Question

If in a triangle, $(1-\frac{r_1}{r_2}\left)\right(1-\frac{r_1}{r_3})=2$, then the triangle is

• A. right angled
• B. isosceles
• C. equilaternal
• D. none of these

Answer 1

The correct option is A right angled

$(1-\frac{r_1}{r_2}\left)\right(1-\frac{r_1}{r_3})=2$

$\Rightarrow (1-\frac{\frac{\Delta }{s-a}}{\frac{\Delta }{s-b}})(1-\frac{\frac{\Delta }{s-a}}{\frac{\Delta }{s-c}})=2$

$\Rightarrow (1-\frac{s-b}{s-a})(1-\frac{s-c}{s-a})=2$

$\Rightarrow \frac{(b-a)(c-a)}{{(s-a)}^2}=2$

$\Rightarrow (b-a)(c-a)=2{(s-a)}^2$

$\Rightarrow (b-a)(c-a)=\frac{1}{2}{(b+c-a)}^2$

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

Hence the triangle is right-angled.