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Question Number 191958 by Rupesh123 last updated on 04/May/23

Commented by Rupesh123 last updated on 04/May/23

x=?

Commented by mr W last updated on 04/May/23

$$x=r=\frac{a+b}{2}=\frac{4+8}{2}=6$$

Answered by mr W last updated on 04/May/23

Commented by mr W last updated on 04/May/23

$$a=4$$$$b=8$$$$r=\frac{a+b}{2}$$$$\alpha =\frac{\pi }{3}$$$$OD=\frac{b-a}{2}$$$$r^2=c^2+{\left(\frac{b-a}{2}\right)}^2-2c\left(\frac{b-a}{2}\right)\cos \frac{2\pi }{3}$$$${\left(\frac{a+b}{2}\right)}^2=c^2+{\left(\frac{b-a}{2}\right)}^2+c\left(\frac{b-a}{2}\right)$$$$\Rightarrow c^2-\left(\frac{a-b}{2}\right)c-ab=0$$$$similarly$$$$\Rightarrow d^2+\left(\frac{a-b}{2}\right)d-ab=0$$$$dand-carerootsof{z}^2+\left(\frac{a-b}{2}\right)z-ab=0.$$$$\Rightarrow d-c=-\frac{a-b}{2}$$$$\Rightarrow -cd=-ab$$$$x^2=c^2+d^2-2cd\cos \frac{\pi }{2}=c^2+d^2-cd={(d-c)}^2+cd$$$$x^2={\left(\frac{a-b}{2}\right)}^2+ab={\left(\frac{a+b}{2}\right)}^2=r^2$$$$\Rightarrow x=r=\frac{a+b}{2}=\frac{4+8}{2}=6$$

Commented by Shlock last updated on 04/May/23

This is perfect ��

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