Question

In right angled isoceles $\Delta ABC$, let $R=$ circumradius, $r=$ inradius.

If r is the distance between the circumcenter and the incenter then the ratio $R:r$ is equal to?

• A. $\sqrt{2}-1$
• B. $\sqrt{3}-1$
• C. $\sqrt{2}+1$
• D. $\sqrt{3}+1$

Answer 1

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

Since, triangle is an isosceles right angled
Therefore, $A=C=\frac{\pi }{4}$ and $B=\frac{\pi }{2}$
Now, $R:r=R:4R\sin \frac{A}{2}\sin \frac{B}{2}\sin \frac{C}{2}=1:4{\sin }^2\frac{\pi }{8}\sin \frac{\pi }{4}$
$\Rightarrow R:r=1:2\sqrt{2}\left(\frac{1-\cos \frac{\pi }{4}}{2}\right)=1:(\sqrt{2}-1)=(\sqrt{2}+1):1$
Ans: C