Question

If $\frac{r}{r_2}=\frac{r_1}{r_3}$ then

• A. $\angle A={90}^{\circ }$
• B. $\angle B={90}^{\circ }$
• C. $\angle C={90}^{\circ }$
• D. $\angle A={60}^{\circ }$

Answer 1

The correct option is C $\angle C={90}^{\circ }$

Given: $\frac{r}{r_2}=\frac{r_1}{r_3}$
$\Rightarrow \frac{r{r}_3}{r_1{r}_2}=1$
We know that,
$\frac{r{r}_3}{r_1{r}_2}=\frac{\frac{\Delta }{s}\times \frac{\Delta }{s-c}}{\frac{\Delta }{s-a}\times \frac{\Delta }{s-b}}$
$\frac{r{r}_3}{r_1{r}_2}=\frac{(s-a)(s-b)}{s(s-c)}={\tan }^2\frac{C}{2}\text{(Using half-angles)}$
$\Rightarrow {\tan }^2\frac{C}{2}=1$
$\Rightarrow \frac{C}{2}={45}^{\circ }\Rightarrow C={90}^{\circ }$