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avatar+1839 

In triangle $ABC$, let the perpendicular bisector of $BC$ intersect $BC$ and $AC$ at $D$ and $E$, respectively. If $BC = 20$ and $\angle C = 15^\circ$, then find the length of $BE$.

 Jul 21, 2024
 #1
avatar+129895 
+1

             A    E

 

 

B                 D                  C  15

      10                  10   

 

EC / sin 90 = DC / sin75

EC = 10/sin75  

 

BD = DC

ED = ED

So....by LL,  right triangle BED  congruent to right triangle CED

 

So BE  = 10/sin75 ≈ 10.35

 

cool cool cool

 Jul 21, 2024
 #2
avatar+16 
+1

I just want to add on that we can represent the answer \(10\sin(75) =  10\tan(15) = 10\tan(\frac{30}{2}).\)
Now we can use the trigonometric half-angle identity \(\tan(\frac{\alpha}{2}) = \frac{\sin(\alpha)}{\cos(\alpha)+1}\).
We can input our normal values: \(10\tan(\frac{30}{2}) = 10(\tan(\frac{30}{2})) = 10(\frac{\sin(30)}{\cos(30)+1}) = 10(\frac{\frac{1}{2}}{\frac{\sqrt{3}}{2}}+1) = 10(\frac{1}{\sqrt{3}+2}) = 10(2-\sqrt{3})= 20 - 10\sqrt{3}.\)
Therefore \(BE = \sqrt{(20 - 10\sqrt{3})^2+10^2} = \sqrt{400 + 300 - 400\sqrt{3} + 100} = \sqrt{800 - 400\sqrt{3}} = \sqrt{400(2-\sqrt{3})} = 20\sqrt{2-\sqrt{3}}\)

 Jul 21, 2024

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