In a triangle ABC,if r_1=3, r_2=4, r_3=6,then | vedclass

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

Question

(A) We know the formulas for the exradii of a triangle: $r_1 = \frac{\Delta}{s-a}, r_2 = \frac{\Delta}{s-b}, r_3 = \frac{\Delta}{s-c}$.Given $r_1=3, r

Vedclass · Accepted Answer

$2 \sqrt{6}$

Vedclass · Answer

(A) We know the formulas for the exradii of a triangle: $r_1 = \frac{\Delta}{s-a}, r_2 = \frac{\Delta}{s-b}, r_3 = \frac{\Delta}{s-c}$.Given $r_1=3, r_2=4, r_3=6$.Taking reciprocals: $\frac{1}{r_1} = \frac{s-a}{\Delta}, \frac{1}{r_2} = \frac{s-b}{\Delta}, \frac{1}{r_3} = \frac{s-c}{\Delta}$.Summing these: $\frac{1}{r_1} + \frac{1}{r_2} + \frac{1}{r_3} = \frac{3s-(a+b+c)}{\Delta} = \frac{3s-2s}{\Delta} = \frac{s}{\Delta} = \frac{1}{r}$.$\frac{1}{3} + \frac{1}{4} + \frac{1}{6} = \frac{4+3+2}{12} = \frac{9}{12} = \frac{3}{4} = \frac{1}{r} \implies r = \frac{4}{3}$.Also,$\Delta^2 = r r_1 r_2 r_3 = \frac{4}{3} 	imes 3 	imes 4 	imes 6 = 96 \implies \Delta = \sqrt{96} = 4\sqrt{6}$.Since $r_2 = \frac{\Delta}{s-b}$,we have $4 = \frac{4\sqrt{6}}{s-b} \implies s-b = \sqrt{6}$.Also $s = \frac{\Delta}{r} = \frac{4\sqrt{6}}{4/3} = 3\sqrt{6}$.Thus,$b = s - (s-b) = 3\sqrt{6} - \sqrt{6} = 2\sqrt{6}$.

In a triangle $ABC$,if $r_1=3, r_2=4, r_3=6$,then $b=$

Similar Questions

In $\triangle ABC$,if $a:b:c = 4:5:6$,then the ratio of the circumradius to its inradius is

In a $\triangle ABC$,$D, E$,and $F$ are the points of contact of the incircle with the sides $AB, BC$,and $CA$ respectively,such that $AD=\alpha, BE=\beta$,and $CF=\gamma$. Then $\frac{\alpha \beta \gamma}{\alpha+\beta+\gamma} = $

In an equilateral triangle of side $2\sqrt{3} \text{ cm}$,the circum-radius is ..... $\text{cm}$.

In an equilateral triangle $ABC$,the ratio $r : R : r_1$ is

If the roots of the equation $x^3-11x^2+36x-36=0$ are the ex-radii of a triangle $ABC$,then the perimeter of the triangle $ABC$ is

Vedclass Test Series

Exam Paper Generator

Online Exam Module