Q3 of 31 Page 234

∠B is a right angle in ΔABC. If AB = 24, BC = 7, then find the radius of the circle which touches all the three sides of ΔABC.


Let radius be r and the centre of the circle touching all the sides of a triangle be I i.e. incentre.


In the figure,


ID = IE = IF = r


Since ΔABC is a right angled triangle, ∠B = 90°.


Also ID ⊥ BC and AB ⊥ BC.


∴ ID || AB and ID || FB


Similarly, IF || BD


∴ IFBD is a parallelogram.


∴ FB = ID = r and BD = IF = r … (1)


∴ Parallelogram IFBD is a rhombus.


Since ∠B = 90°, parallelogram IFBD is a square.


By Pythagoras Theorem,


⇒ AC2 = AB2 + BC2


= 242 + 72


= 576 + 49


= 625


∴ AC = 25


⇒ AB + BC + AC = 24 + 7 + 25


⇒ AF + FB + BD + DC + AC = 56


⇒ AE + r + r + CE + AC = 56


⇒ 2r + (AE + CE) + AC = 56


⇒ 2r + 2AC = 56


⇒ 2r + 2(25) = 56


⇒ r + 25 = 28


⇒ r = 3


∴ The radius of circle is 3.


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