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Question-125084




Question Number 125084 by ajfour last updated on 08/Dec/20
Commented by ajfour last updated on 08/Dec/20
Find x, hence radius of circle.
$${Find}\:{x},\:{hence}\:{radius}\:{of}\:{circle}. \\ $$
Commented by ajfour last updated on 08/Dec/20
Commented by mr W last updated on 08/Dec/20
x can be any value, because the shape  formed by both rectangles is always  cyclic.
$${x}\:{can}\:{be}\:{any}\:{value},\:{because}\:{the}\:{shape} \\ $$$${formed}\:{by}\:{both}\:{rectangles}\:{is}\:{always} \\ $$$${cyclic}. \\ $$
Answered by mr W last updated on 08/Dec/20
Commented by mr W last updated on 08/Dec/20
ABCD is always cyclic.  AB=CD=(√((((x−1)/2))^2 +(1+x)^2 ))  AC=BD=(√((((x+1)/2))^2 +(1+x)^2 ))=(((x+1)(√5))/2)  R=radius  Δ_(ABD) =((1×(1+x))/2)  formula R=((abc)/(4Δ))  ⇒R=((√([(((x−1)/2))^2 +(1+x)^2 ][(((x+1)/2))^2 +(1+x)^2 ]))/(2(1+x)))  ⇒R=((√(5(5x^2 +6x+5)))/8)
$${ABCD}\:{is}\:{always}\:{cyclic}. \\ $$$${AB}={CD}=\sqrt{\left(\frac{{x}−\mathrm{1}}{\mathrm{2}}\right)^{\mathrm{2}} +\left(\mathrm{1}+{x}\right)^{\mathrm{2}} } \\ $$$${AC}={BD}=\sqrt{\left(\frac{{x}+\mathrm{1}}{\mathrm{2}}\right)^{\mathrm{2}} +\left(\mathrm{1}+{x}\right)^{\mathrm{2}} }=\frac{\left({x}+\mathrm{1}\right)\sqrt{\mathrm{5}}}{\mathrm{2}} \\ $$$${R}={radius} \\ $$$$\Delta_{{ABD}} =\frac{\mathrm{1}×\left(\mathrm{1}+{x}\right)}{\mathrm{2}} \\ $$$${formula}\:{R}=\frac{{abc}}{\mathrm{4}\Delta} \\ $$$$\Rightarrow{R}=\frac{\sqrt{\left[\left(\frac{{x}−\mathrm{1}}{\mathrm{2}}\right)^{\mathrm{2}} +\left(\mathrm{1}+{x}\right)^{\mathrm{2}} \right]\left[\left(\frac{{x}+\mathrm{1}}{\mathrm{2}}\right)^{\mathrm{2}} +\left(\mathrm{1}+{x}\right)^{\mathrm{2}} \right]}}{\mathrm{2}\left(\mathrm{1}+{x}\right)} \\ $$$$\Rightarrow{R}=\frac{\sqrt{\mathrm{5}\left(\mathrm{5}{x}^{\mathrm{2}} +\mathrm{6}{x}+\mathrm{5}\right)}}{\mathrm{8}} \\ $$
Commented by ajfour last updated on 08/Dec/20
Very Nice solution, Sir!
$${Very}\:{Nice}\:{solution},\:{Sir}! \\ $$

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