Question

Assertion :In a $△ABC$, if $a<b<c$ and $r$ is inradius and $r_1,r_2,r_3$ are the axradii opposite to angle $A,B,C$ respectively, then $r<r_1<r_2<r_3$ Reason: $△ABC,r_1{r}_2+r_2{r}_3+r_3{r}_1=\frac{r_1{r}_2{r}_3}{r}$

• A. Both the statements are TRUE and reason is the correct expansion of assertion
• B. Both the statements are TRUE but reason is NOT the correct explanation of assertion
• C. Assertion is TRUE and reason is FALSE
• D. Assertion is FALSE and reason is TRUE

Answer 1

The correct option is B Both the statements are TRUE but reason is NOT the correct explanation of assertion

Assertion: $a<b<c$

$\Rightarrow s>s-a>s-b>s-c$

$\Rightarrow \frac{\Delta }{s}<\frac{\Delta }{s-a}<\frac{\Delta }{s-b}<\frac{\Delta }{s-c}$

$\Rightarrow r<r_1<r_2<r_3$

Reason: $r=\frac{\Delta }{s},r_1=\frac{\Delta }{s-a},r_2=\frac{\Delta }{s-b},r_3=\frac{\Delta }{s-c}$

$\frac{1}{r_1}+\frac{1}{r_2}+\frac{1}{r_3}=\frac{s}{\Delta }=\frac{1}{r}\Rightarrow r_1{r}_2+r_2{r}_3+r_3{r}_1=\frac{r_1{r}_2{r}_3}{r}$

$\therefore$ Assertion and reason both are correct but reason is not correct explanation os assertion.