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Question Number 180927 by mr W last updated on 19/Nov/22

Commented by mr W last updated on 19/Nov/22

$f i n d t h e a r e a o f t h e s q u a r e .$
	Answered by mr W last updated on 20/Nov/22

	Commented by mr W last updated on 20/Nov/22

	$M e t h o d I$ $s = s i d e l e n g t h o f s q u a r e . i t s a r e a i s s^{2} .$ $\cos θ = \frac{7^{2} + 9^{2} - {(\sqrt{2} s)}^{2}}{2 \times 7 \times 9}$ $\cos θ = \frac{13^{2} + 11^{2} - {(2 a)}^{2}}{2 \times 13 \times 11}$ $\frac{13^{2} + 11^{2} - {(2 a)}^{2}}{2 \times 13 \times 11} = \frac{7^{2} + 9^{2} - {(\sqrt{2} s)}^{2}}{2 \times 7 \times 9}$ $\frac{145}{143} - \frac{2 a^{2}}{143} = \frac{65}{63} - \frac{s^{2}}{63}$ $\Rightarrow a^{2} = \frac{143 s^{2} - 160}{126} . . . (I)$ $\cos α = \frac{a^{2} + s^{2} - 6^{2}}{2 a s}$ $\cos β = \frac{a^{2} + s^{2} - 2^{2}}{2 a s}$ $α + β = 90 ° \Rightarrow \cos β = \sin α$ ${(\frac{a^{2} + s^{2} - 6^{2}}{2 a s})}^{2} + {(\frac{a^{2} + s^{2} - 2^{2}}{2 a s})}^{2} = 1$ ${(a^{2} + s^{2} - 36)}^{2} + {(a^{2} + s^{2} - 4)}^{2} = 4 a^{2} s^{2}$ $2 {(a^{2} + s^{2})}^{2} - 2 (36 + 4) (a^{2} + s^{2}) + 36^{2} + 4^{2} = 4 a^{2} s^{2}$ $\Rightarrow s^{4} - 40 s^{2} + {(a^{2} - 20)}^{2} + 256 = 0 . . . (I I)$ $p u t (I) i n t o (I I) :$ $s^{4} - 40 s^{2} + {(\frac{143 s^{2} - 160}{126} - 20)}^{2} + 256 = 0$ $36 325 s^{4} - 1 401 520 s^{2} + 11 246 656 = 0$ $\Rightarrow s^{2} = \frac{700 760 \pm \sqrt{700 760^{2} - 36 325 \times 11 246 656}}{36 325}$ $= \frac{700 760 \pm 287 280}{36 325}$ $= \frac{136}{5} o r \frac{82 696}{7 265} (r e j e c t e d)$ $i . e . t h e a r e a o f t h e s q u a r e i s \frac{136}{5} .$
	Answered by mr W last updated on 20/Nov/22

	Commented by mr W last updated on 20/Nov/22

	$M e t h o d I I$ ${(\sqrt{2} s)}^{2} = 7^{2} + 9^{2} - 2 \times 7 \times 9 \cos θ$ ${(\sqrt{2} s)}^{2} = 2^{2} + 6^{2} - 2 \times 2 \times 6 \cos (α + β)$ $α + β = 180 ° - θ \Rightarrow \cos (α + β) = - \cos θ$ $2 s^{2} = 2^{2} + 6^{2} + 2 \times 2 \times 6 \cos θ = 7^{2} + 9^{2} - 2 \times 7 \times 9 \cos θ$ $\Rightarrow \cos θ = \frac{3}{5}$ $\Rightarrow s^{2} = \frac{130}{2} - 63 \times \frac{3}{5} = \frac{136}{5}$ $i . e . a r e a o f s q u a r e i s \frac{136}{5} .$
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