Question

In the figure shown, the bigger circle has radius 1 unit. Therefore the radius of smaller circle must be

• A. $\sqrt{2}+1$
• B. $\frac{1}{2}$
• C. $\frac{1}{\sqrt{2}}$
• D. $\frac{1}{\sqrt{2}+1}$

Answer 1

The correct option is D $\frac{1}{\sqrt{2}+1}$

Let the radius of smaller circle be r and the distance from center of bigger circle to the circummference of smaller circle be x.
Then, $x+2r=1$
and, In smaller triangle, which forms a right triangle,
$(x+r{)}^2=r^2+r^2$
$x+r=\sqrt{2}r$
$x=(\sqrt{2}-1)r$
Now, $(\sqrt{2}-1)r+2r=1$
$r=\frac{1}{\sqrt{2}+1}$