Question

Statement 1: In a $\Delta ABC,r_1{r}_2{r}_3=s^2$

Statement 2: In an equilateral triangle, $r:R:r_1=1:4:5$

• A. Statement-1 is true, Statement-2 is true, Statement-2 is correct explanation of Statement- 1
• B. Statement-1 is true, Statement-2 is true, Statement-2 is not correct explanation for Statement-1
• C. Statement-1 is true, Statement-2 is false
• D. Statement-1 is false, Statement-2 is false

Answer 1

The correct option is D Statement-1 is false, Statement-2 is false

($i$)

for any triangle

$r_1=\left(s\right)\tan \frac{A}{2}$

$r_2=\left(s\right)\tan \frac{B}{2}$

$r_3=\left(s\right)\tan \frac{C}{2}$

now $r_1{r}_2{r}_3$=$\left(s^3\right)\tan \frac{A}{2}\tan \frac{B}{2}\tan \frac{C}{2}$

So first statement is wrong

($ii$)

for any triangle

$R=\frac{a}{2sinA}$

$r=(s-a)\tan \frac{A}{2}$

and $r_1=\left(s\right)\tan \frac{A}{2}$

Given.. $a=b=c$ and $\angle A=\angle B=\angle C=60°$

hence $R=\frac{a}{\sqrt{3}}$ - - - - - - - - - $1$

$r=(\frac{a+b+c}{2}-a)\tan 30°=\frac{a}{2\sqrt{3}}$ - - - - - - $2$

$r_1=\left(\frac{a+b+c}{2}\right)\tan \frac{A}{2}$=$\frac{a\sqrt{3}}{2}$ - - - - - - $3$

From eqn($1$), eqn($2$) and eqn($3$)

$r:R:r_1=1:2:3$

Hence second statement is also wrong