In a triangle ABC,if r_1=2 r_2=3 r_3,then fra | vedclass

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

Question

(D) Given $r_1=2 r_2=3 r_3$.Using the formula $r_1 = \frac{\Delta}{s-a}$,$r_2 = \frac{\Delta}{s-b}$,$r_3 = \frac{\Delta}{s-c}$,we have:$\frac{\Delta}{

Vedclass · Accepted Answer

$\frac{191}{60}$

Vedclass · Answer

(D) Given $r_1=2 r_2=3 r_3$.Using the formula $r_1 = \frac{\Delta}{s-a}$,$r_2 = \frac{\Delta}{s-b}$,$r_3 = \frac{\Delta}{s-c}$,we have:$\frac{\Delta}{s-a} = \frac{2\Delta}{s-b} = \frac{3\Delta}{s-c}$From $\frac{1}{s-a} = \frac{2}{s-b}$,we get $s-b = 2s-2a \Rightarrow s = 2a-b$.From $\frac{1}{s-a} = \frac{3}{s-c}$,we get $s-c = 3s-3a \Rightarrow 2s = 3a-c$.Substituting $s = \frac{a+b+c}{2}$ into these equations:$a+b+c = 4a-2b \Rightarrow 3a-3b = c$.$a+b+c = 3a-c \Rightarrow 2a-b = 2c$.Solving for ratios:From $3a-3b = c$ and $2a-b = 2c$,we get $2(3a-3b) = 2a-b$ $\Rightarrow 6a-6b = 2a-b$ $\Rightarrow 4a = 5b$ $\Rightarrow \frac{a}{b} = \frac{5}{4}$.Then $c = 3a-3b = 3a - 3(\frac{4a}{5}) = 3a - \frac{12a}{5} = \frac{3a}{5} \Rightarrow \frac{c}{a} = \frac{3}{5}$.Since $\frac{a}{b} = \frac{5}{4}$ and $\frac{c}{a} = \frac{3}{5}$,then $\frac{b}{c} = \frac{b}{a} 	imes \frac{a}{c} = \frac{4}{5} 	imes \frac{5}{3} = \frac{4}{3}$.Finally,$\frac{a}{b}+\frac{b}{c}+\frac{c}{a} = \frac{5}{4} + \frac{4}{3} + \frac{3}{5} = \frac{75+80+36}{60} = \frac{191}{60}$.

In a triangle $ABC$,if $r_1=2 r_2=3 r_3$,then $\frac{a}{b}+\frac{b}{c}+\frac{c}{a}=$

Similar Questions

The base of a triangle is $80$ and one of the base angles is $60^{\circ}$. If the sum of the lengths of the other two sides is $90$,then the shortest side is of length

In $\Delta ABC$,$(a-b)^2 \cos^2 \frac{C}{2} + (a+b)^2 \sin^2 \frac{C}{2} =$

In a $\triangle ABC$,if $a \cos^2 \frac{C}{2} + c \cos^2 \frac{A}{2} = \frac{3b}{2}$,then the sides of the triangle are in

In a triangle $ABC$,with usual notations,if $a=5$,$b=7$,and $\sin A=\frac{3}{4}$,then the total number of triangles possible is:

In $\triangle ABC$,if $b+c : c+a : a+b = 7 : 8 : 9$,then the smallest angle (in radians) of that triangle is

Vedclass Test Series

Exam Paper Generator

Online Exam Module