Question

In a $\Delta \mathrm{AB}C$ , if $r_1=2r_2=3r_3,$ , then

• A. $\frac{a}{b}=\frac{4}{5}$
• B. $\frac{a}{b}=\frac{5}{4}$
• C. $a+b-2c=0$
• D. $2a=b+c$

Answer 1

Answer: (B)

Given, $r_1=2r_2=3r_3$ $\therefore$ $\frac{\Delta }{s-a}=\frac{2\Delta }{s-b}=\frac{3\Delta }{s-c}$ $\Rightarrow$ $(s-b)=2(s-a)$ and $(s-c)=3(s-a)$ $\Rightarrow$ $\left(\frac{a+b+c}{2}-b\right)=2\left(\frac{a+b+c}{2}-a\right)$ and $\left(\frac{a+b+c}{2}-c\right)=3\left(\frac{a+b+c}{2}-a\right)$ $\Rightarrow$ $a+c-b=2(-a+b+c)$ and $a+b-c=3(-a+b+c)$ $\Rightarrow$ $3a=3b+c$ and $2a=b+2c$ $\Rightarrow$ $4a=5b$ $\Rightarrow$ $\frac{a}{b}=\frac{5}{4}$