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 .

\frac{1}{4}

B .

\frac{1}{6}

C .

\frac{1}{7}

D .

\frac{1}{8}

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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