In the given figure ΔABC is right angled at A. Semicircles are drawn on AB, AC and BC as diameter. It is given that AB = 3 cm and AC = 4 cm. Find the area of the shaded region.

Let semicircle I, II and III are semicircles with diameters AB, AC and BC respectively


Area of shaded region =


Area of semicircle I + Area of semicircle II + Area of triangle ABC – Area of semicircle III


As, ∠BAC is in semicircle,


∠BAC = 90° [Angle in a semicircle is right angle]


And ABC is a right – angled triangle at A


By Pythagoras Theorem


(Hypotenuse)2 = (Base)2 + (Perpendicular)2


(BC)2 = (AB)2 + (AC)2


⇒ (BC)2 = 32 + 42 = 9 + 16 = 25


⇒ BC = 5 cm


Now, For semicircle I


Diameter = AB = 3 cm


Radius


Area of semicircle of radius r


Area of semicircle I


For semicircle II


Diameter = AC = 4 cm


Radius


Area of semicircle of radius r


Area of semicircle II


For semicircle III


Diameter = BC = 5 cm


Radius,


Area of semicircle of radius r


Area of semicircle I


Area of a right – angled triangle


Area of ΔABC


Required area (From eqn [1])


25