AB is a line segment with P as midpoint. Three semicircles are drawn with AP,PB

Question

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

Options:

A . 14AB

B . 16AB

C . 17AB

D . 18AB

Answer: Option B
:
B
Ans: b)

\frac{1}{6}

AB.

Soln:

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

(r + x)^{2} = (2 r - x)^{2} + r^{2}

.
Solving we get x=

\frac{r}{3}

\frac{A B}{6}

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