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In triangle $ABC,$ $BC = 32,$ $\tan B = \frac{3}{5},$ and $\tan C = \frac{1}{4}.$ Find the area of the triangle.

 Jun 1, 2024
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Let B = (0,0)   C  =(32,0)

To find the height of the triangle we can construct two lines  using the tangents 

y = (3/5)x

And

y = (-1/4) (x -32) =  (-1/4)x + 8

 

 

To find the x coordinate   of their intersection

 

(3/5()x =(-1/4)x + 8

(3/5 + 1/4)x =  8

[(12 + 5] / 20]  x  =  8

[17/20] x = 8

x = (8*20] / 17  =  160/17

 

The height of the triangle (AD)  = (3/5)(160/17)  = 480 / 85 = 96/17

 

Area ABC  = (1/2) BC * AD =  (1/2) 32 * (96/17)  = 1536 / 17

 

cool cool cool

 Jun 1, 2024

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