Question

In a $\Delta$ ABC, let $\angle$ C = $\pi /2$ .If r is the inradius and R is the circumradius of the triangle, then 2 (r +R) is equal to

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

Answer 1

The correct option is A

a + b

The hypotenuse of the right triangle ABC is AB. We take C at the origin and CB along x - axis and CA along y - axis. The mid point $M(\frac{a}{2},\frac{b}{2})$ of AB is the

Therefore,

$R^2=M{C}^2=\frac{1}{4}(a^2+b^2)=\frac{1}{4}{c}^2\Rightarrow R=\frac{c}{2}Next,r=(s-c)tan\frac{C}{2}=(s-c)tan\frac{\pi }{4}=s-cThus,2(r+R)=2r+2R=2s-2c+c=a+b+c-2c+c=a+b$