Question

In the adjoining figure the radius of the inner circle if other circles are of radii m units is

• A. $(\sqrt{2}-1)m$
• B. $\sqrt{2}$ m
• C. $\frac{1}{\sqrt{2}}m$
• D. $\frac{2}{\sqrt{2}+1}m$

Answer 1

The correct option is A $(\sqrt{2}-1)m$

Area of arc=$\frac{90}{360}\pi m^2=\frac{1}{4}\pi m^2$

Then area of 4 arc = $\pi m^2$

The area of square whose side is 2m=$(2m{)}^2=4m^2$

Then area of inner side =$4m^2-\pi m^2$

Let the radius of inner circle is r

Then length of square =a=$\sqrt{2}r$

And area of square=$2r^2$

And area inner circle=$\pi r^2$

$\therefore 2r^2-\pi r^2=4m^2-\pi m^2$

$\Rightarrow r^{2}(2-\pi )=m^{2}(4-\pi

)$

$\Rightarrow r=(\sqrt{2}-1)m$