Given: ∆ABC, m∠C = 90° m∠BAC = 2m∠ABC BC = 24, AL −∠ bisector Find: AL
Accepted Solution
A:
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