AB is a line segment & M is its mid- point. Semi- circles are drawn with AM, MB and AB as diameters on the same side of the line AB. A circle C (0, r) is drawn so that it touches all the three semicircles. The length of AB is 'x' times r. Find x.
Let AB = x cm , Then AM = x2 and PM = PN = x4
Using Pythagoras Theorem,
⇒ OP2=PM2+OM2
⇒ (x/4+r)2=(x/4)2+(x/2−r)2
⇒ (x/4)2+r2+(xr)/2=(x/4)2+(x/2)2+r2–xr
⇒ r=16x
⇒ r=16AB
⇒ AB=6r