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Question-185893




Question Number 185893 by mr W last updated on 29/Jan/23
Commented by mr W last updated on 29/Jan/23
length of common tangent line x=?
$${length}\:{of}\:{common}\:{tangent}\:{line}\:{x}=? \\ $$
Commented by som(math1967) last updated on 29/Jan/23
x=(√7) ?
$${x}=\sqrt{\mathrm{7}}\:? \\ $$
Commented by mr W last updated on 29/Jan/23
yes, thanks!
$${yes},\:{thanks}! \\ $$
Answered by a.lgnaoui last updated on 29/Jan/23
Answered by som(math1967) last updated on 29/Jan/23
Commented by som(math1967) last updated on 29/Jan/23
AB=diagonal of square  =dist between two centre=4(√2)  BC=2+3=5  ∴AC=DE=x=(√((4(√2))^2 −5^2 ))=(√7)
$${AB}={diagonal}\:{of}\:{square} \\ $$$$={dist}\:{between}\:{two}\:{centre}=\mathrm{4}\sqrt{\mathrm{2}} \\ $$$${BC}=\mathrm{2}+\mathrm{3}=\mathrm{5} \\ $$$$\therefore{AC}={DE}={x}=\sqrt{\left(\mathrm{4}\sqrt{\mathrm{2}}\right)^{\mathrm{2}} −\mathrm{5}^{\mathrm{2}} }=\sqrt{\mathrm{7}} \\ $$$$ \\ $$
Answered by a.lgnaoui last updated on 29/Jan/23
Triangle AEF rectangle en E  EF^2 =AF^2 − 4  Truangke DFE rectangle en F  EF^2 =DE^2 −9  AD^2 =AE^2 +DE^2 −AE×DEcos ((π/2)+α)(i)  (4(√2) )^2 =4+DE^2 +4DEsin α  △EFD      EF=(3/(tan α))  sin α=(3/(DE))     DE^2 =9+EF^2   (i)  ⇒32=4+9+EF^2 +4(3/(sin α))sin α              32=25+EF^2                   EF=(√7)
$${Triangle}\:{AEF}\:{rectangle}\:{en}\:{E} \\ $$$${EF}^{\mathrm{2}} ={AF}^{\mathrm{2}} −\:\mathrm{4} \\ $$$${Truangke}\:{DFE}\:{rectangle}\:{en}\:{F} \\ $$$${EF}^{\mathrm{2}} ={DE}^{\mathrm{2}} −\mathrm{9} \\ $$$${AD}^{\mathrm{2}} ={AE}^{\mathrm{2}} +{DE}^{\mathrm{2}} −{AE}×{DE}\mathrm{cos}\:\left(\frac{\pi}{\mathrm{2}}+\alpha\right)\left({i}\right) \\ $$$$\left(\mathrm{4}\sqrt{\mathrm{2}}\:\right)^{\mathrm{2}} =\mathrm{4}+{DE}^{\mathrm{2}} +\mathrm{4}{DE}\mathrm{sin}\:\alpha \\ $$$$\bigtriangleup{EFD}\:\:\:\: \\ $$$${EF}=\frac{\mathrm{3}}{\mathrm{tan}\:\alpha} \\ $$$$\mathrm{sin}\:\alpha=\frac{\mathrm{3}}{{DE}}\:\:\:\:\:{DE}^{\mathrm{2}} =\mathrm{9}+{EF}^{\mathrm{2}} \\ $$$$\left({i}\right)\:\:\Rightarrow\mathrm{32}=\mathrm{4}+\mathrm{9}+{EF}^{\mathrm{2}} +\mathrm{4}\frac{\mathrm{3}}{\mathrm{sin}\:\alpha}\mathrm{sin}\:\alpha \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{32}=\mathrm{25}+{EF}^{\mathrm{2}} \\ $$$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\boldsymbol{{E}}{F}=\sqrt{\mathrm{7}} \\ $$$$ \\ $$
Answered by ajfour last updated on 29/Jan/23
Commented by mr W last updated on 29/Jan/23
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