In a Right Angled Triangle.

Question

In a triangle ABC, Let $\angle$C=$\frac{\pi}{2}$. If $r$ is the inradius and $R$ is the circumradius, then what is the value of $2r+R$.

Options are

1. a+b
2. b+c
3. c+a
4. a+b+c

My approach.

Radius of circumcircle, $R$= $\frac{c}{2}$, if $c$ is the side opposite to $\angle$C

let $\Delta$ be the area of the triangle,

so, radius of incircle, $r$= $\frac{\Delta}{s}$, where, $s=\frac{a+b+c}{2}$

$\Delta$=$\frac{ab}{2}$,

Answer 1

i think the the triangle is a right angle triangle helps: $R = c/2$ because the hypotenuse is the diameter of the circumcircle. now if you look at the tangent from $C$ to the incircle you will see that $r = s - c$ where $s = (a + b + c)/2$ is half of the perimeter. putting these together $$ 2r + R = (a + b + c). $$

edit: as pointed out by @peterwhy, i made an arithmetical error $r = s- c, R = c/2$ gives you $$r + 2R = (a+b+c)/2.$$

here is how you can see that $r = s - c.$ if you look at the four points: center $I$ of the incircle, two contact points where the circle touches the arms of the right angle($BC, AC)$, and the corner $C.$ these four points make up a square. rememeber we have already established the points contacts from $C$ are at a distance $s-c$ from the corner which does not depend $C$ being a right angle. that follows by splitting $b$ as the sum of $s-c$ and $s-a$ and symmetrically for $a$ and $b.$

it is always a good idea to draw a figure on a piece of paper first.(sorry i dont know how to put up a figure here but i did this work on a piece of paper first). the text book, goeometry reviisted by coxeter and greitzer is a good one to have if you can get it around where you live. good luck.