A problem by Shreyash Taori

Level pending

AB is a line segment of length 24 cm "C" is its mid-point. On AB, AC and CB semi-circles are described. The radius of circle which touches all these semi circles is

8cm 6cm 3cm 4cm

Shreyash Taori by Shreyash Taori , 22, India

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1 solution

Arun Jayapal
Apr 19, 2014

For figure, visit https://imgur.com/zHAbEkG

Let A B = l AB = l units.

Therefore, we can easily deduce that,

P C = P X = C Q = Q Y = l 4 PC = PX = CQ = QY = \frac { l }{ 4 }

Assume radius of semicircle in question is k k units.

Considering △ P O C \triangle POC which is rt angle at C C ,

P O 2 = C O 2 + P C 2 { PO }^{ 2 } = { CO }^{ 2 } + { PC }^{ 2 }

i.e. ( l 4 + k ) 2 = ( l 2 − k ) 2 + ( l 4 ) 2 { ( \frac{ l }{ 4 } +k ) }^{ 2 } = { (\frac { l }{ 2 } -k) }^{ 2 }+{ (\frac { l }{ 4 } ) }^{ 2 }

Solving for k k , we get k = l 6 k = \frac { l }{ 6 } units.

Hence for l = 24 c m , k = 4 c m . l = 24 cm , \boxed { k = 4 cm } .

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