Question

The inscribed circle of right angled triangle abc touches the sides ab, bc and ca at d, e and f respectively. If ad = 6 cm and be = 5 cm, then find the length of ac.

Answer 1

Consider ABC be the right angled triangle such that ∠A = 90° and AB = 5cm, AC = 12 cm.  

And O be the centre and r be the radius of the incircle.

AB, BC and CA are tangents to the circle at P, N and M.

∴ OP = ON = OM =  r  (radius of the circle)

Area of ΔABC = ½ × 5 × 12 = 30 cm2

By Pythagoras theorem,

BC2  = AC2  + AB2

⇒ BC2  = 122  + 52

⇒ BC2  = 169

⇒ BC = 13 cm

Area of ∆ABC = Area ∆OAB + Area ∆OBC + Area ∆OCA

30 = 1 2 r × AB + 1 2 r × BC + 1 2 r × CA  

30 = 1 2 r(AB+BC+CA)

⇒ r = 2 × 30 (AB+BC+CA)  

⇒ r = 60 5+13+12  

⇒ r = 60/30 = 2 cm.