+207 In triangle $ABC,$ $BC = 32,$ $\tan B = \frac{3}{5},$ and $\tan C = \frac{1}{4}.$ Find the area of the triangle.
+129852 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
