Question

In \(\Delta\)ABC, if \(r_1=r_2+r_3+r\) then
a) A=\(45^0\)
b) B=\(90^0\)
c) C=\(90^0\)
d) A=\(90^0\)

Answer 1

r1,r2,r3,r ?

Answer 2

r1, r2, r3 are ex-radius opposite to angles A, B and C.
r is in-radius.

Answer 3

i can solve it with some formulas but i dont know if u are familiar with this formulas or not?

Answer 4

we know that \(\large r_{1}=p \tan \frac{A}{2} \) , \(\large r_{2}=p \tan \frac{B}{2} \) , \(\large r_{3}=p \tan \frac{C}{2} \) ,

\(\large r=p \tan \frac{A}{2} \tan \frac{B}{2} \tan \frac{C}{2}\) where p is half of Perimeter of Triangle

Answer 5

r1=r2+r3+r so

\(\large \tan \frac{A}{2}=\tan \frac{B}{2}+\tan \frac{C}{2}+\tan \frac{A}{2}\tan \frac{B}{2} \tan \frac{C}{2} \)

then u have \( \large \tan \frac{A}{2}=\frac{\tan \frac{B}{2}+\tan \frac{C}{2}}{1-\tan \frac{B}{2}\tan \frac{C}{2}} =\tan (\frac{B}{2}+\frac{C}{2})=\tan (\frac{\pi}{2}-\frac{A}{2})=\cot \frac{A}{2} \) so A=90

Answer 6

Thanks a lot! yeah, I was not familiar with those formulas but now i am! Thanks a lot @mukushla

Answer 7

welcome my friend