Question

ABC is triangle right angled at A with AB=AC. bisector of angle A meets BC at D . prove that BC=2AD.

Answer 1

In Δ , ABC right angled at A and AB = AC 

Hence ∠ A = ∠ B 

We know that Sum of angles of a triangle = 180º 

∠A+ ∠B+ ∠C=180º 

90º+∠B+∠B=180º 

2∠B=180º -90º 

2∠B=90º 

∠B=45º………………………………………..(i) 

ALSO , AD is the bisector of BAC 

So , ∠BAD = ∠CAD = 90º/2 = 45º …………………………….(ii) 

∠BAD = ∠ABC 

SO, AD = BD ………………………………(iii) . 

Similarly angle CAD = angle ACD 

So, AD = DC ………………………………..(iv) 

adding equation (iii) and (iv) 

We will get, AD + AD = BD+DC 
2AD = BC