Question

In $\Delta ABC$ if $r_1-r$ $=r_2+r_3$, then

• A. $A={90}^0$
• B. $B={90}^0$
• C. $C={90}^0$
• D. $A={45}^o$

Answer 1

Answer: (A)

$A={90}^0$

$\frac{\Delta }{(s-a)}-\frac{\Delta }{s}=\frac{\Delta }{(s-b)}+\frac{\Delta }{(s-c)}$....(using formulas of $r,r_1,r_2,r_3$)

$\frac{s-s+a}{s(s-a)}=\frac{s-c+s-b}{(s-b)(s-c)}$

$s^2-(b+c)s+bc=s^2-sa$

$bc=s(b+c-a)$

$bc=\left(\frac{a+b+c}{2}\right)(b+c-a)$

$2bc=(b+c{)}^2-a^2$

$2bc=b^2+c^2+2bc-a^2$

$a^2=b^2+c^2$

$\therefore \angle A=\pi /2$