In triangle ABC,find the value of frac{r_1-r} | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(C) We know that $r = \frac{\Delta}{s}$,$r_1 = \frac{\Delta}{s-a}$,$r_2 = \frac{\Delta}{s-b}$,and $r_3 = \frac{\Delta}{s-c}$.Substituting these into t

Vedclass · Accepted Answer

$\frac{c}{r_3}$

Vedclass · Answer

(C) We know that $r = \frac{\Delta}{s}$,$r_1 = \frac{\Delta}{s-a}$,$r_2 = \frac{\Delta}{s-b}$,and $r_3 = \frac{\Delta}{s-c}$.Substituting these into the expression:$\frac{r_1-r}{a} + \frac{r_2-r}{b} = \frac{\frac{\Delta}{s-a} - \frac{\Delta}{s}}{a} + \frac{\frac{\Delta}{s-b} - \frac{\Delta}{s}}{b}$$= \frac{\Delta}{a} \left( \frac{s - (s-a)}{s(s-a)} \right) + \frac{\Delta}{b} \left( \frac{s - (s-b)}{s(s-b)} \right)$$= \frac{\Delta}{a} \left( \frac{a}{s(s-a)} \right) + \frac{\Delta}{b} \left( \frac{b}{s(s-b)} \right)$$= \frac{\Delta}{s(s-a)} + \frac{\Delta}{s(s-b)}$$= \frac{\Delta}{s} \left( \frac{1}{s-a} + \frac{1}{s-b} \right) = \frac{\Delta}{s} \left( \frac{s-b+s-a}{(s-a)(s-b)} \right)$$= \frac{\Delta}{s} \left( \frac{2s-a-b}{(s-a)(s-b)} \right) = \frac{\Delta}{s} \left( \frac{c}{(s-a)(s-b)} \right)$Since $\Delta^2 = s(s-a)(s-b)(s-c)$,we have $(s-a)(s-b) = \frac{\Delta^2}{s(s-c)}$.Substituting this:$= \frac{\Delta}{s} \cdot \frac{c \cdot s(s-c)}{\Delta^2} = \frac{c(s-c)}{\Delta} = \frac{c}{\frac{\Delta}{s-c}} = \frac{c}{r_3}$.

In $\triangle ABC$,find the value of $\frac{r_1-r}{a}+\frac{r_2-r}{b}$.

Similar Questions

In triangle $ABC$,if $a=7, b=10, c=11$,then $\frac{R}{r}=$

The area of the equilateral triangle which contains three coins of unit radius is

In $\triangle ABC$,if $2R + r = r_2$,then $\angle B$ is equal to

In a $\triangle ABC$,if $a: b: c = 4: 5: 6$,then the ratio of the radius of its circumcircle to that of its incircle is:

In a triangle $ABC$,if $r r_2 = r_1 r_3$,then $\cos 2B =$

Vedclass Test Series

Exam Paper Generator

Online Exam Module