AB is a line segment and M is its midpoint Semicircles are drawn with AM, MB and AB as - Maths - Circles

Question

AB is a line segment and M is its midpoint Semicircles are drawn
with AM, MB and AB as diameters on the same side of line AB A
circle is drawn to touch all the semicircles Prove that its radius
r is given by AB/6

Vipin Verma · Accepted Answer

<img alt="" src=
"https://img-nm%20mnimgs%20com/img/study_content/content_ck_images/images/circle%20PNG"
style="width: 300px; height: 188px;">
Let the length of AB=x, and radius of the inner circle be r AM
=MB =x/2AL=PL=QN=x/4OM ⊥ABHence from pythagoras
theoremOL2=OM2+LM2OL =(r+x4)OM =
(RM-OM)=(x2-r)LM=x4(r+x4)2=(x2-r)2+(x4)2r2+rx2+x216=x24-rx+r2+x216rx2=x24-rx⇒r2=x4-r⇒3r2=x4⇒r=x6Hence
Radius r =AB6 Ans

AB is a line segment and M is its midpoint. Semicircles are drawn with AM, MB and AB as diameters on the same side of line AB. A circle is drawn to touch all the semicircles. Prove that its radius r is given by AB/6.