AB is a line segment whose midpoint is M. Taking AM, BM and AB as diameters three semi-circles are drawn on the same side of AB. A circle is drawn with radius 'r' touching all the three semi-circles. Prove that r=AB/6

Asked by abhishekabin12306 | 3rd Jan, 2018, 08:01: PM

Expert Answer:

The image will be as shown below:
 
 
begin mathsize 16px style Since space straight M space is space the space mid minus point space of space AB comma
AM equals MB equals AB over 2
Since space straight P space and space straight S space are space the space midpoints space of space AM space and space MB comma
AP equals PM equals MS equals SB equals AB over 4
Radius space of space straight a space circle equals straight r
rightwards double arrow CP equals left parenthesis straight r plus PQ right parenthesis equals open parentheses straight r plus AB over 4 close parentheses space cm
CM equals DM minus DC equals open parentheses AB over 2 minus straight r close parentheses space cm
In space triangle CMP comma space angle CMP equals 90 degree
rightwards double arrow CM squared plus MP squared equals CP squared
rightwards double arrow open parentheses AB over 2 minus straight r close parentheses squared plus open parentheses AB over 4 close parentheses squared equals open parentheses straight r plus AB over 4 close parentheses squared
Solving comma space we space get
straight r equals AB over 6 end style

Answered by Rashmi Khot | 5th Jan, 2018, 10:22: AM