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Question Number 203480 by mr W last updated on 20/Jan/24

Commented by mr W last updated on 20/Jan/24

$i s i t p o s s i b l e t h a t t h e r e d l i n e s d i v i d e$ $t h e s q u a r e i n t o 4 p a r t s w i t h g i v e n$ $a r e a s ? i f y e s, f i n d t h e l e n g t h o f t h e$ $r e d l i n e s .$
Commented by behi834171 last updated on 20/Jan/24

$1) f r o m t h e a r e a o f s q a r e :$ $s^{2} = 20 + 30 + 40 + 70 = 160 \Rightarrow s = 4 \sqrt{10}$ $2) f r o m t h e a r e a o f t r a p e z i u s s,$ $p : i s t h e v e r t i c a l p a r t o f s e c t i o n w i t h, 20 a r e a$ $q : i s t h e h o r i z o n t a l p a r t o f s e c t i o n w i t h, 20 a r e a$ $\frac{20 p + 40 p}{2} . s = 20 + 40$ $\frac{20 q + 30 q}{2} . s = 20 + 30$ $\frac{30 q + 70 q}{2} . s = 30 + 70$ $\frac{40 p + 70 p}{2} . s = 40 + 70$ $[Σ (r h s)] . s = 2 \times 320 \Rightarrow 4 s . s = 640 \Rightarrow s = 4 \sqrt{10}$ $f r o m 1 & 2, y e s . i t i s p o s s i b l e .$
	Answered by esmaeil last updated on 20/Jan/24

	$a^{2} = 160 \to a = 4 \sqrt{10}$ $(\frac{m + n}{2}) \times 4 \sqrt{10} = 50 (A r e a o f t h e t o p t r a p i z o i d) \to$ $m + n = \frac{5 \sqrt{10}}{2}$ $r s i n α = n - m < m + n < \frac{5 \sqrt{10}}{2}$ $α = (a n g l e o f r e d l i n e w i t h t h e$ $h o r i z o n)$ $r > a > 4 \sqrt{10}$ $m a x (s i n α) = 1 \to r s i n α ≮ \frac{5 \sqrt{10}}{2} \to$ $n o t p o s s i b l e$
	Commented by mr W last updated on 21/Jan/24

	$w h a t i f o n e a r e a i s n o t 70 b u t 50 ?$ $20, 30, 40 r e m a i n .$
	Commented by mr W last updated on 21/Jan/24

	Commented by esmaeil last updated on 21/Jan/24

	$a^{2} = 140$ $(\frac{m + n}{2}) \times 2 \sqrt{35} = 50 \to m + n = \frac{10 \sqrt{35}}{7} \to$ $n - m < \frac{10 \sqrt{35}}{7}$ $s i n α = \frac{n - m}{r} \to r s i n α = n - m < \frac{10 \sqrt{35}}{7}$ $\approx 8.4515$ $b u t r > \sqrt{140} = a \approx 12$ $i t h i n k o n l y f o r (α = 0) \to$ $m = n = \frac{a}{2} i s p o s s i b l e$
	Answered by mr W last updated on 21/Jan/24

	Commented by mr W last updated on 21/Jan/24

	$s i d e l e n g t h o f s q u a r e = s$ $s^{2} = 20 + 30 + 40 + 70$ $\Rightarrow s = \sqrt{20 + 30 + 40 + 70}$ $x y - \frac{x^{2} \tan θ}{2} + \frac{y^{2} \tan θ}{2} = 20$ $2 x y - (x^{2} - y^{2}) \tan θ = 40 . . . (i)$ $s i m i l a r l y$ $2 y (s - x) - [y^{2} - {(s - x)}^{2}] \tan θ = 60 . . . (i i)$ $2 x (s - y) - [{(s - y)}^{2} - x^{2}] \tan θ = 80 . . . (i i i)$ $f r o m (i) :$ $\tan θ = \frac{2 x y - 40}{x^{2} - y^{2}}$ $p u t t h i s i n t o (i i) a n d (i i i) :$ $2 y (s - x) - [y^{2} - {(s - x)}^{2}] (\frac{2 x y - 40}{x^{2} - y^{2}}) = 60 . . . (I)$ $2 x (s - y) - [{(s - y)}^{2} - x^{2}] (\frac{2 x y - 40}{x^{2} - y^{2}}) = 80 . . . (I I)$ $i f t h e a r e a o f t h e f o u r t h p a r t i s 70,$ $t h e r e i s n o s o l u t i o n f o r t h e e q u a t i o n$ $s y s t e m . i f t h e f o u r t h p a r t i s 60, w e$ $g e t a s o l u t i o n :$ $x \approx 4.3225, y \approx 5.0431, θ \approx - 28.06 °$
	Commented by mr W last updated on 21/Jan/24

	Answered by ajfour last updated on 21/Jan/24

	Commented by ajfour last updated on 21/Jan/24

	$l e t 2 s = a + b = c + d = \sqrt{P + Q + R + S}$ $\tan θ = 2 t$ $Q - a^{2} t + c^{2} t = a c$ $P - c^{2} t + b^{2} t = b c$ $S - b^{2} t + d^{2} t = b d$ $R - d^{2} t + a^{2} t = d a$
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