Question

If the diameter of any excircle of a triangle is equal to its perimeter, then the triangle is

• A. equilateral
• B. isosceles
• C. right angled
• D. scalene

Answer 1

The correct option is C right angled

$a+b+c=2r_1$...(given in the qs)

$S=r_1$

$r_1=\frac{\Delta }{(s-a)}$

$S=\frac{\Delta }{(s-a)}$

$s(s-a)=\sqrt{s(s-a)(s-b)(s-c)}$

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

$\tan A/2=\sqrt{\frac{(s-b)(s-c)}{s(s-c)}}=1$

$A/2=\pi /4$

$A=\pi /2$

$\therefore$ Right angled triangle.