Question

lf $2R+r=r_1$, then

• A. $A={90}^0$
• B. $B={90}^0$
• C. $C={90}^0$
• D. $A={60}^o$

Answer 1

Answer: (A)

$A={90}^0$

$2R=r_1-r$
$2R=\frac{\Delta }{(s-a)}-\frac{\Delta }{s}$
$\frac{2R}{\Delta }=\frac{s-s+a}{s(s-a)}$
$\frac{a}{\sin A\Delta }=\frac{a}{s(s-a)}$
$\sin A\cdot \sqrt{s(s-a)(s-b)(s-c)}=s(s-a)$
$\sin A=\sqrt{\frac{s(s-a)}{(s-b)(s-c)}}$
$\sin A=\frac{\cos A/2}{\sin A/2}$
$2\sin A/2\cos A/2=\frac{\cos A/2}{\sin A/2}$
${\sin }^{2}A/2=\frac{1}{2}$
$\sin A/2=\frac{1}{\sqrt{2}}$
$A/2=\pi /4$
$A=\pi /2$