Question

In a right -angled isosceles triangle the ratio of the circumradius and inradius is?

• A. $2\left(\sqrt{2+1}\right):1$
• B. $\left(\sqrt{2+1}\right):1)$
• C. $2:1$
• D. $\sqrt{2}:1$

Answer 1

Answer: (B)

$\left(\sqrt{2+1}\right):1)$

Since, triangle is an isosceles right angled
Therefore, $A=C=\frac{\pi }{4}$ and $B=\frac{\pi }{2}$
Now, $R:r=R:4R\sin \frac{A}{2}\sin \frac{B}{2}\sin \frac{C}{2}$

$=1:4{\sin }^2\frac{\pi }{8}\sin \frac{\pi }{4}$
$\Rightarrow R:r=1:2\sqrt{2}\left(\frac{1-\cos \frac{\pi }{4}}{2}\right)$

$=1:(\sqrt{2}-1)$

$=(\sqrt{2}+1):1$