In any triangle ABC,r_1 r_2 + r_2 r_3 + r_3 r | vedclass

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

Question

(A) We know that the exradii are given by $r_1 = \frac{\Delta}{s-a}$,$r_2 = \frac{\Delta}{s-b}$,and $r_3 = \frac{\Delta}{s-c}$.Substituting these into

Vedclass · Accepted Answer

$\frac{\Delta^2}{r^2}$

Vedclass · Answer

(A) We know that the exradii are given by $r_1 = \frac{\Delta}{s-a}$,$r_2 = \frac{\Delta}{s-b}$,and $r_3 = \frac{\Delta}{s-c}$.Substituting these into the expression $r_1 r_2 + r_2 r_3 + r_3 r_1$:$= \frac{\Delta^2}{(s-a)(s-b)} + \frac{\Delta^2}{(s-b)(s-c)} + \frac{\Delta^2}{(s-c)(s-a)}$$= \frac{\Delta^2(s-c + s-a + s-b)}{(s-a)(s-b)(s-c)}$Since $2s = a+b+c$,we have $s-a+s-b+s-c = 3s - (a+b+c) = 3s - 2s = s$.Also,we know that $\Delta^2 = s(s-a)(s-b)(s-c)$,so $(s-a)(s-b)(s-c) = \frac{\Delta^2}{s}$.Substituting these values:$= \frac{\Delta^2 \cdot s}{\frac{\Delta^2}{s}} = s^2$Since $r = \frac{\Delta}{s}$,we have $s = \frac{\Delta}{r}$,so $s^2 = \frac{\Delta^2}{r^2}$.

In any $\triangle ABC$,$r_1 r_2 + r_2 r_3 + r_3 r_1$ is equal to

Similar Questions

For a chemical reaction,$A + 2B \to C + D,$ the rate of reaction increases $3$ times when the concentration of $A$ only is increased $9$ times. While when the concentration of $B$ only is increased $2$ times,the rate of reaction also increases $2$ times. The order of this reaction is:

The required amount of $KBr$ (molar mass $= 119 \, g/mol$) in grams to start the precipitation of $AgBr$ in $500 \, mL$ solution of $0.05 \, M \, AgNO_3$ will be ($K_{sp}$ of $AgBr = 5 \times 10^{-13}$)

$A$ proton, a deuteron, and an $\alpha$-particle having the same momentum enter a region of uniform electric field between the parallel plates of a capacitor. The electric field is perpendicular to the initial path of the particles. The ratio of deflections suffered by them is

If the internal energy of an ideal gas decreases by the same amount as the work done by the system,the process is

Vedclass Test Series

Exam Paper Generator

Online Exam Module