Q65 / 160 · AP EAMCET 2023 (16 May Shift 1) · Mathematics

In $\triangle A B C$, if $r_1=2 r_2=3 r_3$, then

  • A
    $b+c=2 a$
  • B
    $\mathrm{a}+\mathrm{b}=2 \mathrm{c}$
  • Correct answer
    C
    a + c $=2 b$
  • D
    $\Delta=a b c$

Solution

Given, In a $\triangle A B C$, if $r_1=2 r_2=3 r_3$ Then, we have $\frac{\Delta}{s-a}=2 \times \frac{\Delta}{s-b}=3 \times \frac{\Delta}{s-c}$ $\Rightarrow \frac{1}{s-a}=\frac{2}{s-b}=\frac{3}{s-c}=k$ (let) $\Rightarrow \frac{1}{s-a}=k \Rightarrow s-a=\frac{1}{k} \Rightarrow a=s-\frac{1}{k}$ ...(i) and $\frac{2}{s-b}=k \Rightarrow \frac{s-b}{2}=\frac{1}{k} \Rightarrow b=s-\frac{2}{k}$ ...(ii) and $\frac{3}{s-c}=k \Rightarrow \frac{s-c}{3}=\frac{1}{k} \Rightarrow c=s-\frac{3}{k}$ ...(iii) From eqn. (i), (ii) and (iii) $a+c=2 b$.

Hence, the correct answer is option: 'C'.

Relevant Questions