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Question Number 131665 by naka3546 last updated on 07/Feb/21

Answered by mr W last updated on 07/Feb/21

Commented by mr W last updated on 07/Feb/21

$$\frac{{20}^2+{22}^2-{\left(\sqrt{2}s\right)}^2}{2\times 20\times 22}=\frac{{30}^2+{26}^2-{\left(2a\right)}^2}{2\times 30\times 26}$$$$\Rightarrow \frac{442-s^2}{22}=\frac{788-2a^2}{39}...\left(i\right)$$$$\cos \alpha =\frac{a^2+s^2-{10}^2}{2as}$$$$\cos \beta =\frac{a^2+s^2-4^2}{2as}=\sin \alpha$$$${\left(\frac{a^2+s^2-100}{2as}\right)}^2+{\left(\frac{a^2+s^2-16}{2as}\right)}^2=1$$$${(a^2+s^2-100)}^2+{(a^2+s^2-16)}^2=4a^2{s}^2$$$$a^4+s^4-116(a^2+s^2)+5128=0...\left(ii\right)$$$$from\left(i\right):$$$$a^2=\frac{39(s^2-442)}{44}+394=\frac{39s^2}{44}+\frac{49}{22}$$$$putthisinto\left(ii\right):$$$${(\frac{39s^2}{44}+\frac{49}{22})}^2+s^4-116(\frac{39s^2}{44}+\frac{49}{22}+s^2)+5128=0$$$$\frac{3457}{4}{s}^4-103997s^2+2359305=0$$$$S=s^2=2\left(\frac{103997\pm 51568}{3457}\right)=90or30\frac{1148}{3457}$$

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