Geometry

Question

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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.

Pythagorearn Jan 23, 2025

CPhill · Answer 1 · 2025-01-23T06:35:39

Let AD be the altitude and BC the base

tan B = AD / BD = 3/5 = 3 / 5

tan C = AD / CD = 1/4 = 3 / 12

BD = 5 / (12 + 5) *32 = (5/17) * 32 = 160 / 17

AD / BD = AD / (160/17) = 3/5

AD = (160 / 17) *( 3 / 5) = 96 / 17

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