AB is line segment of length 24 cm. C is its midpoint. On AB, AC and BC semicircles are described. Find the radius of the circle which touches all the three semicircles.
Let the required radius be r cm.
O1O3 = Radius of smaller semicircle + r = 24/4 + r = 6 + r
O1C = Radius of smaller semicircle = 24/4 = 6 cm
In right triangle O1O3C:
O1O32 = O1C2 + O3C2
or O3C2 = O1O32 – O1C2
= 36 + r2 + 12r – 36
= r2 + 12r
or O3C = (r2 + 12r)1/2
DC = DO3 + O3C
or 24/2 = 12 = r + (r2 + 12r)1/2
or 12 – r = (r2 + 12r)1/2
Squaring both sides
144 + r2 – 24r = r2 + 12r
or 36r = 144
or r = 144/36 = 4
Hence, radius of the circle which touches all three semicircles is 4 cm.