Question

In a triangle abc ad is perpendicular from a to bc meets bc at d. If bd=8 ,dc=2, ad=4 then prove that abc is a right angled at a

Answer 1

As AD is drawn perpendicular to BC in right angled ΔABC, it is apparent that ΔABC is right angled at ∠Aas shown below (not drawn to scale).

 

As can be seen ∠B is common in ΔABC as well as ΔDBA (here we have written two triangles this way as ∠A=∠D, ∠B=∠B and ∠C=∠BAD) - as both are right angled (obviously third angles too would be equal) and therefore we have

ΔABC≈ΔDBA and hence

BCAB=ABBD=ACAD..............(1)

therefore, we have BCAB=ABBD or

AB2=BC×BD=9×4=36

Hence AB=6