{2} {a^{2} - a c - a b - a c + b c + a^{2} - a c - a b + b c$ $+ (a^{2} - 2 a b + b^{2} - a^{2} + 2 a c - c^{2}) \tan θ}$ $= (a^{2} - a b - a c + b c)$ $Missing \left or extra \right$ $= (a - c) (a - b)$ $+ (c - b) (a - \frac{b + c}{2}) \frac{(d - b)}{\sqrt{a^{2} - {(d - b)}^{2}}} .$
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