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Question Number 189114 by normans last updated on 12/Mar/23

Answered by mr W last updated on 12/Mar/23

Commented by mr W last updated on 12/Mar/23

$$c=\sqrt{a^2+{(b-a)}^2}$$$$\frac{(b-a)a}{c}+c+\frac{ab}{c}=13$$$$\Rightarrow a^2+b^2=13c...\left(i\right)$$$$\frac{(b-a)b}{c}+\frac{ab}{c}=9$$$$\Rightarrow b^2=9c...\left(ii\right)$$$$a^2=\frac{4b^2}{9}\Rightarrow a=\frac{2b}{3}$$$$b^2=9\sqrt{a^2+{(b-a)}^2}$$$$\Rightarrow b=3\sqrt{5}\Rightarrow a=2\sqrt{5}$$$$\Rightarrow b^2=45,a^2=20$$

Answered by HeferH last updated on 12/Mar/23

Commented by HeferH last updated on 12/Mar/23

$$\ast \frac{x}{4}=\frac{b}{a};x=\frac{4b}{a}$$$$\ast \frac{b}{b+a}=\frac{9-\frac{4b}{a}}{5}$$$$5b=9(b+a)-\frac{4b(b+a)}{a}$$$$5ab=9ab+9a^2-4b^2-4ab$$$$4b^2=9a^2$$$$\frac{b}{a}=\sqrt{\frac{9}{4}}=\frac{3}{2}$$$$\Rightarrow x=6$$$$\ast \mathrm{now}:$$$${(5-3)}^2+4^2=a^2=20u^2$$$${(9-6)}^2+6^2=b^2=45u^2$$

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