Question

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

• A. Right angled
• B. Isoceles
• C. Equilateral
• D. None of these

Answer 1

The correct option is A Right angled

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

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

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

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

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

therefore the triangle is right angle