Question

Corresponding to a triangle $A B C$, match the items given in List-I with the items given in List II.

	List I		List I
(A)	$r r_{2}=r_{1} r_{3}$	(I)	$\angle A=90^{\circ}$
(B)	$r_{1}+r_{2}=r_{3}-r$	(II)	$b^{2}=c^{2}+a^{2}$
(C)	$r_{1}=r+2 R$	(III)	$\angle C=90^{\circ}$
		(IV)	$\angle B=120^{\circ}$

The correct match is

• A. A - II , B - III ,C - I
• B. A - II, B - I ,C - III
• C. A - I , B - IV ,C - III
• D. A - III , B - I,C - IV

Answer 1

Answer: (A)

In $\Delta A B C$
(A) We have,
$r r_{2}=r_{1} \,r_{3}$
$\therefore \frac{\Delta}{s} \cdot \frac{\Delta}{s-b}=\frac{\Delta}{s-a} \cdot \frac{\Delta}{s-c} \Rightarrow \frac{(s-a)(s-c)}{s(s-b)}=1$
$\Rightarrow \tan ^{2} \frac{B}{2}=1 $
$ \Rightarrow \tan \frac{B}{2}=\tan 45^{\circ} $
$\Rightarrow B=90^{\circ}$
$\therefore b^{2}=a^{2}+c^{2}$
$\therefore A \rightarrow II$
(B) We have,
$r_{1}+r_{2}=r_{3}-r$
$\frac{\Delta}{s-a}+\frac{\Delta}{s-b}=\frac{\Delta}{s-c}-\frac{\Delta}{s}$
$\Rightarrow \frac{s-b+s-a}{(s-a)(s-b)}=\frac{s-s+c}{s(s-i)}$
$\Rightarrow \frac{2 s-(a+b)}{(s-a)(s-b)}=\frac{c}{s(s-c)}=\tan ^{2} \frac{C}{2}=1$
$\Rightarrow \angle C=90^{\circ}$
$\therefore B \rightarrow III$
(C) $r_{1}=r+2 R$
$\frac{\Delta}{s-a}=\frac{\Delta}{s}+\frac{a}{\sin A} $
$\Rightarrow \sin A=\frac{s(s-a)}{\Delta}$
$\Rightarrow 2 \sin \frac{A}{2} \cos \frac{A}{2}=\sqrt{\frac{s(s-a)}{(s-b)(s-c)}}=\cot \frac{A}{2}$
$\Rightarrow \sin ^{2} \frac{A}{2}=\frac{1}{2} $
$\Rightarrow \sin \frac{A}{2}=\frac{1}{\sqrt{2}} \Rightarrow \angle A=90^{\circ}$
$\therefore C \rightarrow I$