Given: ∆ABC, m∠C = 90° m∠BAC = 2m∠ABC BC = 24, AL - ∠ bisector Find: AL

The answer is that AL is equal to 16.

Given that in triangle ABC, m∠C=90,

it means m∠A + ∠B = m∠BAC + m∠ABC = 90

m∠ABC = 90 - m∠BAC

Also given that; m∠BAC = 2m∠ABC

So,

m∠BAC = 2 (90 - m∠BAC) = 180 - 2m∠BAC

m∠BAC + 2m∠BAC = 180

3m∠BAC = 180

m∠BAC=180/3 = 60

m∠ABC = 60/3 = 30

thus,ΔBAC is a 30-60-90 right triangle, in which the ratio of the side lengths is 1:√3:2AC:BC=1:√3, AC=BC/√3BC=24, So,

AC=24/√3=8√3AL bisects angle A = >m∠LAC=30

ΔALC is a 30-60-90 right triangle, in which the ratio of the side lengths is 1:√3:2AC:AL=√3:2AL=2AC/√3=2x8√3/√3=16