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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 26, 2024
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Let B = (0,0)

Let C = (32,0)

Let A  be the apex

 

The line through AB has the equation

y = (3/5)x

The line through CA has the equation

y = (-1/4) (x - 32)  =  (-1/4)x + (-1/4)

 

To find the x coordiante of A, set the lines equal

 

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

 

(3/5 + 1/4)x  = 8

 

(17/20) x = 8

 

x = 160 / 17

 

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

 

[ABC ]  =  (1/2) (32) (96/17)  = 1536/17 ≈  90.35

 

cool cool cool

 Jun 26, 2024

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