Q. 573.7( 6 Votes )

In the given figure, ΔABC is right-angled at A. Find the area of the shaded region if AB = 6 cm, BC = 10 cm and 0 is the centre of the incircle of ΔABC. [Take π = 3.14.]


Answer :

Given AB = 6 cm, BC = 10 cm


Consider ∆ABC, BAC = 90°



102 = 62 + AC2 (putting given values)


100 = 36 + AC2


100 – 36 = AC2


AC2 = 64



AC = 8 cm



Join OB, OA, OC, OE, OF, OD


Here OE = OF = OD = radius of circle = r cm


OEC = ODB = OFB = 90° (angle at the point of contact of radius & tangent)


Area ∆ABC = Area of ∆OAC + Area of ∆OCB + Area of ∆OAB eqn1




Area of ∆OAC = 4r eqn2




Area of ∆OCB = 5r eqn3




Area of ∆OAB = 3r eqn4




Area of ∆ABC = 3×8


Area of ∆ABC = 24 cm2 eqn5


Putting all the values in equation we get;


24 = 4r + 5r + 3r


24 = 12r



r = 2 cm


Area of circle = πr2


Put the value of r, we get,


Area of circle = π×22


Area of circle = 3.14×4 (putting π = 3.14)


Area of circle = 12.56 cm2 eqn6


Area of shaded region = Area of triangle – Area of circle


Area of shaded region = 24 – 12.56 (from eqn5 and eqn6)


Area of shaded region = 11.44 cm2


Area of shaded region is 11.44 cm2.


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