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 triangle.
• B. isosceles triangle.
• C. equilateral triangle.
• D. none of these

Answer 1

Answer: (A)

right-angled triangle.

$(1-\frac{r_1}{r_2})(1-\frac{r_1}{r_3})=2$
$\Rightarrow (1-\frac{\frac{△}{(s-a)}}{\frac{△}{(s-b)}})(1-\frac{\frac{△}{(s-a)}}{\frac{△}{(s-c)}})=2$
$\Rightarrow (a-b)(a-c)=2{(s-a)}^2$
$\Rightarrow a^2-ab-ac+bc=2{(\frac{a+b+c}{2}-a)}^2=\frac{{(b+c-a)}^2}{2}$
$\Rightarrow b^2+c^2=a^2$
Therefore, the triangle is right angled.
Ans: A