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In triangle $ABC$, $\angle A = 30^\circ$ and $\angle B = 90^\circ$. Point $X$ is on side $\overline{AC}$ such that line segment $\overline{BX}$ bisects $\angle ABC$. If $BC = 12$, then find the area of triangle $BXA$.

 Feb 8, 2024
 #1
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A

 

              X

 

B       12            C

 

AB =  12sqrt 3

AC =  24

sin (angle A)  = sin (30)  =  (1/2)

Angle ABX = 45

Angle BXA = 180 - 30 - 45  =  105

 

Law of Sines

 

BX  /sin (30)  = AB / sin (105)

 

BX  /  sin (30) = 12 sqrt 3  / sin (105)

 

BX =  12sqrt (3) sin (30)  /sin (105)  = 12sqrt (6) / (sqrt (3) + 1)

 

[BXA ]  =  (1/2)AB*BX *sin (45)  =   (1/2) (12sqrt 3)(12sqrt6) / (sqrt (3)+ 1) *sqrt (2) / 2  =

 

216  / (1 + sqrt 3)  =   216 ( 1 -sqrt 3)  / [ 1 - 3]  =     216 [ sqrt (3) - 1] / 2  =  108 (sqrt (3) - 1)

 

 

cool cool cool

 Feb 8, 2024

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