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.

O_{1}O_{3} = Radius of smaller semicircle + r = 24/4 + r = 6 + r

O_{1}C = Radius of smaller semicircle = 24/4 = 6 cm

In right triangle O_{1}O_{3}C:

O_{1}O_{3}^{2} = O_{1}C^{2} + O_{3}C^{2}or O_{3}C^{2} = O_{1}O_{3}^{2} – O_{1}C^{2} = 36 + r^{2} + 12r – 36

= r^{2} + 12r

or O_{3}C = (r^{2} + 12r)^{1/2}

Also,

DC = DO_{3} + O_{3}C

or 24/2 = 12 = r + (r^{2} + 12r)^{1/2}

or 12 – r = (r^{2} + 12r)^{1/2}

Squaring both sides

144 + r^{2} – 24r = r^{2} + 12r

or 36r = 144

or r = 144/36 = 4

Hence, radius of the circle which touches all three semicircles is 4 cm.

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