Question

In ∆ ABC, m angle B= 90° . seg AD is perpendicular to side BC. If AC= 20,AB=15 , BC= 12 ; than BD = ?​

Answer 1

Answer:

Since , ∆ABC is right triangle (right angled at A) , so by Pythagoras theorem , we say :

AB² + AC² = BC²….(1)

Also AD is perpendicular to BC which implies that ∆ BDA and ∆ CDA are also right triangles : so we again apply Pythagoras theorem to get following equations :

AD² + DC² = AC²….(2)

BD² + AD² = AB²….(3)

Adding equations (2) and (3) :

BD² + DC² + 2AD² = AC² + AB²

From (1) we replace RHS of above equation to get :

BD² + DC² + 2AD² = BC²

But BC = (BD + CD)

So we again replace :

BD² + DC² + 2AD² = (BD + DC)²

Expanding we get :

BD² + DC² + 2AD² = BD² + CD² + 2BD×CD

Cancelling and simplifying we get :

AD² = BD × CD

Hence, proved.

However it is a long method and not as effecient as the other two answers are by using similarity and trigonometry.