Question-188208

Question Number 188208 by Rupesh123 last updated on 26/Feb/23

Answered by a.lgnaoui last updated on 26/Feb/23

∡ADB=150 ⇒∡BCD=30 ∡C=120 △ABC ((sin 20)/8)=((sin 60)/(x+CD)) (1) CD=((8sin 60)/(sin 20))−x (2) △BCD ((sin 50)/(CD))=((sin 30)/8) (3) CD=(( 8sin 50)/(sin 30)) (4) (2)=(4) x=((8sin 60)/(sin 20))−((8sin 50)/(sin 30)) x =((8(sin 60sin 30−sin 20sin 50))/(sin 20sin 30)) x=8

$$\measuredangle{ADB}=\mathrm{150}\:\:\Rightarrow\measuredangle{BCD}=\mathrm{30}\:\:\measuredangle{C}=\mathrm{120} \\ $$$$\bigtriangleup{ABC}\:\:\:\frac{\mathrm{sin}\:\mathrm{20}}{\mathrm{8}}=\frac{\mathrm{sin}\:\mathrm{60}}{{x}+{CD}}\:\:\:\:\:\:\:\:\:\:\:\left(\mathrm{1}\right) \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{CD}=\frac{\mathrm{8sin}\:\mathrm{60}}{\mathrm{sin}\:\mathrm{20}}−{x}\:\:\:\:\:\:\:\left(\mathrm{2}\right) \\ $$$$\bigtriangleup{BCD}\:\:\:\:\frac{\mathrm{sin}\:\mathrm{50}}{{CD}}=\frac{\mathrm{sin}\:\mathrm{30}}{\mathrm{8}}\:\:\:\:\:\:\:\:\:\:\:\:\left(\mathrm{3}\right) \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{CD}=\frac{\:\mathrm{8sin}\:\mathrm{50}}{\mathrm{sin}\:\mathrm{30}}\:\:\:\:\:\left(\mathrm{4}\right)\:\:\:\:\:\:\: \\ $$$$\left(\mathrm{2}\right)=\left(\mathrm{4}\right)\:\:\:\:\:\:{x}=\frac{\mathrm{8sin}\:\mathrm{60}}{\mathrm{sin}\:\mathrm{20}}−\frac{\mathrm{8sin}\:\mathrm{50}}{\mathrm{sin}\:\mathrm{30}} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\: \\ $$$${x}\:=\frac{\mathrm{8}\left(\mathrm{sin}\:\mathrm{60sin}\:\mathrm{30}−\mathrm{sin}\:\mathrm{20sin}\:\mathrm{50}\right)}{\mathrm{sin}\:\mathrm{20sin}\:\mathrm{30}} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\boldsymbol{{x}}=\mathrm{8} \\ $$

Commented by Rupesh123 last updated on 26/Feb/23

Perfect ��

Answered by mr W last updated on 27/Feb/23

((AB)/(sin 100))=(8/(sin 20)) ⇒AB=((8 cos 10)/(sin 20)) ((AD)/(sin 10))=((AB)/(sin 150))=((8 cos 10)/(sin 20 sin 30)) ⇒x=((8 cos 10 sin 10)/(sin 20 sin 30)) =((sin 20×8)/(2×sin 30×sin 20))=8

$$\frac{{AB}}{\mathrm{sin}\:\mathrm{100}}=\frac{\mathrm{8}}{\mathrm{sin}\:\mathrm{20}}\:\Rightarrow{AB}=\frac{\mathrm{8}\:\mathrm{cos}\:\mathrm{10}}{\mathrm{sin}\:\mathrm{20}} \\ $$$$\frac{{AD}}{\mathrm{sin}\:\mathrm{10}}=\frac{{AB}}{\mathrm{sin}\:\mathrm{150}}=\frac{\mathrm{8}\:\mathrm{cos}\:\mathrm{10}}{\mathrm{sin}\:\mathrm{20}\:\mathrm{sin}\:\mathrm{30}} \\ $$$$\Rightarrow{x}=\frac{\mathrm{8}\:\mathrm{cos}\:\mathrm{10}\:\mathrm{sin}\:\mathrm{10}}{\mathrm{sin}\:\mathrm{20}\:\mathrm{sin}\:\mathrm{30}} \\ $$$$\:\:\:\:\:\:\:=\frac{\mathrm{sin}\:\mathrm{20}×\mathrm{8}}{\mathrm{2}×\mathrm{sin}\:\mathrm{30}×\mathrm{sin}\:\mathrm{20}}=\mathrm{8} \\ $$

Commented by Rupesh123 last updated on 27/Feb/23

Perfect ��

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