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Question

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.


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Solution

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/2r)2

(x/4)2+r2+(xr)/2=(x/4)2+(x/2)2+r2xr

r=16x

r=16AB

AB=6r


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