Question

In a triangle $ABC$, $\angle C={90}^{\circ },r$ and $R$ are the in-radius and circum-radius of the triangle $ABC$ respectively, then $2(r+R)$ is equal to

• A. $b+c$
• B. $c+a$
• C. $a+b$
• D. $a+b+c$

Answer 1

The correct option is C $a+b$

The figure below represents the triangle.
In the above figure, $OECD$ is a square since all angles are right angle.
$BE=a-r$
$AD=b-r$
Also, tangents drawn from an external point are equal.
Therefore, $BF=BE$
and $AF=AD$

$a-r+b-r=2R$
$\Rightarrow 2(r+R)=a+b$