Question

AB is a line segment with P as mid-point. Three semicircles are drawn with AB, BC and AC as diameters. Another circle is drawn so as to touch all the three semicircles. What’s the radius of this smaller circle?

• A. AB
• B. AB
• C. AB
• D. AB

Answer 1

The correct option is D AB

$\frac{1}{3}$AB.

Let PB=r=PQ. QR=x. In triangle PRB, RB=2r-x. PR= r+x.

Since PRB is right angled, (r+x)${}^2$=(2r-x)${}^2$+r${}^2$ .

Solving we get x=$\frac{r}{3}=\frac{AB}{3}$