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

BRAlNBOLT Jun 26, 2024

#1**+2 **

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

CPhill Jun 26, 2024